John Zarka
email: john.zarka@ens.fr

Louis Thiry
email: louis.thiry@ens.fr

Tomás Angles
email: tomas.angles@ens.fr

Stéphane Mallat

DEEP NETWORK CLASSIFICATION BY SCATTERING AND HOMOTOPY DICTIONARY LEARNING

We introduce a sparse scattering deep convolutional neural network, which provides a simple model to analyze properties of deep representation learning for classification. Learning a single dictionary matrix with a classifier yields a higher classification accuracy than AlexNet over the ImageNet 2012 dataset. The network first applies a scattering transform that linearizes variabilities due to geometric transformations such as translations and small deformations. A sparse 1 dictionary coding reduces intra-class variability while preserving class separation through projections over unions of linear spaces. It is implemented in a deep convolutional network with a homotopy algorithm having an exponential convergence. A convergence proof is given in a general framework that includes ALISTA. Classification results are analyzed on ImageNet.

INTRODUCTION

Deep convolutional networks have spectacular applications to classification and regression [START_REF] Lecun | Deep learning[END_REF], but they are black boxes that are hard to analyze mathematically because of their architecture complexity. Scattering transforms are simplified convolutional neural networks with wavelet filters which are not learned [START_REF] Bruna | Invariant scattering convolution networks[END_REF]. They provide state-of-the-art classification results among predefined or unsupervised representations, and are nearly as efficient as learned deep networks on relatively simple image datasets, such as digits in MNIST, textures [START_REF] Bruna | Invariant scattering convolution networks[END_REF] or small CIFAR images [START_REF] Oyallon | Deep roto-translation scattering for object classification[END_REF]Mallat, 2016). However, over complex datasets such as ImageNet, the classification accuracy of a learned deep convolutional network is much higher than a scattering transform or any other predefined representation [START_REF] Oyallon | Scattering networks for hybrid representation learning[END_REF]. A fundamental issue is to understand the source of this improvement. This paper addresses this question by showing that one can reduce the learning to a single dictionary matrix, which is used to compute a positive sparse 1 code.

Figure 1: A sparse scattering network is composed of a scattering transform S followed by an optional linear operator L that reduces its dimensionality. A sparse code approximation of scattering coefficients is computed in a dictionary D. The dictionary D and the classifier are jointly learned by minimizing the classification loss with stochastic gradient descent.

The resulting algorithm is implemented with a simplified convolutional neural network architecture illustrated in Figure 1. The classifier input is a positive 1 sparse code of scattering coefficients calculated in a dictionary D. The matrix D is learned together with the classifier by minimizing a classification loss over a training set. We show that learning D improves the performance of a scattering representation considerably and is sufficient to reach a higher accuracy than AlexNet (Krizhevsky et al., 2012) over ImageNet 2012. This cascade of well understood mathematical operators provides a simplified mathematical model to analyze optimization and classification performances of deep neural networks.

Dictionary learning for classification was introduced in [START_REF] Mairal | Supervised dictionary learning[END_REF] and implemented with deep convolutional neural network architectures by several authors [START_REF] Sulam | Multilayer convolutional sparse modeling: Pursuit and dictionary learning[END_REF][START_REF] Mahdizadehaghdam | Deep dictionary learning: A parametric network approach[END_REF][START_REF] Sun | Supervised deep sparse coding networks[END_REF]. To reach good classification accuracies, these networks cascade several dictionary learning blocks. As a result, there is no indication that these operators compute optimal sparse 1 codes. These architectures are thus difficult to analyze mathematically and involve heavy calculations. They have only been applied to small image classification problems such as MNIST or CIFAR, as opposed to ImageNet. Our architecture reaches a high classification performance on ImageNet with only one dictionary D, because it is applied to scattering coefficients as opposed to raw images. Intra-class variabilities due to geometric image transformations such as translations or small deformations are linearized by a scattering transform [START_REF] Bruna | Invariant scattering convolution networks[END_REF], which avoids unnecessary learning.

