Jiasong Wu

Shijie Qiu

Youyong Kong

Longyu Jiang

Yang Chen

Wankou Yang

Lotfi Senhadji

Huazhong Shu

PCANet: An energy perspective

Keywords: Deep learning, Convolutional neural networks, PCANet, Error rate, Energy, Image recognition

The principal component analysis network (PCANet), which is one of the recently proposed deep learning architectures, achieves the state-of-the-art classification accuracy in various databases. However, the visualization or explanation of the PCANet is lacked. In this paper, we try to explain why PCANet works well from energy perspective point of view based on a set of experiments. The paper shows that the error rate of PCANet is qualitatively correlated with the inverse of the logarithm of BlockEnergy, which is the energy after the block sliding process of PCANet, and also this relation is quantified by using curve fitting method. The proposed energy explanation approach can also be used as a testing method for checking if every step of the constructed networks is necessary.

A C C E P T E D M

A N U S C R I P T

Introduction

Deep learning [START_REF] Hinton | A fast learning algorithm for deep belief nets[END_REF][START_REF] Hinton | Reducing the dimensionality of data with neural networks[END_REF][START_REF] Bengio | Greedy layer-wise training of deep networks[END_REF][START_REF] Lee | Convolutional deep belief networks for scalable unsupervised learning of hierarchical representations[END_REF][START_REF] Vincent | Extracting and composing robust features with denoising autoencoders[END_REF][START_REF] Lecun | Deep learning[END_REF][START_REF] Bengio | Learning deep architectures for AI[END_REF][START_REF] Deng | Deep Learning: Methods and Applications[END_REF][START_REF] Bengio | Representation learning: A review and new perspectives[END_REF][START_REF] Schmidhuber | Deep learning in neural networks: An overview[END_REF], especially convolutional neural networks (CNNs) [START_REF] Lecun | Backpropagation applied to handwritten zip code recognition[END_REF][START_REF] Lecun | Gradient-based learning applied to document recognition[END_REF], is a hot research topic that achieves the state-of-the-art results in many image classification tasks, including ImageNet large scale visual recognition [START_REF] Russakovsky | ImageNet large scale visual recognition challenge[END_REF][START_REF] Krizhevsky | ImageNet classification with deep convolutional neural network[END_REF][START_REF] Szegedy | Going deeper with convolutions[END_REF][START_REF] Simonyan | Very deep convolutional networks for large-scale image recognition[END_REF][START_REF] He | Deep residual learning for image recognition[END_REF], Labeled Faces in the Wild (LFW) face recognition [START_REF] Learned-Miller | Labeled faces in the wild: A survey[END_REF][START_REF] Sun | DeepID3: Face recognition with very deep neural networks[END_REF][START_REF] Schroff | FaceNet: A unified embedding for face recognition and clustering, Computer Vision and Pattern Recognition[END_REF], handwritten digit recognition [START_REF] Lecun | Backpropagation applied to handwritten zip code recognition[END_REF][START_REF] Ciresan | Multi-column deep neural networks for image classification[END_REF], and other applications [START_REF] Zhang | Constructing deep sparse coding network for image classification[END_REF][START_REF] Zu | Robust multi-atlas label propagation by deep sparse representation[END_REF][START_REF] Ohn-Bar | Multi-scale volumes for deep object detection and localization[END_REF][START_REF] Lopes | Facial expression recognition with Convolutional Neural Networks: Coping with few data and the training sample order[END_REF][START_REF] Zhou | Recurrent convolutional neural network for answer selection in community question answering[END_REF][START_REF] Chen | Accurate seat belt detection in road surveillance images based on CNN and SVM[END_REF][START_REF] Liu | A survey of deep neural network architectures and their applications[END_REF][START_REF] Yu | Convolutional neural networks for hyperspectral image classification[END_REF][START_REF] Qayyum | Medical image retrieval using deep convolutional neural network[END_REF][START_REF] Barros | Emotion-modulated attention improves expression recognition: A deep learning model[END_REF], etc. The great success of deep learning systems is impressive, but a fundamental question still remains: Why do they work [START_REF] Patel | A probabilistic theory of deep learning[END_REF]? In the recent years, several attempts have been made for explaining the deep learning systems. These attempts can be roughly categorized into two classes: theoretical explanation and experimental explanation.

Theoretical explanation method tries to elucidate deep learning systems by using various theories, which can be classified into seven subclasses: 1) Renormalization Theory. Mehta and Schwab [START_REF] Mehta | An exact mapping between the variational renormalization group and deep learning[END_REF] constructed an exact mapping from the variational renormalization group (RG) scheme [START_REF] Kadanov | Statistical Physics: Statics, Dynamics and Renormalization[END_REF] to deep neural networks (DNNs) based on Restricted Boltzmann Machines (RBMs) [START_REF] Hinton | A fast learning algorithm for deep belief nets[END_REF][START_REF] Hinton | Reducing the dimensionality of data with neural networks[END_REF], and thus explained DNNs as a RG-like procedure to extract relevant features from structured data. 2) Probabilistic Theory. Patel et al. [START_REF] Patel | A probabilistic theory of deep learning[END_REF] developed a new probabilistic framework for deep learning based on a Bayesian generative probabilistic model. By relaxing the generative model to a discriminative one, their models recover two of the current leading deep learning systems: deep CNNs and random decision forests (RDFs). 3) Information Theory. Tishby and Zaslavsky [START_REF] Tishb | Deep learning and the information bottleneck principle[END_REF] analyzed DNNs via the theoretical framework of the information bottleneck principle. Steeg and Galstyan [START_REF] Steeg | The information sieve[END_REF] further introduced a framework for unsupervised learning of deep representations based on a hierarchical decomposition of information. 4) Developmental robotic perspective. Sigaud and Droniou [START_REF] Sigaud | Towards deep developmental learning[END_REF] scrutinized deep learning techniques under the light of their capability to construct a hierarchy of multimodal representations from the raw sensors of robots. 5) Geometric viewpoint. Lei et al. [START_REF] Lei | A geometric view of optimal transportation and generative model[END_REF] showed the intrinsic relations between optimal transportation and convex geometry and gave a geometric interpretation to generative models. Dong et al. [START_REF] Dong | How deep learning works -The geometry of deep learning[END_REF] draw a geometric picture of the deep learning system by finding its analogies with two existing geometric structures, the geometry of quantum computations and the geometry of the diffeomorphic template matching. 6) Group Theory. Paul and Venkatasubramanian [START_REF] Paul | Why does unsupervised deep learning work?-A perspective from group theory[END_REF] explained deep learning system from the group-theoretic perspective point of view and showed why higher layers of deep learning framework tend to learn more abstract features. Shaham et al. [START_REF] Shaham | Provable approximation properties for deep neural networks[END_REF] discussed the approximation of wavelet functions using deep neural nets. Anselmi et al. [START_REF] Anselmi | On invariance and selectivity in representation learning[END_REF] explained deep CNNs by invariant and selective theory, whose ideas come from i-Theory [START_REF] Anselmi | Unsupervised learning of invariant representations in hierarchical architectures[END_REF], which is a

A C C E P T E D M

A N U S C R I P T 3 recent theory of feedforward processing in sensory cortex. 7) Energy perspective. The mathematical analysis of CNNs was performed by Mallat in [START_REF] Mallat | Group invariant scattering[END_REF], where wavelet scattering network (ScatNet) was proposed. The convolutional layer, nonlinear layer, pooling layer were constructed by prefixed complex wavelets, modulus operator, and average operator, respectively. Owning to its characteristic of using prefixed filters which are not learned from data, ScatNet was explained in [START_REF] Mallat | Group invariant scattering[END_REF] from energy perspective both in theory and experiment aspect, that is, ScatNet maintains the energy of image in each layer although using modulus operator. ScatNet achieves the state-of-the-art results in various image classification tasks [START_REF] Bruna | Invariant scattering convolution networks[END_REF] and was then extended to semi-discrete frame networks [START_REF] Wiatowski | Deep convolutional neural networks based on semi-discrete frames[END_REF] as well as complex valued convolutional nets [START_REF] Bruna | A theoretical argument for complex-valued convolutional networks[END_REF].

Experimental explanation methods tend to understand deep learning systems by inverting them to visualize the "filters" learned by the model [START_REF] Erhan | Visualizing higher-layer features of a deep network[END_REF]. For example, Larochelle et al. [START_REF] Larochelle | An empirical evaluation of deep architectures on problems with many factors of variation[END_REF] presented a series of experiments on Deep Belief Networks (DBN) [START_REF] Hinton | A fast learning algorithm for deep belief nets[END_REF] and stacked autoencoder networks [START_REF] Bengio | Greedy layer-wise training of deep networks[END_REF] by using artificial data and indicate that these models show promise in solving harder learning problems that exhibit many factors of variation. Goodfellow et al. [START_REF] Goodfellow | Measuring invariances in deep networks[END_REF] examined the invariances of stacked autoencoder networks [START_REF] Bengio | Greedy layer-wise training of deep networks[END_REF] and also convolutional deep belief networks (CDBNs) [START_REF] Lee | Convolutional deep belief networks for scalable unsupervised learning of hierarchical representations[END_REF] by using natural images and natural video sequences. Erhan et al. [START_REF] Erhan | Visualizing higher-layer features of a deep network[END_REF] studied three filter visualization methods (Maximizing the activation, Sampling from a unit of a network, Linear combination of previous layers" filters) on DBN [START_REF] Hinton | A fast learning algorithm for deep belief nets[END_REF] and Stacked Denoising Autoencoders [START_REF] Vincent | Extracting and composing robust features with denoising autoencoders[END_REF]. Szegedy et al. [START_REF] Szegedy | Intriguing properties of neural networks[END_REF] reported two intriguing properties of deep neural networks. Zeiler and Fergus [START_REF] Zeiler | Visualizing and understanding convolutional networks[END_REF] proposed DeConvNet method in which the network computations were backtracked to identify which image patches are responsible for certain neural activations. DeConvNet uses AlexNet [START_REF] Lecun | Backpropagation applied to handwritten zip code recognition[END_REF] as an example to observe the evolution of features during training and to diagnose potential problems by using such a model. Simonyan et al. [START_REF] Simonyan | Deep inside convolutional networks: Visualising image classification models and saliency maps[END_REF] demonstrated how saliency maps can be obtained from a Convnet by projecting back from the fully connected layers of the network. Girshick et al. [START_REF] Girshick | Rich feature hierarchies for accurate object detection and semantic segmentation[END_REF] showed visualizations that identify patches within a dataset that are responsible for strong activations at higher layers in the model. Mahendran and Vedaldi [START_REF] Mahendran | Understanding deep image representations by inverting them[END_REF] gave a general framework to invert CNNs and they tried to answer the following question: given an encoding of an image, to which extent is it possible to reconstruct the image itself? Recently, Chan et al. [START_REF] Chan | PCANet: A simple deep learning baseline for image classification?[END_REF] proposed a new deep learning algorithm called principal component analysis network (PCANet), whose convolutional layer, nonlinear processing layer, and feature pooling layer consist of principal component analysis (PCA) filter bank, binarization, and block-wise histogram, respectively. Chan et al. [START_REF] Chan | PCANet: A simple deep learning baseline for image classification?[END_REF] also

A C C E P T E D M A N U S C R I P T 4
visualize the filters of PCANet like in [START_REF] Erhan | Visualizing higher-layer features of a deep network[END_REF] and [START_REF] Zeiler | Visualizing and understanding convolutional networks[END_REF]. Although PCANet is constructed with most basic units, it surprisingly achieves the state-of-the-art performance for most image classification tasks. PCANet arouses the interest of many researchers in this field. For example, Gan et al. proposed a graph embedding network (GENet) [START_REF] Gan | A deep graph embedding network model for face recognition[END_REF] for image classification. Wang and Tan [START_REF] Wang | Unsupervised feature learning with C-SVDDNet[END_REF] presented a C-SVDDNet for unsupervised feature learning. Feng et al. [START_REF] Feng | DLANet: A manifold-learning-based discriminative feature learning network for scene classification[END_REF] presented a discriminative locality alignment network (DLANet) for scene classification. Ng and Teoh [START_REF] Ng | DCTNet: A simple learning-free approach for face recognition[END_REF] proposed discrete cosine transform network (DCTNet) for face recognition. Gan et al. [START_REF] Gan | A PCA-based convolutional network[END_REF] presented a PCA-based convolutional network by combining the structure of PCANet and the LeNet-5 [START_REF] Lecun | Backpropagation applied to handwritten zip code recognition[END_REF][START_REF] Lecun | Gradient-based learning applied to document recognition[END_REF]. Zhao et al. [START_REF] Zhao | Multi-level modified finite radon transform network for image upsampling[END_REF] proposed multi-level modified finite radon transform network (MMFRTN) for image upsampling. Lei et al. [START_REF] Lei | Learning stacked image descriptor for face recognition[END_REF] developed stacked image descriptor for face recognition. Li et al. [START_REF] Li | Feature learning based on SAE-PCA network for human gesture recognition in RGBD images[END_REF] proposed SAE-PCA network for human gesture recognition in RGBD (Red, Green, Blue, Depth) images. Zeng et al. [START_REF] Zeng | Color image classification via quaternion principal component analysis network[END_REF] proposed a quaternion principal component analysis network (QPCANet) for color image classification. Wu et al. [START_REF] Wu | Multilinear principal component analysis network for tensor object classification[END_REF] proposed a multilinear principal component analysis network (MPCANet) for tensor object classification. Although PCANet has been extensively investigated, the question still remains: Why it works well by using the most basic and simple units? To the best of our knowledge, no attempt to explain every step of the PCANet is available in the literature.