Learning a dictionary in a deep neural network requires to implement a sparse 1 code. We show that homotopy iterative thresholding algorithms lead to more efficient sparse coding implementations with fewer layers. We prove their exponential convergence in a general framework that includes the ALISTA [START_REF] Liu | ALISTA: Analytic weights are as good as learned weights in LISTA[END_REF] algorithm. The main contributions of the paper are summarized below:

• A sparse scattering network architecture, illustrated in Figure 1, where the classification is performed over a sparse code computed with a single learned dictionary of scattering coefficients. It outperforms AlexNet over ImageNet 2012.

• A new dictionary learning algorithm with homotopy sparse coding, optimized by gradient descent in a deep convolutional network. If the dictionary is sufficiently incoherent, the homotopy sparse coding error is proved to convergence exponentially.

We explain the implementation and mathematical properties of each element of the sparse scattering network. Section 2 briefly reviews multiscale scattering transforms. Section 3 introduces homotopy dictionary learning for classification, with a proof of exponential convergence under appropriate assumptions. Section 4 analyzes image classification results of sparse scattering networks on Ima-geNet 2012.

SCATTERING TRANSFORM

A scattering transform is a cascade of wavelet transforms and ReLU or modulus non-linearities. It can be interpreted as a deep convolutional network with predefined wavelet filters (Mallat, 2016). For images, wavelet filters are calculated from a mother complex wavelet ψ whose average is zero.

It is rotated by r -θ , dilated by 2 j and its phase is shifted by α:

ψ j,θ (u) = 2 -2j ψ(2 -j r -θ u) and ψ j,θ,α = Real(e -iα ψ j,θ )
We choose a Morlet wavelet as in [START_REF] Bruna | Invariant scattering convolution networks[END_REF] to produce a sparse set of non-negligible wavelet coefficients. A ReLU is written ρ(a) = max(a, 0).

Scattering coefficients of order m = 1 are computed by averaging rectified wavelet coefficients with a subsampling stride of 2 J :

Sx(u, k, α) = ρ(x ψ j,θ,α ) φ J (2 J u) with k = (j, θ)
where φ J is a Gaussian dilated by 2 J [START_REF] Bruna | Invariant scattering convolution networks[END_REF]. The averaging by φ J eliminates the variations of ρ(x ψ j,θ,α ) at scales smaller than 2 J . This information is recovered by computing their variations at all scales 2 j < 2 J , with a second wavelet transform. Scattering coefficients of order two are:

Sx(u, k, k , α, α ) = ρ(ρ(x ψ j,θ,α ) ψ j ,θ ,α ) φ J (2 J u) with k, k = (j, θ), (j , θ )
To reduce the dimension of scattering vectors, we define phase invariant second order scattering coefficients with a complex modulus instead of a phase sensitive ReLU:

Sx(u, k, k ) = ||x ψ j,θ | ψ j ,θ | φ J (2 J u) for j > j
The scattering representation includes order 1 coefficients and order 2 phase invariant coefficients.

In this paper, we choose J = 4 and hence 4 scales 1 ≤ j ≤ J, 8 angles θ and 4 phases α on [0, 2π]. Scattering coefficients are computed with the software package Kymatio [START_REF] Andreux | Kymatio: Scattering transforms in python[END_REF]. They preserve the image information, and x can be recovered from Sx [START_REF] Oyallon | Scattering networks for hybrid representation learning[END_REF]. For computational efficiency, the dimension of scattering vectors can be reduced by a factor 6 with a linear operator L that preserves the ability to recover a close approximation of x from LSx.

The dimension reduction operator L of Figure 1 may be an orthogonal projection over the principal directions of a PCA calculated on the training set, or it can be optimized by gradient descent together with the other network parameters.

The scattering transform is Lipschitz continuous to translations and deformations [START_REF] Mallat | Group invariant scattering[END_REF].

Intra-class variabilities due to translations smaller than 2 J and small deformations are linearized. Good classification accuracies are obtained with a linear classifier over scattering coefficients in image datasets where translations and deformations dominate intra-class variabilities. This is the case for digits in MNIST or texture images [START_REF] Bruna | Invariant scattering convolution networks[END_REF]. However, it does not take into account variabilities of pattern structures and clutter which dominate complex image datasets. To remove this clutter while preserving class separation requires some form of supervised learning. The sparse scattering network of Figure 1 computes a sparse code of scattering representation β = LSx in a learned dictionary D of scattering features, which minimizes the classification loss. For this purpose, the next section introduces a homotopy dictionary learning algorithm, implemented in a small convolutional network.