In this paper, 1) We present a new way to visualize, explain and understand every step of PCANet from an energy perspective on experiment aspect by using five image databases: Yale database [67], AR database [START_REF] Martinez | The AR face database[END_REF], CMU PIE face database [START_REF] Sim | The CMU pose, illumination, and expression (PIE) database[END_REF], ORL database [START_REF]ORL datasets[END_REF], and CIFAR-10 database [START_REF] Krizhevsky | Learning multiple layers of features from tiny images[END_REF]. The proposed energy explanation approach can provide more information than the filter visualization method reported in [START_REF] Chan | PCANet: A simple deep learning baseline for image classification?[END_REF]; 2) We shows qualitatively that the error rate of PCANet is correlated with the inverse of the logarithm of BlockEnergy, which is the energy after the block sliding process of PCANet, and then we try to find quantitatively their relations by using curve fitting method; 3) we show that the proposed energy explanation approach can be used as a testing method for checking if every step of the constructed networks is necessary; and 4) The energy explanation approach proposed in this paper can be extended to other PCANet-based networks [START_REF] Gan | A deep graph embedding network model for face recognition[END_REF][START_REF] Wang | Unsupervised feature learning with C-SVDDNet[END_REF][START_REF] Feng | DLANet: A manifold-learning-based discriminative feature learning network for scene classification[END_REF][START_REF] Ng | DCTNet: A simple learning-free approach for face recognition[END_REF][START_REF] Gan | A PCA-based convolutional network[END_REF][START_REF] Zhao | Multi-level modified finite radon transform network for image upsampling[END_REF][START_REF] Lei | Learning stacked image descriptor for face recognition[END_REF][START_REF] Li | Feature learning based on SAE-PCA network for human gesture recognition in RGBD images[END_REF][START_REF] Zeng | Color image classification via quaternion principal component analysis network[END_REF][START_REF] Wu | Multilinear principal component analysis network for tensor object classification[END_REF].

The paper is organized as follows. PCANet is reviewed in section 2. Section 3 presents an energy method to visualize, explain and understand every step of PCANet. Discussion is given in Section 4 and Section 5 concludes the work.

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A N U S C R I P T 5

Review of Principal Component Analysis Network

In this section, we first review the PCANet [START_REF] Chan | PCANet: A simple deep learning baseline for image classification?[END_REF], whose architecture is shown in Fig. 1 and can be divided into three stages, including 10 steps. Suppose that we have N input training images   , 1, 2,..., ,

mn i i iN   II
, and that the patch size (or two-dimensional filter size) of all stages is k 1 ×k 2 , where k 1 and k 2 are odd integers satisfying 1≤k 1 ≤m, 1≤k 2 ≤n. We further assume that the number of filters in layer i is L i , that is, L 1 for the first stage and L 2 for the second stage. In the following, we describe the structure of PCANet in detail.

Let the N input images   (1)

The first stage of PCANet

As shown in Fig. 1, the first stage of PCANet includes the following 3 steps:

Step 1: the first patch sliding process.

The images are padded to

12 ( 1) ( 1) m k n k i        I
before sliding operation. Out-of-range input pixels are taken to be zero. This can ensure all weights in the filters reach the entire images. We use a patch of size k 1 ×k 2 to slide each pixel of the ith image

( 1) ( 1) m k n k i        I
, then reshape each k 1 ×k 2 matrix into a column vector, which is then concatenated to obtain a matrix 12 ,1 ,2 , , 1, 2,..., ,

k k mn i i i i mn iN       X x x x (2)
where x i,j denotes the jth vectorized patch in I i .

Therefore, for all the input training images   , 1, 2,..., i iN  I , we can obtain the following matrix   12

12

,

k k Nmn N   X X X X (3)
Step 2: the first mean remove process.

In this step, we subtract patch mean from each patch and obtain 12 ,1 ,2 , , 1, 2,..., ,

k k mn i i i i mn iN       X x x x (4) where , , , 1 1
,

mn i j i j i j j mn    x x
x is a mean-removed vector. .

k k Nmn N     X X X X ( 5
)
Step 3: the first PCA process.

In this step, we get the eigenvalues and eigenvectors of X in (5) by using PCA algorithm, which in fact minimizes the reconstruction error in Frobenius norm as follows:

1 1 2 1 2 min , s.t. , k k L TT L F    U X UU X U U D (6)
I I m n i l i l l L i N       I I W (8)
where * denotes two-dimensional (2-D) convolution, and the boundary of I i is zero-padded before convolving with I l W in order to make ,

I il

I having the same size as I i .

For each input image

mn i   I , we obtain L 1 output images   , 1 ,
, 1, 2,..., ,

I I m n i l i l lL   II
after the first stage of PCANet. We denote I I as

1 1 1 1,1 1, ,1 , m NL n I I I I I L N N L        I I I I I . ( 9
)

The second stage of PCANet

As shown in Fig. 1, the second stage of PCANet also includes 3 steps:

Step 4: the second patch sliding process.

A C C E P T E D M

A N U S C R I P T ,

k k Nmn l l l N l     Y Y Y Y (11)
We concatenate the matrices of all the L 1 filters and obtain .

k k L Nmn L      Y Y Y Y ( 12
)
Step 5: the second mean remove process.

Similar to

Step 2, we obtain the mean-removed version of ( 12) by

1 2 1 1 1 2 1
, 1, 2,..., ,

k k L Nmn L lL       Y Y Y Y (13) where 12 1, 2, ,
,

k k Nmn l l l N l     Y Y Y Y (14) 12 , , ,1 , ,2 , , 1 , 1, 2,
..., ; 1, 2,..., ,

k k mn i l i l i l i l mn l L i N          Y y y y (15) , , , , , , 1 1 1
, 1, 2,..., ; 1, 2,..., .

mn i l j i l j i l j j l L i N mn       y y y (16)
Step 6: the second PCA process.

Similar to

Step 3, we use the PCA algorithm to minimize the following optimization problem:

2 1 2 2 2 min , s.t. , k k L TT L F    V Y VV Y V V D (17)
where 2

L

D is an identity matrix of size L 2 ×L 2 . Eq. ( 17) can also be solved by the eigenvalue decomposition method shown in Appendix in which we replace , , , , , , , 1, 2,...,

I i L N i L   X U u C by 2 2 1 2 , , , , , , , 1, 2,
...,

II i L L N i L   Y V v C
, respectively. The PCA filter of the second stage of PCANet is then obtained from (A.5) by . Note that the italic superscript II in [START_REF] Learned-Miller | Labeled faces in the wild: A survey[END_REF] denotes the second stage. Therefore, the output of the second stage of PCANet is given by 12 , , ,

  12
..., ; 1, 2,..., ; 1, 2, , ,

kk II I II i l i l l L L i N        I I W (19)
where * denotes 2-D convolution.

For each input image ,

I m n il   I , we obtain L 2 output images   , , 2 , , , 1, 2,
..., ,

II II m n i l i l L   II
after the second stage of PCANet. Thus, we obtain after the first and second stages.

NL 1 L 2 images   , , 1 2 , , 1, 2,
We denote

I II as 2 1 1 2 2 1 1 2 1,1,1 1,1, 1, , 1 1, , ,1,1 ,1, , ,1 , ,
.

II II II II II II II II II L L L L N N L N L N L L      I I I I I I I I I (20)

The output stage of PCANet

Step 7: binary quantization.

In this step, we binarize the outputs ,,

II il I of the second stage of PCANet and obtain   , , , , 1 2
, 1, 2,..., ; 1, 2,..., ; 1, 2, , , [START_REF] Ciresan | Multi-column deep neural networks for image classification[END_REF] where H(•) is a Heaviside step function whose value is one for positive entries and zero otherwise. We denote P as

II i l i l H l L L i N     PI
2 1 1 2 2 1 1 2 1,1, 1 1,1, 1, ,1 1, , ,1,1 ,1, , ,1 , ,
.

L L L L N N L N L N L L   
Step 8: weight and sum.

Around each pixel, we view the vector of L 2 binary bits as a decimal number. This converts the binary images  

,, il P back into integer-valued images as follows:

2 1 , , , 1 2,
L i l i l     TP (23)
We denote T as

1 1 1 1,1 1, ,1 , m NL n L N N L        T T T T T . ( 24
)
Step 9: block sliding.

A C C E P T E D M A N U S C R I P T 9
We use a block of size h 1 ×h 2 to slide each of the L 1 images , 1 , 1,..., , il lL  T with overlap ratio R, and then reshape each h 1 ×h 2 matrix into a columnvector, which is then concatenated to obtain a matrix

12 , , ,1 , ,2 , , , 1,2,
..., ,

h h B i l i l i l i l B iN       Z z z z (25)
where z i,l,j denotes the jth vectorized patch in , 1 , 1,..., il lL  T . B is the number of blocks when using a block of size h 1 ×h 2 to slide each , 1 , 1,..., , il lL  T with overlap ratio R and given by

    1 1 2 2 1 1 / stride1 1 1 / stride2 B m k h n k h                   , (26)
where stride1 and stride2 are vertical and horizontal steps, respectively,

    1 stride1 1 round R h    , (27)
    2 stride2 1 round R h    , (28)
and round (.) means round off. As we can see from ( 26)-( 28), the number of blocks B is increasing as the overlap ratio R is increasing.

For L 1 images, we concatenate , il Z to obtain a matrix

1 2 1 ,1 ,2 ,
, 1,2,..., ,

h h L B i i i i B iN       Z Z Z Z (29)
We denote Z as

1 2 1 1,1 1, , 1 ,
.

h h L BN B N N B        Z Z Z Z Z ( 30
)
Step 10: histogram.

We compute the histogram (with 2 2 L bins) of the decimal values in each column of Z i and concatenate all the histograms into one vector and obtain

2 1 1 1 (2 ) ,1,1 ,1, , ,1 , , Hist( ) Hist( ) Hist( ) Hist( )
. L T LB i i i B i L i L B       f z z z z (31)
which is the "feature" of the input image I i and Hist(.) denotes the histogram operation. We denote f as

  2 1 (2 ) 1 L L BN N  f f f . ( 32
)
The feature vector is then sent to a classifier, for example, support vector machine (SVM) [START_REF] Fan | LIBLINEAR: A library for large linear classification[END_REF], k-nearest neighbors algorithm (KNN) [START_REF] Altman | An introduction to kernel and nearest-neighbor nonparametric regression[END_REF], etc.

A C C E P T E D M

A N U S C R I P T 10

The parameters of PCANet

The dimension of PCANet feature is related to the core parameters shown in Table 1. For a given dataset, m and n are fixed. In this paper, we always set k 1 =k 2 =3 since in this case the consumption of memory is tolerable for an ordinary computer (for example, 64 GB RAM). Note also that we set a constraint

21 / h nh m    , (33)
which is a common choice in practice. After the above setting, there are four free parameters left: L 1 , L 2 , h 1 , and R.

Energy Analysis and Energy Explanation of PCANet

In this section, we introduce the details of the proposed energy explanation approach method. We first present the motivation of the proposed approach, and then we describe the experiment methods and the used databases. The following experiments were implemented in Matlab programming language.

Motivations

CNNs are visualized and understood by using back-trace or inverted method to show the filters (or weights) that are learned by every layer [START_REF] Erhan | Visualizing higher-layer features of a deep network[END_REF][START_REF] Zeiler | Visualizing and understanding convolutional networks[END_REF][START_REF] Mahendran | Understanding deep image representations by inverting them[END_REF]. The filters show much information since CNNs generally have many layers and the number of these filters are very large. Chan et al. [START_REF] Chan | PCANet: A simple deep learning baseline for image classification?[END_REF] also visualize the filters of PCANet like in [START_REF] Erhan | Visualizing higher-layer features of a deep network[END_REF], [START_REF] Zeiler | Visualizing and understanding convolutional networks[END_REF], and [START_REF] Mahendran | Understanding deep image representations by inverting them[END_REF]. However, PCANet does not have such large number of filters like CNNs, therefore, visualizing the filters only provides limited information about two filter layers (Steps 3 and 6) of PCANet. Furthermore, how to effectively visualize other implemental steps (Steps 1, 2, 4, 5, and 7-9) of PCANet is still an open problem since these simple operations have a prominent impact on the performance of PCANet. For example, as depicted in [START_REF] Chan | PCANet: A simple deep learning baseline for image classification?[END_REF], the performance of PCANet is degraded if we delete the simple mean-remove process.

On the one hand, PCA, also known as Karhunen-Loeve transform (KLT), is very popular in data compression domain since it has very good energy concentration capability, that is, few coefficients of KLT capture vast majority of energy of the original signal. On the other hand, Mallat and Bruna [START_REF] Mallat | Group invariant scattering[END_REF][START_REF] Bruna | Invariant scattering convolution networks[END_REF] also prove and explain ScatNet from energy perspective. However, there are two main differences between PCANet and ScatNet [START_REF] Mallat | Group invariant scattering[END_REF][START_REF] Bruna | Invariant scattering convolution networks[END_REF]:

Experiment databases

In the following, we use five databases, whose Matlab formats are provided in [START_REF] Krizhevsky | Learning multiple layers of features from tiny images[END_REF] and [START_REF] Cai | Orthogonal laplacianfaces for face recognition[END_REF], AR database [START_REF] Martinez | The AR face database[END_REF]. We use a non-occluded subset of AR database containing 700 images (50 male subjects, 14 images pixels. For the cross validation, we use the simplest hold-out validation due to the following two reasons: 1) some cases of the experiments are time and memory consuming; 2) the objective of the paper is not to achieve the lowest error rate but to establish the one-to-one correspondence of error rate e and the inverse of the logarithm of BlockEnergy g, and then find the parameters to achieve the relatively low error rate by the guidance of energy. Therefore, we need e and g are directly correspondent.

Experiment methods

In this section, we exploit the above-mentioned databases to analyze the PCANet by using energy method. We first introduce the definition of the energy of a two-dimensional image I (i, j) of size m×n:

2 11 ( ) ( , ) mn ij E i j    II . ( 34
)
Fig. 1 shows that we record 10 energies for PCANet. It is convenient for us to divide these energies into two parts:

Energies 1-9 belong to the first part and Energy 10 belongs to the second part. The reason for this partitioning is that Energies 1-9 are not related to the block size h 1 × h 2 and the overlap ratio R. The values that we recorded in the whole process of PCANet are shown in Table 2, which includes 10 energies and the error rate. Note that we do not take the

A C C E P T E D M A N U S C R I P T 12
energy of the feature f in (32) into consideration, since the block-wise histogram is only the statistic (or occurrence frequency) of energy value and we consider that this process does not change the energy.

The analysis of the second part

In this subsection, we first introduce two metrics of the experiments. The first metric is the inverse of the logarithm of BlockEnergy E(Z), shown in Table 2, that is

  10 1/ log ( ) gE  Z . ( 35
)
The second metric is the error rate

The number of false classified testing images The total number of testing images e  .

In the second part, there are 4 free parameters: h 1 (the number of rows of block), R (the overlap ratio), L 1 (the number of filters of the 1 st stage), L 2 (the number of filters of the 2 nd stage). Therefore, the aim of this subsection is to explore qualitatively the metric g and also the metric e with regards to the parameters h 1 , R, L 1 , and L 2 .

1) How the g and the e qualitatively vary according to h 1 , R, L 1 , and L 2 ?

In this experiment, we use Yale database for analyzing the g and also the e in relation with The results of the variations of g according to h 1 , R, L 1 , L 2 are shown in Fig. 2, whose vertical axis is h 1 {1:1:32}

h 1 , R, L 1 , L 2 ,
and horizontal axis is related to R and L 2 . Specifically, the horizontal axis can be divided into 9 blocks corresponding to L 2 {1:1:9}, each block can further be divided into 10 sub-blocks corresponding to R{0:0.1:0.9}, where each sub-block has the size of 32×1. Therefore, the range of horizontal axis value is from 1 to 90 (=10×9). b) The impact of L 2 on g and e. As the value of L 2 increases, the first five figures corresponding to L 2 {1:1:5} change a lot, but when L 2 ≥6, the left four figures only make little changes. Therefore, we can think that when L 2 = 6, both g and e are stable.

c) The impact of R on g and e. There are some differences of the changes of g and e according to R. From Fig. 2, we can see that g is gradually decreased as the value of R increases. However, from Fig. 3, the probability of lower error rate e is increased as the value of R increases from 0 to 0.9 with interval of 0.1. Nonetheless, the appropriate values of R in fact belong to {0:0.1:0.6}, due to the following two reasons: Firstly, from the experimental aspect, when R≥0.7, it may encounter the cases where PCANet can not work; When R≥0.7, the result is also good if it does not encounter the cases where PCANet can not work, but both the computational and the memory complexities are very high from experiment aspect since they lead to too much overlap. Let us take Yale database for example, setting k 1 =k 2 =3, 3 shows the accuracy, time consumption, and dimension of features when R varies from 0 to 0.9. Secondly, from the theory or explanation aspect, the overlap ratio R has a similar role with the eigenvector in PCA filtering as shown in Appendix. In the case of realizing dimension reduction by PCA, we want to use as few as possible eigenvectors while keeping enough useful features of original image. Similarly, we want to use as small as possible overlap ratio R.

L 1 =L 2 =8, h 1 =h 2 =8, Table
d) The impact of h 1 on g and e.

As we can see, when h 1 is higher than a critical value (m+k 1 -1)/2+10R, each block has an obvious trapezoidal area, where the values of g and e are evidently higher than other parts of the block. The reason is that the block sliding process (Step 9) leads to "great reduction in energy" when h 1 >(m+k 1 -1)/2+10R. Why this phenomenon appears? Let us take R=0 as an example, in this case, the critical value (m+k 1 -1)/2+10R=(32+3-1)/2+10×0=17. 18×18 block on the upper left corner, other information (or energy) of the image is not used. That is to say, the trapezoidal area is caused by "not making full use of energy". Therefore, the appropriate choice of the number of rows of block is h 1 ≤(m+k 1 -1)/2+10R since it can avoid the trapezoidal area.

In total, we can see the following things:

i) The changes of g and e have similar overall consistent trends, for example, the values of g and e are gradually decreased as the value of L 1 and L 2 increase.

ii) The changes of g are relatively smooth, while the changes of e seem more varied than that of g. Furthermore, e

is not simply directly proportional to g.

2)

How the e varies quantitatively according to g ?

In the following, we use the Curve Fitting Tool of Matlab to further investigate the correlation between e and g.

We fit the function e = f(g), by using the linear model Poly1 and the nonlinear model Poly3 in Matlab with 95% confidence bounds, that is, we used e = p 1 g+p 2 , and e =p 3 g 3 +p 4 g 2 +p 5 g+p 6 , where p i , i =1,2,…,6, are the coefficients to fit, to find the relationship of e and g, respectively. The curve fitting results of five datasets are shown in Fig. 5 and the quantitative results are shown in Table 4. The criterions include:

a) The sum of error squares (SSE), sum of regression squares (SSR), and total sum of squares (SST) are defined respectively as

2 1 ( ( )) ii i M SSE e f g    , ( 37
) 2 1 ( ( ) ) i M i SSR f g e    , ( 38
) 2 1 () i i M SST SSE SSR e e       , (39)
A M ee    , and i is the index of the number of e or g, and M is the total number of e or g. In our experiment, M=32×10×9×9=25920.

b) Coefficient of determination or R-square is defined as

2 1 SSE r SST  , (40)
which is a number that indicates how well data fit a statistical model, for example, a curve. c) Root mean squared error (RMSE) is defined as

2 1 ( ( )) N i i i e f g RMSE N     . ( 41
)
As we can see from Table 4, the R-square of the data of Yale database by using linear model Poly1 is 0.6104, which means 61.04% of the variance in the response variable can be explained by the explanatory variables. Therefore, e and g are fairly linearly correlated. As we mentioned above that changes of g and e have similar overall consistent trends but the changes of e seem more varied than that of g. Can we use a little more complicated model for describing the correlation between e and g? We then use a nonlinear polynomial function of degree 3 (Poly3) to model the relationships of e and g. The R-square of the data of Yale database by using model Poly3 is 0.6288, which means 62.88% of the variance in the response variable can be explained by the explanatory variables. The remaining 37.12% can be attributed to unknown, lurking variables or inherent variability. Therefore, e and g are fairly non-linearly correlated. We also give the curve fitting results for the data of five databases (Yale, AR, CMU PIE, ORL, CIFAR-10) in Table 4. Similar conclusion can be obtained for the other databases. In fact, if we cancel the constraint in [START_REF] Mehta | An exact mapping between the variational renormalization group and deep learning[END_REF] and do the similar experiment on Yale database, the R-square of the data of Yale database by using Poly3 increases further to 0.7637, which is even larger than 0.6288 that has the constraint in [START_REF] Mehta | An exact mapping between the variational renormalization group and deep learning[END_REF].

3) How to explain the block sliding process?

We can in fact explain this process as an "overlap filtering", which is a term that we proposed in this paper to explain and better understand the sliding process, including the patch sliding process and the block sliding process. The idea of "overlap filtering" is in fact from the PCA filter. The comparison and the connection of "overlap filtering" and PCA filter are given in Appendix and also in Table 5.

A C C E P T E D M

A N U S C R I P T 16

The analysis of the first part

In this subsection, we will analyze the changes of Energies 1-9 of the first part of PCANet shown in Fig. 1. Since the first part is not related to h 1 , h 2 , and R, so in the experiment of this section, we simply set the parameters h 1 = h 2 = 8, and R= 0.5, due to the following three reasons: (1) they are near one of the recommended parameters h 1 =8, h 2 = 6, and R= 0.5 in PCANet [START_REF] Chan | PCANet: A simple deep learning baseline for image classification?[END_REF];

(2) they have the moderate computational complexity and memory complexity; (3) they satisfy the constraint 21 / h nh m    in [START_REF] Mehta | An exact mapping between the variational renormalization group and deep learning[END_REF] in the analysis of the second part. Let us take Yale dataset as an example, Fig. 6 shows the energy values of every stage of the first part of PCANet versus the ith energy. The results of 9 filters, corresponding to L 1 {1:1:9}, are shown in different colors. For example, the lowest line is the result that we choose only the largest eigenvector corresponding to L 1 =1. The second lowest line is the result that we choose the 2 largest eigenvectors corresponding to L 1 =2, and so on. Note that the vertical axis is shown in logarithmic coordinate. From Fig. 6, for all the 9 filters, we can easily see that the energy changes are as follows

Step 1

Step 2 or Step 7 3

Step 4

Step 5 or Step [START_REF] Lecun | Deep learning[END_REF] Step [START_REF] Anselmi | On invariance and selectivity in representation learning[END_REF] where the arrow means that the energy values change from the left-hand side to the right-hand side. "+", "-", "0" above the arrow denote the energy increase, decrease, and equal process, respectively. Steps 1-8 below the arrow correspond to the eight steps of Fig. 1.