HOMOTOPY DICTIONARY LEARNING FOR CLASSIFICATION

Task-driven dictionary learning for classification with sparse coding was proposed in [START_REF] Mairal | Task-driven dictionary learning[END_REF]. We introduce a small convolutional network architecture to implement a sparse 1 code and learn the dictionary with a homotopy continuation on thresholds. The next section reviews dictionary learning for classification. Homotopy sparse coding algorithms are studied in Section 3.2.

SPARSE CODING AND DICTIONARY LEARNING

Unless specified, all norms are Euclidean norms. A sparse code approximates a vector β with a linear combination of a minimum number of columns D m of a dictionary matrix D, which are normalized D m = 1. It is a vector α 0 of minimum support with a bounded approximation error Dα 0 -β ≤ σ. Such sparse codes have been used to optimize signal compression [START_REF] Mallat | Matching pursuits with time-frequency dictionaries[END_REF] and to remove noise, to solve inverse problems in compressed sensing [START_REF] Candes | Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information[END_REF], and for classification [START_REF] Mairal | Task-driven dictionary learning[END_REF]. In this case, the dictionary learning optimizes the matrix D in order to minimize the classification loss. The resulting columns D m can be interpreted as classification features selected by the sparse code α 0 . To enforce this interpretation, we impose that sparse code coefficients are positive, α 0 ≥ 0.

Positive sparse coding Minimizing the support of a code α amounts to minimizing its 0 "norm", which is not convex. This non-convex optimization is convexified by replacing the 0 norm by an 1 norm. Since α ≥ 0, we have α 1 = m α(m). The minimization of α 1 with Dα -β ≤ σ is solved by minimizing a convex Lagrangian with a multiplier λ * which depends on σ:

α 1 = argmin α≥0 1 2 Dα -β 2 + λ * α 1 (1)
One can prove [START_REF] Donoho | On the stability of the basis pursuit in the presence of noise[END_REF]) that α 1 (m) has the same support as the minimum support sparse code α 0 (m) along m if the support size s and the dictionary coherence satisfy:

s µ(D) < 1/2 where µ(D) = max m =m |D t m D m | (2)
The sparse approximation Dα 1 is a non-linear filtering which preserves the components of β which are "coherent" in the dictionary D, represented by few large amplitude coefficients. It eliminates the "noise" corresponding to incoherent components of β whose correlations with all dictionary vectors D m are typically below λ * , which can be interpreted as a threshold.

Supervised dictionary learning with a deep neural network Dictionary learning for classification amounts to optimizing the matrix D and the threshold λ * to minimize the classification loss on a training set {(x i , y i )} i . This is a much more difficult non-convex optimization problem than the convex sparse coding problem (1). The sparse code α 1 of each scattering representation β = LSx depends upon D and λ * . It is used as an input to a classifier parametrized by Θ. The classification loss i Loss(D, λ * , Θ, x i , y i ) thus depends upon the dictionary D and λ * (through α 1 ), and on the classification parameters Θ. The dictionary D is learned by minimizing the classification loss. This task-driven dictionary learning strategy was introduced in Mairal et al. ( 2011).

An implementation of the task-driven dictionary learning strategy with deep neural networks has been proposed in [START_REF] Papyan | Convolutional neural networks analyzed via convolutional sparse coding[END_REF][START_REF] Sulam | Multilayer convolutional sparse modeling: Pursuit and dictionary learning[END_REF][START_REF] Mahdizadehaghdam | Deep dictionary learning: A parametric network approach[END_REF][START_REF] Sun | Supervised deep sparse coding networks[END_REF]. The deep network is designed to approximate the sparse code by unrolling a fixed number N of iterations of an iterative soft thresholding algorithm. The network takes β as input and is parametrized by the dictionary D and the Lagrange multiplier λ * , as shown in Figure 2. The classification loss is then minimized with stochastic gradient descent on the classifier parameters and on D and λ * . The number of layers in the network is equal to the number N of iterations used to approximate the sparse code. During training, the forward pass approximates the sparse code with respect to the current dictionary, and the backward pass updates the dictionary through a stochastic gradient descent step.