From [START_REF] Anselmi | On invariance and selectivity in representation learning[END_REF], we can see that:

1) For Steps 1 and 4, both of them are energy increasing processes, which can be easily understood since these two steps perform the overlapping patch sliding process. However, how can we explain the patch sliding process? In fact, the patch sliding process is a special case of block sliding process with the overlap ratio R=0.9 shown in the second part.

2) For the Steps 2 and 5, both of them are mean removing processes. Step 2 is energy decreasing process, which is reasonable since the mean removing leads to the reduction of image energy. However, it is strange that Step 5 keeps the energy. Therefore, we think this step may not be needed in the construction of PCANet. We then perform some experiments on four databases to verify our inference. We compute the differences of error rates of PCANet with mean remove Step 5 and that of without Step 5 for all the parameter setting of h 1 {1:1:32}, R{0:0.1:0.9}, and L 1 , L 2 {1:1:9}. The mean of differences of error rates for Yale database, AR database, CMU PIE database, and ORL database are 1.1762×10 -6 , 1.5878×10 -7 , -1.4886×10 -7 , -4.9619×10 -7 , respectively, all of which are very small. Therefore, when taking the computational complexity into consideration, we recommend to not use the mean remove process of Step 5.

3) For the Steps 3 and 6, both of them are PCA processes. Both Step 3 and Step 6 are energy increasing processes except the case of 1 filter, but the energy change is very small. Table 6 shows the energy of each of the 9 filters (the 1 st PCA), h 1 =h 2 =8, R=0.5, L 2 =1, and L 1 varies from 1 to 9, and the energy of every filter (the 2 nd PCA), h 1 =h 2 =8, R=0.5, L 1 =8, and L 2 varies from 1 to 9. Table 7 shows the error rate corresponding to L 1 , L 2 ∈{1:1:9} when h 1 =h 2 =8, R=0.5.

From the two Tables, we can see that the cumulative percentage of eigenvalue can be a guideline for the choice of L 1 and L 2 , for example, the error rate of Yale database is low when the cumulative percentage of eigenvalue is larger than 99.7%, which corresponds to L 1 =8 and L 2 =8. Note that L 1 =L 2 =8 is just one of the parameter settings recommended by Chan"s paper [START_REF] Chan | PCANet: A simple deep learning baseline for image classification?[END_REF]. 4) For the Steps 7 and 8, which cause the sharpest energy changes of PCANet. It is also reasonable since the binarization operation is a very powerful quantization process (nonlinear processing), which loses much energy.

Meanwhile, the weighted and summed step is a very powerful energy increasing step. It is interesting to see from Fig. 6 that the energy in Step 8 gradually increases when L 2 varies from 1 to 9. Let's focus on the two horizontal dashed lines: the first one denotes the MaxEnergyLinear (maximum energy of linear layer of PCANet) including either PatchEnergy2 or PCAEnergy2, and the second one denotes the TrainEnergy. We can see from Fig. 6 and Table 7: a) When WeightSumEnergy<TrainEnergy, corresponding to Fig. 6 (a)-(e), the error rate is gradually decreased when L 1 ∈{1:1:9} and L 2 ∈{1:1:5}. b) When TrainEnergy < WeightSumEnergy ≤ MaxEnergyLinear, corresponding to Fig. 6 (f)-(h), the error rate is flattened when L 1 ∈{1:1:9} and L 2 ∈{6:1:8}.

A C C E P T E D M

A N U S C R I P T 18 c) When WeightSumEnergy>MaxEnergyLinear, corresponding to Fig. 6 (i), the error rate is increased when L 1 ∈ {1:1:9} and L 2 =9. In this case, we think that the WeightSumEnergy includes "pseudo energy" so that the WeightSumEnergy is even larger than the maximum energy of linear layer of PCANet. Note that the above experiments performed in other four databases (AR database, CMU PIE face database, ORL database, CIFAR-10 database), the results of which correspond to the Fig. 2, Fig. 3, Fig. 5, and Tables 5-7 of YALE database are provided in the supplement document due to the limited space. The results of other three databases are similar to that of Yale database.

Therefore, we can conclude from the aforementioned analysis:

1) For the linear process of PCANet (Steps 1 to 6), repeated use of the combination of patch sliding process, mean removing process, and PCA increases the energy and seems beneficial for the recognition performance of PCANet.

It may explain that two-layer PCANet can generally obtain better recognition performance than one-layer PCANet.

2) For the nonlinear process of PCANet (Steps 7 and 8), the choice of the value of L 2 is very important. When we choose a proper L 2 value, which makes WeightSumEnergy comparable to the TrainEnergy (like Fig. 6 (f)), in this case, PCANet can generally lead to good results.

3) We can eliminate the mean removing process of Step 5 when taking the computational complexity into consideration since there is no energy changes in this step and the recognition performances of PCANet with or without Step 5 are very similar.

4) The changes of e (error rate) seem more varied than that of g (the inverse of the logarithm of BlockEnergy), however, correlations between e and g are fairly strongly non-linear as underlined when using the model Poly3 in Matlab. Therefore, BlockEnergy may be seen as a wind arrow for the error rate of PCANet.

Discussion

This section discusses some issues related to the proposed energy explanation method.

Parameter settings of k 1 and k 2 . In this paper, we always set k 1 =k 2 =3 due to the following two reasons:

(1) The parameters setting of k 1 =k 2 =3 is a representative choice since filter kernels with the size 3×3 are the most common choices in the design of CNNs, for example, VGGNet [START_REF] Simonyan | Very deep convolutional networks for large-scale image recognition[END_REF], ResNet [START_REF] He | Deep residual learning for image recognition[END_REF], etc.

(2) In this case, we have 1≤L 1 ≤9 and 1≤L 2 ≤9, that is to say, the number of outputs of PCANet is at most 81 times of that of input after the Step 6. The consumption of memory is tolerable for an ordinary computer (for example, 64 GB

A C C E P T E D M

A N U S C R I P T 19 RAM). Let us take Yale dataset for an example (m=n=32), if we set L 1 =L 2 =9, h 1 =h 2 =6, R=0.9, the dimension of the resulting PCANet feature after Step 10 is 3875328, which is tolerable for SVM and KNN classifiers. However, if we set k 1 =k 2 =5, the most time and memory consumption case is L 1 = L 2 =25, that is to say, the number of outputs of PCANet is 625 times of that of input after the Step 6. Let us still take Yale dataset for an example (m=n=32), if we set L 1 =L 2 =25, h 1 =h 2 =6, R=0.9, Step 10 (HashingHist function) will apply a memory for the matrix of size 33554432×961 (240.3GB), which is unbearable for Matlab and also a very challenge dimension for SVM and KNN classifiers. As a further direction, we will try to extend to other parameter settings (k 1 =k 2 =5, k 1 =k 2 =7, etc.) when higher performance machines will be available.

The impact of classifier. Besides SVM, we also give the results of KNN classifier of three datasets (Yale dataset, AR dataset, ORL dataset) in the supplement document. Results show that the error rates of PCANet by using the SVM and the KNN classifiers can be nearly fitted by linear function.

The size of dataset. The number of images in the database affects the speed and memory consumption of PCANet. Specifically, when the number of images increases, the covariance matrix computation in the PCA filter becomes slow and the training of SVM and KNN classifiers also becomes slow; meanwhile, it needs training more parameters and thus more memory consumption when training SVM and KNN classifiers. Therefore, the number of images is also not allowed to be big (for example, less than 4000 images for a color image database) in order to ensure the SVM and KNN classifications.

The impact of cross validation. Besides the simplest hold-out validation (or 2-fold cross validation) in the above experiments, we also do the 5-fold cross validation on Yale dataset, we find the curve fitting results are slightly worse than that of the hold-out validation. The reason is that both the error rate and the energy are the average values of 5 experiments, however, the relation of the average values of error rate is in fact not actually corresponding to the average of values of the energy. Therefore, 5-fold cross validation can make the error rate of a dataset more reasonable, but it is not suitable for our case since our method needs the direct correspondences of the energy and the error rate but not their average values.

Conclusions

In this paper, an energy method is proposed to visualize, explain and understand every step of PCANet. The paper shows that the error rate of PCANet is qualitatively correlated with the inverse of the logarithm of BlockEnergy, and

A C C E P T E D M A N U S C R I P T 20
then their relations are further quantified by using curve fitting methods. The role and the explanation of every step of PCANet have been investigated, for example, the patch sliding process and the block sliding process are explained by "overlap filtering". The proposed energy explanation approach could be used as a testing method for checking if every step of the constructed networks is necessary. For example, we find that the second mean remove step is not needed in the construction of PCANet since the energy is not changed. The proposed energy explanation approach can provide more information than the filter visualization method shown in [START_REF] Chan | PCANet: A simple deep learning baseline for image classification?[END_REF]. Furthermore, the proposed energy explanation approach can be extended to other PCANet-based networks [START_REF] Gan | A deep graph embedding network model for face recognition[END_REF][START_REF] Wang | Unsupervised feature learning with C-SVDDNet[END_REF][START_REF] Feng | DLANet: A manifold-learning-based discriminative feature learning network for scene classification[END_REF][START_REF] Ng | DCTNet: A simple learning-free approach for face recognition[END_REF][START_REF] Gan | A PCA-based convolutional network[END_REF][START_REF] Zhao | Multi-level modified finite radon transform network for image upsampling[END_REF][START_REF] Lei | Learning stacked image descriptor for face recognition[END_REF][START_REF] Li | Feature learning based on SAE-PCA network for human gesture recognition in RGBD images[END_REF][START_REF] Zeng | Color image classification via quaternion principal component analysis network[END_REF][START_REF] Wu | Multilinear principal component analysis network for tensor object classification[END_REF].

Appendix Comparisons of PCA Filter and Overlap filtering PCA Filter

Overlap filtering

Objective

Minimize the reconstruction error in Frobenius norm as

1 1 2 1 2 min , s.t. , k k L TT L F    U X UU X U U D (A.1)
where DL is an identity matrix of size L×L, the superscript T denotes transposition, and

12 k k Nmn   X .
How to explain the sliding process, including the patch sliding process and the block sliding process? How to choose an appropriate overlap ratio R?

Solve

Eigenvalue decomposition method Eq. (A.1) can be solved by eigenvalue decomposition method 11 ,

T  CU Λ U (A.2)
where C1 is a covariance matrix given by

1 2 1 2 1 1 , k k k k T Nmn   C XX (A.3)
and 1 Λ is a diagonal matrix composed of the first L1 largest eigenvalues of C1,

1 1 1 2
, .

I I I L diag       Λ (A.
k k L L     U u u u (A.5)
Explain the sliding process as an "overlap filtering"

1) Suppose we use a block h1×

1 / nh m   , 1≤h1≤m, to slide an image, we can get at most 10 images. The energy of each image has a similar effect with the eigenvalue in PCA filtering. The overlap ratio R=0,0.1,0.2,...,0.9, corresponds to the 1 st , 2 nd , 3 rd , ..., 10 th eigenvectors in PCA filtering.

2) Rearrange the energy of the 10 images, that is, sort the energy values in the descending order. This is shown in the 3th row of Table 5.

How to choose the parameter?

How to choose the number of filters L? Cumulative energy method 1) Compute the cumulative energy content g as 12 1 , 1,...,

j I ji i g j k k     .
2) Use the vector g as a guide in choosing an appropriate value for L. The goal is to choose a value of L as small as possible while achieving a reasonably high value of g on a percentage basis, for example, the cumulative energy g is above a certain threshold, like 90 percent, that is, 12 / 0.9

L k k gg .

How to choose the overlap ratio R? Cumulative energy method

We show the ratio of energy in the 5 th row. As shown in Section 3.3.1, R{0:0.1:0.6} is appropriate for PCANet. We can see from Table 5, the energy we keep is about 95.75% when R=0.6, which seems the point that comprises the training error and the generalized error. Therefore, the energy can be used as a guidance for the choice of an appropriate overlap ratio R. k k mn

1,1 Y 1 1 , L Y , 1 N Y 1 , N L Y 1 2 k k mn 1,1 Y 1 1 , L Y , 1 N Y 1 , N L Y m n 1,1,1 I I I 2 1,1, I I L I 1 , , 1
I I N L I 1 2
, ,

I I N L L I m n 1,1,1 P 2 1,1,L P 1 , ,1 N L P 1 2 , , N L L P m n 1,1 T 1 1 , L T , 1 N T 1 , N L

T

Step 8: weight and sum

1 h 2 h 1 2 h h B 1,1 Z 1 1 , L Z , 1 N Z 1 , N L Z 2 1 (2 ) 1 L L B  f
Step 4: the 2 nd patch sliding

Step 5: the 2 nd mean remove

1 2 k k mn 1 X 2 X 1 N  X N X 1 2 k k mn 1 X 2 X 1 N  X N X
Step 1: the 1 st patch sliding

Step 2: the 1 st mean remove The second stage of PCANet

The output stage of PCANet

2 1 / h nh m      1 2 h h 1 L B 1 Z 2 Z 1 N  Z N Z 2 1
(2 )

R

The overlap ratio of block. R∈{0:0.1:0.9}which means R varies from 0 to 0.9 with the interval 0.1.

Table 2

The energy and error rate of PCANet we recorded and their descriptions.

Two parts

Energy or Error rate Descriptions The first part TrainEnergy The total energy of N input training images.

PatchEnergy1

The total energy of N images after the first patch sliding process (Step 1).

 

PatchEnergyRed1

The total energy of N images after the first mean remove process (Step 2).

 

XX

, X is defined in Eq. ( 5).

PCAEnergy1

The total energy of NL1 images after the first PCA process (Step 3)

    1 2 11 ( , ) L Nn m II jk E j k   

II

, I I is defined in Eq. ( 13).

PatchEnergy2

The total energy of NL1 images after the second patch sliding process (Step 4).

 

  1 2 1 2 11 ( , ) k k L Nmn jk E j k   

YY

, Y is defined in Eq. ( 16).

PatchEnergyRed2

The total energy of NL1 images after the second mean remove process (Step 5).

   

PCAEnergy2

The total energy of NL1L2 images after the second PCA process (Step 6)

   

ErrorRate

The error rate e of PCANet.

12

 12

where 1 LD.

 1 is an identity matrix of size L 1 ×L 1 , and the superscript T denotes transposition. Eq. (6) can be solved by eigenvalue decomposition method shown in Appendix. The PCA filter of the first stage of PCANet is then obtained from (A.5) by Note that the superscript of capital Roman number I denotes the first stage, The output of the first stage of PCANet is given by

  [START_REF] Hinton | A fast learning algorithm for deep belief nets[END_REF] the convolutional layer of PCANet is constructed by data-dependent PCA filter while the convolutional layer of ScatNet is constructed by data-free complex wavelet filter; (2) the nonlinearity of PCANet is binarization while the nonlinearity of ScatNet is modulus. The aforementioned two aspects lead to a question: can we explain every step of PCANet from an energy A C C E P T E D M A N U S C R I P T 11 perspective?

  to analyze the energy change as well as the error rate change of PCANet: 1) Yale database [67]. It contains 165 grayscale images (15 individuals, 11 images per individual), which are randomly divided into 30 training images and 135 testing images; 2)

  per subject), which are randomly divided into 200 training images and 500 testing images; 3) The CMU PIE face database[START_REF] Sim | The CMU pose, illumination, and expression (PIE) database[END_REF]. We use pose C27 (a frontal pose) subset of CMU PIE face database which has 3400 images (68 persons, 50 images per person), which are randomly divided into 1000 training images and 2400 testing images; 4) ORL database[START_REF]ORL datasets[END_REF]. It contains 400 grayscale images (40 individuals, 10 images per individual), which are randomly divided into 80 training images and 320 testing images. 5) CIFAR-10 database[START_REF] Krizhevsky | Learning multiple layers of features from tiny images[END_REF]. The original database contains 60000 color images (10 classes), including 50000 training images and 10000 test images. In our experiment, we randomly choose 3000 images for training and 1000 images for testing. Note that the size of all the images is 32×32

where h 1

 1 {1:1:32}, R{0:0.1:0.9}, and L 1 , L 2 {1:1:9}. Note that {a:b:c} means the set contains the values that vary from a to c with increment b. Each combination of h 1 , L 1 , L 2 , and R has a e value or g value. Therefore, if we denote M as the total number of e or g, then, M=32×10×9×9=25920.

Fig. 2 ( 13 a)

 213 figures have notable changes, this phenomenon implies that the first few number of filters captures most of the energies of the original image, leading to the severe changes of g and e.

Figs. 4

 4 (a) and (b) show the case of block sliding process of h 1 =17 and h 1 =18, respectively. From Fig.4(a), the image of 34×34 becomes an image of 4×289 by block sliding process with block of 17×17. In this case, all the information (or energy) of the imageA C C E P T E D M A N U S C R I P T 14is used through 3 times of block sliding process. However, as we can see from Fig.4(b), the image of 34×34 becomes an image of 1×324 by block sliding process with block of 18×18. Note that both the right side and the bottom side of the 18×18 block are not enough for a new block sliding process. In this case, we only use the information (or energy) of

  superscript of capital Roman number I denotes the first stage, and U is the matrix composed of the first L1 principal eigenvectors

21 Lan

 21 Project and the "333" project (BRA2015288), and by the Short-term Recruitment Program of Foreign Experts (WQ20163200398). The authors are also grateful to the anonymous reviewers for their constructive comments and suggestions to greatly improve the quality of this work and the clarity of the presentation.

Fig. 1 . 25 (Fig. 2 .Fig. 3 .

 12523 Fig. 2. The inverse of logarithm of BlockEnergy of Yale dataset in relation with h1, R, L1, L2, where h1∈{1:1:32}, R∈{0:0.1:0.9}, L1∈{1:1:9}, L2∈{1:1:9}.

Fig. 4 .Fig. 5 .Fig. 6 .

 456 Fig. 4. The illustration of trapezoidal area. (a) h1=(m+k1-1)/2; (b) h1=(m+k1-1)/2+1.

  defined in Eq. (1).

,

  X is defined in Eq. (3).

,

  Y is defined in Eq.[START_REF] He | Deep residual learning for image recognition[END_REF].

  II is defined in Eq.[START_REF] Liu | A survey of deep neural network architectures and their applications[END_REF].BinaryEnergyThe total energy of NL1L2 images after the binary process (Step 7) defined in Eq.[START_REF] Qayyum | Medical image retrieval using deep convolutional neural network[END_REF].WeightSumEnergyThe total energy of NL1 images after the weight and sum process (Step 8) defined in Eq.[START_REF] Patel | A probabilistic theory of deep learning[END_REF].The second part BlockEnergThe total energy of NL1 images after the block sliding process (Step 9). In relation with the parameters h1, h2, and R defined in Eq. (35).

7

  Similar toStep 1, we use a patch of size k 1 ×k 2 to slide each pixel of the ith image

		I	, I il	12 kk   , 1, 2,..., lL 1	, and
	obtain a matrix
		12
	,	, ,

Table 1 .

 1 The core parameters of PCANet.

	Parameters	Descriptions
	m× n	The size of input image
	k1 × k2	The patch size. k1 and k2 are odd integers and satisfy 1≤k1≤m, 1≤k2≤n. In this
		paper, we always set k1=k2=3.
	L1, L2	The number of filters of two stages. 1≤L1≤k1k2, 1≤L2≤k1k2
	h1 × h2	The block size.1≤h1≤m, 1≤h2≤n. Constraint: 21 / h nh m   

Table 3 .

 3 The accuracy, time consumption, and dimension of features of PCANet when R varies from 0 to 0.9 in the case of k1=k2=3, L1=L2=8, h1=h2=8 (YALE database)

	R	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
	Accuracy(%)	0.8074 0.7926 0.7926 0.7926 0.8074 0.8074	0.8148	0.8148	0.8148	0.8148
	Time consumption(s) 0.8563 0.8426 1.0479 1.1019 1.4804 1.7399	2.7093	5.9280	6.1109	22.2064
	Dimension of features 32768	32768	51200	51200	73728 100352 165888 401408 401408 1492992

Table 4 .

 4 The data fitting result and its criterions for five databases. Note that e=f(g), g=1/log10(E(Z)).

			Linear model Poly1			Nonlinear model Poly3
	Databases	Model obtained by	Error sum of	R-square Root mean	Model obtained by	Error sum of	R-square Root mean
		data fitting	squares		square error	data fitting	squares		square error
			(SSE)		(RMSE)		(SSE)		(RMSE)
	Yale	e=5.117g-0.008589	69.23	0.6104	0.06276	e=-518.1g 3 +159.1g 2	65.95	0.6288	0.06126
	database					-9.032g+0.367
	AR	e= 13.78g-0.5269	239.4	0.4302	0.1751	e=355.1g 3 +54.51g 2	236.9	0.4363	0.1742
	database					+1.935g-0.1016
	CMU PIE	e=0.08365g+0.06158	188.9	0.3943	0.1037	e=0.01084g 3 +0.029	137.6	0.5587	0.0885
	database					8g 2 +0.03435g+0.02
						453
	ORL	e= 0.08561g+ 0.06158 183.1	0.413	0.1021	e=0.00658g 3 +0.030	137.2	0.56	0.08837
	database					5g 2 +0.04161g+0.02
						586
	CIFAR-10	e= 11.77g+ 0.006023	65.1	0.7456	0.06262	e=728.1g 3 -340.7g 2 +	57.87	0.7739	0.05904
	database					40.56g-0.6522

Table 5 .

 5 The energy and the cumulative percentage of energy of BlockEnergy corresponding to R∈{0:0.1:0.9} . The parameters are L1 = L2 = 8, h1=h2=8.

	(YALE

Table 6 .

 6 The cumulative percentage of energy and the cumulative percentage of eigenvalue of 9 filters (the 1st and the 2nd PCAs). (YALE database)

	Parameters	The	Energy	Cumulative	Eigenvalue	Eigenval	Cumulative	Error
		ith	sum(×10 10 )	percentage of	(×10 4 )	ue sum	percentage	Rate
		filter		energy		(×10 4 )	of
							eigenvalue
	h1=h2=8	1	0.07984	0.6386	2.1544	2.1544	0.6820	0.5037
	R=0.5	2	0.09794	0.7834	0.3884	2.5428	0.8049	0.4519
	L2=1	3	0.11223	0.8977	0.3330	2.8757	0.9103	0.4370
	L1∈{1:1:9}	4	0.11785	0.9427	0.1151	2.9908	0.9468	0.4370
		5	0.12160	0.9727	0.0789	3.0697	0.9717	0.4148
		6	0.12339	0.9870	0.0466	3.1163	0.9865	0.4000
		7	0.12410	0.9927	0.0189	3.1352	0.9925	0.4000
		8	0.12470	0.9975	0.0152	3.1504	0.9973	0.3926
		9	0.12502	1.0000	0.0086	3.1590	1.000	0.4074
		1	0.6907	0.6632	2.4729	2.4729	0.6912	0.3926
		2	0.8211	0.7883	0.4369	2.9098	0.8134	0.3185
	h1=h2=8	3	0.9271	0.8902	0.3658	3.2755	0.9156	0.2593
	R=0.5	4	0.9772	0.9383	0.1232	3.3987	0.9500	0.2519
	L1=8	5	1.0129	0.9726	0.0871	3.4858	0.9744	0.2370
	L2∈{1:1:9}	6	1.0279	0.9869	0.0514	3.5372	0.9887	0.2074
		7	1.0339	0.9927	0.0188	3.5560	0.9940	0.2000
		8	1.0388	0.9974	0.0155	3.5715	0.9983	0.1926
		9	1.0415	1.0000	0.0060	3.5775	1.0000	0.2000

Table 7 .