For computational efficiency the main issue is to approximate α 1 with as few layers as possible and hence find an iterative algorithm which converges quickly. Next section shows that this can be done with homotopy algorithms, that can have an exponential convergence.

HOMOTOPY ITERATED SOFT THRESHOLDING ALGORITHMS

Sparse 1 codes are efficiently computed with iterative proximal gradient algorithms [START_REF] Combettes | Proximal splitting methods in signal processing[END_REF]. For a positive sparse code, these algorithms iteratively apply a linear operator and a rectifier which acts as a positive thresholding. They can thus be implemented in a deep neural network. We show that homotopy algorithms can converge exponentially and thus lead to precise calculations with fewer layers.

Iterated Positive Soft Thresholding with ReLU Proximal gradient algorithms compute sparse 1 codes with a gradient step on the regression term x -Dz 2 followed by proximal projection which enforces the sparse penalization [START_REF] Combettes | Proximal splitting methods in signal processing[END_REF]. For a positive sparse code, the proximal projection is defined by:

prox λ (β) = argmin α≥0 1 2 α -β 2 + λ α 1 (3) Since α 1 = m α(m) for α(m) ≥ 0, we verify that prox λ (β) = ρ(β -λ)
where ρ(a) = max(a, 0) is a rectifier, with a bias λ. The rectifier acts as a positive soft thresholding, where λ is the threshold. Without the positivity condition α ≥ 0, the proximal operator in (3) is a soft thresholding which preserves the sign.

An Iterated Soft Thresholding Algorithm (ISTA) [START_REF] Daubechies | An iterative thresholding algorithm for linear inverse problems with a sparsity constraint[END_REF] computes an 1 sparse code α 1 by alternating a gradient step on Dx -z 2 and a proximal projection. For positive codes, it is initialized with α 0 = 0, and:

α n+1 = ρ(α n + D t (β -Dα n ) -λ * ) with < 1 D t D 2,2 (4)
where . 2,2 is the spectral norm. The first iteration computes a non-sparse code α 1 = ρ( D t β -λ * ) which is progressively sparsified by iterated thresholdings. The convergence is slow:

α n -α 1 = O(n -1
). Fast Iterated Soft Thresholding Agorithm (FISTA) [START_REF] Beck | A fast iterative shrinkage-thresholding algorithm for linear inverse problems[END_REF] accelerates the error decay to O(n -2 ), but it remains slow. Each iteration of ISTA and FISTA is computed with linear operators and a thresholding and can be implemented with one layer [START_REF] Papyan | Convolutional neural networks analyzed via convolutional sparse coding[END_REF]. The slow convergence of these algorithms requires to use a large number N of layers to compute an accurate sparse 1 code. We show that the number of layers can be reduced considerably with homotopy algorithms.

Homotopy continuation Homotopy continuation algorithms introduced in [START_REF] Osborne | A new approach to variable selection in least squares problems[END_REF], minimize the 1 Lagrangian (1) by progressively decreasing the Lagrange multiplier. This optimization path is opposite to ISTA and FISTA since it begins with a very sparse initial solution whose sparsity is progressively reduced, similarly matching pursuit algorithms [START_REF] Davis | Adaptive greedy approximations[END_REF][START_REF] Donoho | Fast solution of l 1 -norm minimization problems when the solution may be sparse[END_REF]. Homotopy algorithms are particularly efficient if the final Lagrange multiplier λ * is large and thus produces a very sparse optimal solution. We shall see that it is the case for classification.

Homotopy proximal gradient descents [START_REF] Xiao | A proximal-gradient homotopy method for the sparse least-square problem[END_REF] are implemented with an exponentially decreasing sequence of Lagrange multipliers λ n for n ≤ N . Jiao, Jin and Lu [START_REF] Jiao | Iterative soft/hard thresholding with homotopy continuation for sparse recovery[END_REF] have introduced an Iterative Soft Thresholding Continuation (ISTC) algorithm with a fixed number of iterations per threshold. To compute a positive sparse code, we replace the