 7 The error rate corresponds to L1, L2∈{1:1:9} when h1=h2=8, R=0.5. (YALE database)

	L1/Error	1	2	3	4	5	6	7	8	9
	rate/ L2
	1	0.5037	0.4370	0.3259	0.3185	0.2963	0.2667	0.2444	0.2444	0.2444
	2	0.4519	0.3926	0.2963	0.3111	0.2741	0.2370	0.2370	0.2000	0.2074
	3	0.4370	0.3630	0.2963	0.2889	0.2593	0.2519	0.2222	0.2148	0.2000
	4	0.4370	0.3481	0.2741	0.2741	0.2593	0.2444	0.2000	0.2074	0.2000
	5	0.4148	0.3259	0.2741	0.2815	0.2667	0.2148	0.2222	0.2148	0.2000
	6	0.4000	0.3333	0.2815	0.2593	0.2593	0.2222	0.2296	0.2000	0.2074
	7	0.4000	0.3259	0.2741	0.2370	0.2519	0.2148	0.2222	0.1852	0.1926
	8	0.3926	0.3185	0.2593	0.2519	0.2370	0.2074	0.2000	0.1926	0.2000
	9	0.4074	0.3259	0.2741	0.2444	0.2370	0.2148	0.2074	0.2000	0.2000

Acknowledgement

This work was supported by the National Key R&D Program of China (2017YFC0107900), by the National Natural Science Foundation of China (Nos. 61271312, 61201344, 61773117, 61401085, 31571001, 31640028, 31400842, 61572258, 11301074), by the Natural Science Foundation of Jiangsu Province under Grant BK20150650, by the Qing


Dr Mickaël Ménand
email: mickael.menand@upmc.fr

Matthieu Sollogoub

Dr Bernard Boitrel

Dr Stéphane Le Gac
email: stephane.legac@univ-rennes1.fr

Prof M Sollogoub

Cyclodextrin-Sandwiched Hexaphyrin Hybrids. Side-to-Side Cavity Coupling Switched by a Temperature and Redox Responsive Central Device

Access to allosteric enzyme mimics that ideally associate communicating compartments for catalysis and regulation is still challenging. Whereas a sandwich "cavity-catalyst-cavity" approach, developed mainly with cyclodextrins and porphyrins, appears promising, its counterpart with hexaphyrins featuring rich conformation, aromaticity and coordination behavior has not been prospected at all. We thus developed sandwich hybrids made of two cyclodextrins triply-linked on each side of a hexaphyrin. The latter displays switchable oxidation states with interconvertible conformations (28π antiaromatic and 26π aromatic each adopting rectangular and dumbbell forms). These four states are connected by two orthogonal switches under redox [aromaticity] and thermal [shape] control. It leads to twin compartmentalized confined spaces either locked or unlocked depending on the hexaphyrin conformation, the reversibility of the lock↔unlock transition relying on the hexaphyrin aromaticity. The sandwiched heteroannulene thus behaves as an unprecedented dual-responsive double-latches device.

Such hybrid systems open interesting perspectives in allosteric regulation of receptors, catalysts and machineries.

Introduction

The design of allosteric receptors and catalysts able to compete with the efficiency of allosteric enzymes remains a major challenge in supramolecular catalysis. [1] An ultimate design for a synthetic allosteric catalyst would combine distinct recognition sites displaying asymmetrical confined environments conformationally connected. Each cavity would ensure substrates and effectors selection towards distinct catalytic and regulation functions. In this respect, macromolecular edifices integrating a catalytic center surrounded by two hosting sub-units in a sandwiched manner offer several inherent advantages (Figure 1a, right): (i) they enable compartmentalization of catalytic and regulation functions, (ii) afford a protection of a catalytic metal center against irreversible deactivation (e.g., formation of µ-oxo complexes), and (iii) strengthen a side-to-side allosteric communication between the sandwiching cavities. Recently, these kinds of architectures featuring "compartmentalized-andcoupled" catalytic and regulation functions have been proposed as blueprint for molecular Turing machines, capable of processing information on a threaded polymer chain. [2] However, access to sandwiched architectures is synthetically challenging, which explains why most of the "cavity-catalyst- cavity" associations (Figure 1a, left) involve singly linked cavities serving as anchoring points for a substrate presenting a specific domain to the catalytic site. As a result, these cavities rather operate a passive conformational restriction deprived of active allosteric regulation. [3] Still, an effective sandwich approach involving cavities with distinct roles remains to be explored. Cyclodextrins are well known to meet the enzyme pockets requirements in hybrid architectures. [4] Their cyclic oligosaccharide skeleton provides naturally occurring chiral and water soluble structure with a confined hydrophobic environment. [5] Besides, porphyrins are key catalysts extensively developed and used to support enzyme modelling studies. [6] However, only a small number of sandwiched receptors/catalysts integrating either cyclodextrins and/or a central porphyrin have been reported and generally with modest yields. [START_REF] İşci | For cavitand-porphyrinoid hybrids[END_REF][START_REF] Tabushi | For cyclodextrin-sandwiched porphyrin[END_REF][START_REF] Bonar-Law | For steroid-sandwiched porphyrin[END_REF][START_REF] Rudkevich | For calixarene-sandwiched porphyrin[END_REF][START_REF] Nakazawa | For resorcinarene-sandwiched porphyrin[END_REF][START_REF] Thordarson | For glycoluril-sandwiched porphyrin[END_REF] Furthermore, only a couple of examples from Naruta and Nolte groups showed allosteric communication between the porphyrin-separated cavities, owing to induced-fit guest inclusion processes. [START_REF] Nakazawa | For resorcinarene-sandwiched porphyrin[END_REF][START_REF] Thordarson | For glycoluril-sandwiched porphyrin[END_REF] Although the porphyrin macrocycle displays some flexibility, we reasoned that such a sandwich design would be more modular with other porphyrinoids able to adopt major conformational changes, [13] thus allowing more effective coupling of the cavities. Hexaphyrins are good candidates since they can adopt different conformations 1d), [13d,f] with a possible control over their interconversion by external stimuli: (de)protonation, [14] solvent polarity, [15] temperature, [15c], [16] metal coordination, [17] and more importantly, redox control of their π-conjugated circuits (i.e. 26π vs. 28π electrons: R [26]H/ M [28]H Figure 1b, left). [15b] As such, hexaphyrins have a great, yet unexplored potential as switchable components in the design of molecular machineries. [18], [START_REF] Suzuki | For hexaphyrins undergoing a so-called "caterpillar motion[END_REF] In this context, we recently prepared a new type of hybrid edifices made of a hexaphyrin unit triply-linked on one side by a cyclodextrin (HCD, Figure 1b, right). [20] This unique association generates a confined dissymmetrical environment between the rectangular hexaphyrin and the primary rim of the cyclodextrin. Interestingly, a rectangular-to-triangular shape shifting process was observed with protonated forms, allowing to tune the size and shape of the confined space and thus the molecular recognition properties of HCD (Figure 1c). [21] In another recent example, a hexaphyrin triply-linked to a tris(2-aminoethyl)amine (tren) unit showed rectangular-to-Möbius conformational changes triggered by a cooperative two-guest binding process to a buried zinc metal ion. [22] In spite of their triple linkage, the dynamic behavior of these two capped hexaphyrins stresses the high propensity of this heteroannulene to undergo profound and rapid conformational changes affording new types of adaptative molecular receptors. We thus envisioned that hexaphyrins doubly-capped by cyclodextrin units in a sandwich manner (CDHCD, Figure 1d) may undergo similar conformational reorganization allowing to change the shape of the sandwiched edifice, and thus the coupling of the cavities. We therefore describe herein the first synthesis of doubly-capped hexaphyrins, affording compartmentalized-yet-coupled confined spaces matching a unique orthogonal control of the central hexaphyrin conformation and aromaticity.

(rectangular [R], dumbbell [D], triangular [T], Möbius [M], figure- eight [8], Figure

Results and Discussion

Synthesis

The targeted cyclodextrin-sandwiched hexaphyrins (CDHCDs) possess an (AB)3-type meso-substitution pattern. In the case of "unconstrained" hexaphyrins, this pattern is achieved by reacting an equimolar mixture of an aldehyde A and a dipyrromethane B. [23] The hexaphyrin frame is the major one among porphyrin and higher homologues, without scrambling (Scheme 1a, [3+3] macrocyclization). In the case of HCD hybrids, a related strategy consisting in the condensation of a cyclodextrin-tris-aldehyde platform (ACE pattern) with three equivalents of dipyrromethane (Scheme 1b top, [1+3] macrocyclization), afforded a hexaphyrin capped on one side by a cyclodextrin unit. [20] This reaction was also attempted the other way, i.e. by reacting a cyclodextrin-trisdipyrromethane platform with three equivalents of aldehyde (Scheme 1b, bottom), leading to the same HCD hybrids. [24] The present study investigates the ensuing strategy consisting in a Scheme 1. Macrocyclization strategies for (AB)3-type meso-substituted hexaphyrins, (a) without tripodal unit, (b) with a single tripodal cavity on one face, (c) with tripodal cavities on both faces. The notation [x+y] refers to the number of subunits A and B involved in the macrocyclization step.

[1+1] macrocyclization between two complementary cyclodextrin platforms (Scheme 1c). The corresponding doubly-capped hexaphyrins are thus obtained through the condensation of the cyclodextrin-tris-aldehyde 1 and its related cyclodextrin-trisdipyrromethane adduct 2 (Scheme 2). [25] The cyclodextrin-tris-aldehyde 1 was synthesized from the native α-cyclodextrin following a five steps procedure as previously described. [20] Condensation of 1 with a large excess of pyrrole, catalyzed by trifluoroacetic acid (TFA), afforded the cyclodextrintris-dipyrromethane adduct 2 in 66% yield. The [1+1] macrocyclization between 1 and 2 was achieved in dichloromethane at 0.01 M concentration using a catalytic amount of methanesulfonic acid ([MSA] = 0.002 M). The monitoring of the reaction by mass spectrometry indicated rapid completion (ca. 1h). Subsequent oxidation of the cyclodextrin-sandwiched hexaphyrinogens with 2,3-dichloro-5,6-dicyano-1,4benzoquinone (DDQ) followed by chromatography purification afforded the CD R [28]HCD hybrid (Scheme 2) which was however difficult to separate from oligomeric side products. To facilitate its purification, this 28π form was further oxidized in its 26π one by a second treatment with DDQ. Surprisingly, two fractions of different polarity were isolated by chromatography (0.7% and 2.7% overall yield). HRMS (ESI-TOF) of both fractions afforded the same mass ([M+2H] 2+ : m/z 2910.2570) corresponding to CD[26]HCD hybrids. As described below, their 1 H NMR spectra display different signatures corresponding to the rectangular (major) and the dumbbell (minor) conformers (CD R [26]HCD and CD D [26]HCD respectively, Scheme 2). Subsequent reduction of these fractions with NaBH(OAc)3 afforded their corresponding 28π reduced forms with analytical purity (81% yield for CD R [28]HCD, and 40% yield for CD D [28]HCD following a specific pathwayvide infra). It is worth to note that conformational transitions occur between the rectangular and dumbbell CDHCDs for both oxidation states (Section 3), lengthening the purification steps. Although CDHCDs are obtained with modest yields, access to sandwiched expanded porphyrins is unprecedented and competes with strategies used for doubly-capped porphyrin edifices. [8d], [START_REF] Bonar-Law | For steroid-sandwiched porphyrin[END_REF], [START_REF] Rudkevich | For calixarene-sandwiched porphyrin[END_REF]

Characterization

The UV-vis-NIR absorption spectra of these CDHCDs show strong similarities with those of the closely related HCDs (Figure 2b,d). Indeed, the rectangular form of both families in their ( 28 2d). [26] Compared to their rectangular isomers, the 26π and 28π dumbbell conformations exhibit broader UV-vis spectra of similar signatures. CD D [26]HCD displays a less intense Soret-like band at 580 nm and broad Qlike bands between 700 and 1000 nm, while CD D [28]HCD displays large and smeared absorption bands at λmax = 500 and 568 nm. The erosion of intensity and broadening of the Q-like bands might arise from a distortion of the π-conjugation pathway of the hexaphyrin core (see below). [27] Hence, CD D [26]HCD and CD D [28]HCD also exhibit aromatic and antiaromatic character, respectively, but of weaker intensity compared to their rectangular forms. The circular dichroism spectra of all CDHCD hybrids show Cotton effects involving the hexaphyrin Soret-like bands, indicating a clear chirality transfer from the surrounding cyclodextrins (Figure 2a,c). In the case of the rectangular conformations of singly-and doubly-capped hybrids, similar patterns are observed with bisignate signal splitting of the Soret-like bands broader for 28π antiaromatic vs 26π aromatic compounds. [START_REF]hexaphyrin and a close-lying chromophore obliquely oriented[END_REF] Besides, doublycapped rectangular hexaphyrin chromophores experience an enhanced chiral environment compared to their singly-capped ones (for R [26]HCD vs. CD R [26]HCD: Δε585 ≃ 40 vs. 100 M -1 .cm - 1 , Figure 2a; and for R [28]HCD vs. CD R [28]HCD: Δε573 ≃ 30 vs. 40 M -1 .cm -1 , Figure 2c). In stark contrast, the dumbbell conformation of CDHCD hybrids display less intense Cotton effects that are in line with the weaker absorption intensity and/or a less efficient chirality transfer. Analysis of the 1 H NMR spectra (CDCl3) of the four isolated CDHCDs allowed unambiguous structural and aromaticity characterization (Figure 3). The four CDHCDs display C2 symmetrical 1 H NMR spectra, the C2 axes being located within the average plane of the hexaphyrins with different orientation according to the conformation. Whereas the rectangular species exhibit a transversal C2 axis across the two proximal inverted pyrroles (π2/π5 inward), the dumbbell ones exhibit a longitudinal C2 axis across the two distal pyrroles located on the short sides (π2/π5 outward, Scheme 2, simplified views in inset). The NMR spectra display two characteristic sets of signals, a first one accounting for the two C2-symmetrically related cyclodextrins (2 × 6 different glucose units [A to F], 2-6 ppm crowded region) and a second one spanning over ~25 ppm corresponding to half of the hexaphyrin (2 × 6 -pyrrolic protons and 2 × 1 or 2 NH protons). For the sake of conciseness, NMR description focuses on characteristic groups of protons discussed hereafter.

(i) rectangular species

The hexaphyrin protons of the doubly-capped hybrids experience the same field effects as their singly-capped homologues (Figure 3, b vs. a and e vs. f, red and blue boxes).The 1 H NMR spectrum of CD R [26]HCD (Figure 3b) displays two upfield shifted singlets (π-in: -3.35 and -3.21 ppm) and four downfield doublets (π-out: 8.42 to 9.34 ppm) corresponding to the four inward -pyrrolic protons (π2 and π5) and the eight outward -pyrrolic ones (π1, π3, π4, and π6) respectively. The reverse distribution is observed for CD R [28]HCD (Figure 3e), with the two singlets shifted at 20.43 and 20.01 ppm (π-in) and the four doublets ranging from 2.49 to 3.12 ppm (π-out). These chemical shifts witness pronounced diatropic and paratropic ring current for respectively the 26π and 28π conjugated states, shielding(/deshielding) the inside and In addition, the chemical shifts of the sandwiching cyclodextrins follow the same trend as those of HCDs. [20] Notably, the marked characteristic shifting of the protons belonging to the O-6 benzyl group of unit A ("CH2OBn-A", Figure 3, b vs. a and e vs. f, green boxes) indicates an offset of the cavities from the center of the hexaphyrin, the latter protruding over the two C2-symmetrically related A glucose units (Scheme 2, cartoon in inset). [20] This particular topology inherently associates an NNCC coordination box to a specific cavity. Indeed, the two faces of a given NNCC coordination box experience different environments: one axial coordination is hindered to some extent by the cyclodextrin's wall whereas the opposite axial coordination is surrounded by a confined space (see below). As a consequence of the C2symmetry of CDHCDs, the "left" and "right" NNCC coordination sites are thus coupled to divergent cavities ("bottom" and "top"), offering a unique opportunity to study potential side-to-side communication across the hexaphyrin through allosteric events.

(ii) dumbbell species

In the case of the dumbbell CDHCDs (Figure 3c,d This indicates that the introverted meso-aromatic groups stand almost orthogonally to the hexaphyrin mean plane. The inner orientation is expected to generate important steric hindrance, likely compensated by a distortion of the hexaphyrin, and this may explain the weaker diatropic and paratropic ring currents of the dumbbell species. [29] Thus, in accordance with the UV-vis spectra, a marked deviation from planarity likely reduces effective π- In contrast to the rectangular CDHCDs, no evidence was observed for an offset of the sandwiching cyclodextrins vs the dumbbell [26/28] hexaphyrins. The molecular modelling of the rectangular and dumbbell conformers (26π oxidation state) showed no significant variation with the geometric features deduced from UV-vis and NMR analysis (Figure 4). Connolly solvent-accessible surfaces were calculated, evidencing a major difference in hosting potential. For CD R [26]HCD, the hexaphyrin is sandwiched by two identical and dissymmetrical confined spaces, each delineated by the hexaphyrin plane, the three linkers, and the cyclodextrin primary rim (green color, Figure 4a). These confined spaces are roughly similar to that of R [26]HCD, large enough to include two molecules of MSA -. [20], [21] In contrast, the self-inclusion of two meso-aryl groups in the dumbbell severely restricts this solvent-accessible volume (red color, Figure 4b). Although the cyclodextrin cavities are preserved, their "direct" interaction with the hexaphyrin is no longer possible. Thus, the meso-aryl groups act as latches locking or unlocking (dumbbell or rectangular conformation respectively) the access to both the hexaphyrin core and the associated confined spaces. Were transitions between these conformations to be controlled, the central hexaphyrin would become a switchable device for tuning the CDHCD hostguest properties in a coupled manner.

Conformational switching

As stated in Section 1, the separation of the rectangular and dumbbell conformations of CDHCDs was made difficult because of conformational transition between the two forms. We thus investigated the dynamic behavior of these conformations through 1 H NMR spectroscopy in CDCl3. VT-NMR of CD R [26]HCD and CD D [26]HCD (230 to 330 K) showed little changes of their chemical shifts indicating minor deviation from the rectangular and dumbbell conformations over the explored range of temperature (SI). However, for each conformer, traces of the other one were produced at high temperature (330 K) during the analysis. Evolution of these separated samples (CD R [26]HCD or CD D [26]HCD) for a prolonged period of time at 330 K allowed to reach the same ratio of the two conformers, evidencing thermodynamic equilibrium (R/D = 70:30, Figure 5b and SI). In contrast, heating a pure sample of CD R [28]HCD at 330 K in CDCl3 for 15 hours did not produce any trace of CD D [28]HCD conformer, whereas heating CD D [28]HCD in the same conditions led progressively to a full conversion of the dumbbell form into the rectangular one (Figure 5a and SI). Accordingly, the two conformers are not in equilibrium but rather correspond to kinetic (dumbbell) and thermodynamic (rectangular) products. The kinetic profile of the latter transformation was then examined monitoring the time dependent conversion of CD D [28]HCD into CD R [28]HCD. Surprisingly, the transformation did not follow a first order rate law, the kinetic profile being "too linear" (Figure 5c and SI). Instead, the data better fitted with a putative model involving a catalytic step with a catalyst loading estimated at 0.5 mol% (SI). Further experiments should be performed to identify the nature of the expected catalyst and its mode of action. [START_REF]Water as catalyst was ruled out as additional experiments performed in CDCl3 saturated with D2O did not change the kinetic rate[END_REF] Table 1. Kinetic constants (k / h -1 ) and activation free energies (ΔG ‡ / kcal.mol -1 ) for CD D [26]HCD ⇄ CD R [26]HCD equilibrium.

Solvent

T (K) D→R [a] R→D [b] k1 ΔG [a] CDCl3 -6.8 ± 0.5 -0.019 ± 0.001 6.3 -0.5 THF-d8 -7.0 ± 0.6 -0.021 ± 0.002 7.0 0.0

ΔG ‡ k-1 ΔG ‡
[a] Calculated at 330 K.

Conversely, the kinetic profiles of the equilibration pathways concerning CD D [26]HCD and CD R [26]HCD (D⇄R starting from D or R) fitted well by a first-order rate law (Figure 5d and SI). The corresponding kinetic constants (k1 and k-1 respectively) measured in both CDCl3 and THF-d8 allowed to calculate the free energies of activation (Table 1, ΔG ‡ , SI). Indeed, the isomerization pathway between the rectangular and dumbbell forms involves a formal in/out inversion of two opposite pyrroles (red pyrroles in Figure 5b) assisted by NH hydrogen bonding. Concomitantly, their neighboring meso-aromatic groups facing each other undergo a reverse inversion. The energy barriers of these transformations (Table 1, ΔG ‡ 330 = 23.8 to 25.6 kcal.mol -1 ) are of the same amplitude as the one previously determined for the related singly-capped R [26]HCD (ΔG ‡ 360 = 23.3 kcal.mol -1 ). This latter undergoes a so-called isomorphic fluxionality converting the rectangular conformation into another degenerate rectangular one through the formal rotation of four pyrroles. [20] The energy barriers of the D⇄R equilibrium are sufficiently high to allow the isolation of each conformer at room temperature and keep them kinetically stable at low temperature (4°C).

We then examined the equilibrated proportion of the two conformers between 330 and 360 K in CDCl3 and THF-d8 (Figure 5e). [START_REF]THF-d8 leads to faster equilibration rates at 330 K[END_REF] Interestingly, in both solvents, the D⇄R equilibrium is pushed towards the dumbbell conformer with increasing temperature. The effect is more pronounced in THF-d8 with R/D ratios at 360 K up to 30:70 vs 50:50 in CDCl3. The equilibrium constants (K = [R]/[D]) derived from the measured ratios allowed to draw the van't Hoff plot (SI) and determine the energy parameters engaged in the rearrangement (Table The corresponding enthalpic gain is probably arising from a more efficient double hydrogen bonding network (hexaphyrin N-core) in the less distorted rectangular conformation, while the entropic cost might be ascribed to the higher capability of the rectangular conformer to encapsulate solvent molecules (locked vs. unlocked cavities, Figure 4). Remarkably, this thermal tuning of the hexaphyrin conformation transforms CDHCD architecture into the first device in which the dynamical control of the open and closed states is performed by an expanded porphyrin.

Once these dynamic processes analyzed, we devised a strategy to access the CD D [28]HCD isomer (kinetic product) otherwise difficult to obtain through direct synthesis. The strategy involves an equilibration of the 26π isomers at high temperature followed by a rapid quenching and a reduction of the dumbbellenriched mixture (SI). Hence, CD R [26]HCD was heated at 90°C for 15 min in THF pushing the ratio towards the dumbbell conformer. A subsequent rapid cooling from 90°C to -78°C allowed to freeze the equilibrium and kinetically trap the dumbbellenriched mixture (R/D ≈ 30:70 [NMR ratio, SI]). The latter was further reduced at 0°C with tetra-n-butylammonium borohydride (TBABH4) affording the corresponding mixture of 28π conformers with a ca. 1:1 dumbbell/rectangular ratio. Upon separation, CD D [28]HCD was isolated in 40% yield (analytical sample). The combination of the temperature-controlled equilibrium of 26π conformers with their 28π redox conversions not only gives an access in reasonable yield to kinetic species (CD D [28]HCD) but also delineate a global reaction network connecting all CDHCDs as shown below.

Global reaction network and one-way cycling

Figure 6 summarizes the aromaticity and conformational changes connecting the four CDHCDs. On the one hand, the 28π and 26π states are redox interconvertible regardless of their conformation. This allows independent antiaromatic-aromatic transitions (left/right). On the other hand, transitions between the locked and unlocked states (D/R conformations) depend on the aromaticity of the hexaphyrin (top/bottom). Both states of the 26π aromatic hybrids are in equilibrium and temperature controlled, whereas those of the 28π antiaromatic hybrids are irreversibly connected (kinetic locked state and thermodynamic unlocked state). Thus, aromaticity and conformational transitions can be orthogonally addressed (redox vs. temperature control) offering an interesting combination for the development of multiaddressable allosteric receptors. Obviously, a mutual influence of such a reaction network with hosting and coordination properties of the CDHCD hybrids is expected. In addition, these four hybrids exhibit markedly different chiroptical signatures in circular dichroism (Figure 6), offering an additional opportunity for orthogonal chiroptical switches. [18b] To exemplify these orthogonal transitions, we attempted to realize a one-way cycle over the global reaction network (Figure 7 and SI). The following experiment was monitored by 1 H NMR spectroscopy in a sealed tube: first, CD R [28]HCD was fully converted to CD R [26]HCD by treatment with DDQ (Figure 7a,b); then, equilibration following the heating/freezing procedure described above, afforded CD D [26]HCD as the major species (R/D = 40:60, Figure 7c); subsequent reduction with TBABH4 gave CD D [28]HCD without modification of the ratio (R/D = 40:60, Figure 7d); finally, heating the sample converted the last mixture into the initial CD R [28]HCD as a single product (Figure 7e). This process corresponds to the following sequential transitions: antiaromatic/unlocked -> aromatic/unlocked -> aromatic/locked -> antiaromatic/locked -> antiaromatic/unlocked.

Conclusions

In conclusion, the grafting of a cyclodextrin unit on each side of a hexaphyrin, leading to 2:1 sandwich edifices, has been successfully realized. Such doubly-capped hybrid structures are unprecedented with expanded porphyrins. On the one hand, the CDHCDs share similar properties with their parent 1:1 HCD hybrids: (i) the hexaphyrin preferably adopts a rectangular shape for both the 26/28π-conjugated states. Thus, aromatic and antiaromatic compounds, featuring markedly different chiroptical properties, are accessible and interconvertible; (ii) the cyclodextrin units are offset from the hexaphyrin center, in opposite directions, and thus specifically coupled to adjacent NNCC coordination sites. This particular arrangement affords two apical confined spaces, both compartmentalized and connected by the central hexaphyrin unit. On the other hand, CDHCDs exhibit unique conformational features characterized by the dumbbell conformation observed for both 26/28π-conjugated states. A second set of aromatic and antiaromatic compounds is thus accessible with, again, different chiroptical properties. In addition, the dumbbell conformations severely restrict the confined spaces due to the introversion of two meso-aromatic groups. Four different CDHCDs are accessible and connected through a reaction network orthogonally addressed by temperature changes (dumbbell-to-rectangular transitions triggering locked-unlocked states) and/or redox reactions (26π-to-28π-conjugated transitions controlling aromaticity-antiaromaticity and thus the reversibility of the above conformational transitions). Remarkably, a one-way cycle through the reactions network allowed the sequential transition over the four CDHCDs. These architectures are thus switchable devices controlled by a thermal and redox responsive central hexaphyrin unit. The associated conformational reconfigurations of the whole structure connect the cavities in different manners. Hosting and coordination properties of these various hybrids are currently under investigation with a particular interest towards their mutual influence to target multi-addressable allosteric receptors and catalysts.

Experimental Section

Full experimental details and spectral data are given in the Supporting Information.

Figure 1 .

 1 Figure 1. (a) Illustration of linear and sandwiched cavity-catalyst-cavity architectures. (b) Structures of the rectangular R [26]H, Möbius twisted M [28]H and hexaphyrin-cyclodextrin hybrids [26]HCD and [28]HCD (NH into brackets). (c) Rectangular-to-triangular topological reconfiguration of perprotonated [26]HCD‧ 4H + and [28]HCD‧ 2H + . (d) Simplified representation of the targeted cyclodextrin-sandwiched hexaphyrin hybrids (CDHCD) with the various conformations offered by hexaphyrins(1.1.1.1.1.1) (dashed inset). [26/28] indicate the number of electrons of the π-conjugated system.

  )26π states exhibit characteristic Hückel (anti)aromaticity. CD R [26]HCD shows a strong Soret-like band (λmax = 568 nm) with four Q-like bands reaching the NIR region (λmax = 723, 784, 919 and 1050 nm). This characteristic pattern denoting an aromatic character is almost superimposable to those of R [26]HCD, R [26]H and the Möbius aromatic M [28]H (Figure 2b,d). Conversely, CD R [28]HCD displays a UV-vis spectrum typical of antiaromatic hexaphyrins in line with its related R [28]HCD (i.e. broad and smeared absorption bands at λmax = 488 and 583 nm, deprived of Q-like bands; Figure

Figure 2 .

 2 Figure 2. (a) Circular dichroism and (b) UV-vis-NIR spectra of R [26]H, R [26]HCD, CD R [26]HCD and CD D [26]HCD in CH2Cl2. (*) Contamination with traces of CD R [26]HCD. (c) Circular dichroism and (d) UV-vis-NIR spectra of M [28]H, R [28]HCD, CD R [28]HCD and CD D [28]HCD in CH2Cl2. Blue spectra denote aromatic character and red spectra denote antiaromatic character.

  ), the twelve βpyrrolic protons (two singlets and four doublets) are located at the periphery of the hexaphyrin and experience deshielding(/shielding) due to diatropic(/paratropic) ring currents (π-out: 7.36 to 8.57 ppm for CD D [26]HCD and 4.70 to 5.71 ppm for CD D [28]HCD). The dumbbell conformation is also supported by the characteristic shifts of the two introverted meso-aromatic substituents located in the pinched region of the macrocycle (Figure 3c,d, brown boxes). These aromatic protons are unevenly shielded(/deshielded) with a strong effect towards the protons in ortho position to the hexaphyrin link [δ(Ho H ) = 1.71 ppm for CD D [26]HCD and 13.42 ppm for CD D [28]HCD].

Figure 3 .

 3 Figure 3. 1 H NMR spectra (600MHz, CDCl3, 300K) of (b) CD R [26]HCD, (c) CD D [26]HCD, (d) CD D [28]HCD and (e) CD R [28]HCD. For the sake of comparison, selected signals of the four CDHCD species and their HCD analogues (a and f) are depicted as ticks in coloured boxes.

Figure 4 .

 4 Figure 4. Side view and apical view from the cyclodextrin secondary rim of the optimized geometry of (a) CD R [26]HCD and (b) CD D [26]HCD (Avogadro software, UFF parameters; benzyl groups were omitted for the modeling). The calculated Connolly solvent-accessible surface areas are depicted in green and red. The latches cartoons at the center illustrate the locking/unlocking principle.

Figure 6 .

 6 Figure 6. Global network showing the pathways between the four states of CDHCD and the triggering action. Individual circular dichroism signatures are depicted in the middle in association with the locking and aromatic states.Figure 5. Simplified illustrations of (a) the D→R conversion of CD[28]HCD and (b) the D⇄R equilibrium of CD[26]HCD. Rotation of the pyrroles for both transformations are depicted in the insets. Kinetic plots following the time dependant concentration variations of D () and R () forms in CDCl3 at 330 K for (c) CD[28]HCD and (d) CD[26]HCD (fitting curve in red). (e) VT-1 H NMR stacks (600 MHz, sealed tube) of R and D equilibrated mixture of CD[26]HCD from 330 to 360 K in CDCl3 (red box) and THF-d8 (blue box). The regions selected show π-1/3 Ar-in (D: ) and π-4/6 S (R: ) evolution (labelling in Scheme 2).

Figure 5 .

 5 Figure 6. Global network showing the pathways between the four states of CDHCD and the triggering action. Individual circular dichroism signatures are depicted in the middle in association with the locking and aromatic states.Figure 5. Simplified illustrations of (a) the D→R conversion of CD[28]HCD and (b) the D⇄R equilibrium of CD[26]HCD. Rotation of the pyrroles for both transformations are depicted in the insets. Kinetic plots following the time dependant concentration variations of D () and R () forms in CDCl3 at 330 K for (c) CD[28]HCD and (d) CD[26]HCD (fitting curve in red). (e) VT-1 H NMR stacks (600 MHz, sealed tube) of R and D equilibrated mixture of CD[26]HCD from 330 to 360 K in CDCl3 (red box) and THF-d8 (blue box). The regions selected show π-1/3 Ar-in (D: ) and π-4/6 S (R: ) evolution (labelling in Scheme 2).

Figure 7 .

 7 Figure 7. One screw cycle over the four CDHCDs (bottom), monitored by 1 H NMR spectroscopy (top; THF-d8, 276 K). Conditions: see the SI. (*) residual solvents and impurities.

Table 2 .

 2 Thermodynamic values (kcal.mol -1 ) extracted from the van't Hoff plots of CD

	CDCl3	330	0.87	24.9	0.27	25.6
	THF-d8	310	0.43	23.8	0.17	24.3
	[a] Dumbbell to rectangular pathway; [b] Rectangular to dumbbell
	pathway.
	Solvent	ΔH	ΔS		-TΔS [a]

D [26]HCD ⇄ CD R [26]HCD equilibrium.

Scheme 2. Synthesis of CD[26]HCD and CD[28]HCD through a [1+1] condensation of 1 and 2. Representations of the equilibrated rectangular and dumbbell conformations are depicted in detailed and simplified views (perspective and top views in grey insets; C2 symmetrically related parts colored in red and blue). Labelling of the two conformers is differing according to the location of the C2 symmetry axis (see labels on the right).

Acknowledgements

We are grateful to the Agence Nationale de la Recherche for financial support (ANR research program PRALLOCAT, ANR-16-CE07-0014).

Keywords: Aromaticity • conformational analysis • cyclodextrin • hexaphyrin • molecular device

Entry for the Table of Contents

FULL PAPER

Royale with cheese. New sandwich compounds made of a hexaphyrin doubly-capped by cyclodextrins have been developed. The central hexaphyrin behaves as a dualresponsive device, coupling the cavities with different operating modes in switchable manners, depending on both its (anti)aromaticity and conformation. The accessibility of two compartmentalized-and-coupled confined spaces is locked/unlocked in a reversible way or not, opening interesting perspectives in allosteric regulation.

Mickaël


Antoine Legouhy

Olivier Commowick

François Rousseau

Christian Barillot

Anisotropic similarity, a constrained affine transformation: Application to brain development analysis

published or not. The documents may come

Introduction

Approach: introduce an affine transformation with constrained scaling directions (9 degrees of freedom).

Objective: quantify regional brain growth in 3 orthogonal directions by linearly registering subjects on a common basis.

Block-matching registration [1][2]:

1. Matching: find homologous blocks in both images best satisfying a similarity criterion 2. Aggregation into a global transformation by minimizing the distance between the sets of blocks

Results

Chosen scaling directions fixed on the reference image: 1. Orthogonal to the mid-sagittal plane (blue) [4] 2. and 3. Principal directions of voxel coordinates projected on mid-sagittal plane (red and green)

We registered whole brain and lobes from 308 subjects onto For an anisotropic similarity:

an atlas based on this population using anisotropic similarity with chosen U.

Then we extracted the associated scaling factors describing brain growth.

[b] C-mind database: cmind.research.cchmc.org

[a] Developing human connectome project (dHCP) database: http://www.developingconnectome.org

[3] [b] [a]
We developed a method to quantify regional brain growth in 3 orthogonal directions through anisotropic similarity registration. An interesting continuation is to compare the results among groups of population.

Scaling factors along chosen directions

Black curve represents average model Fitted using first order rational regression: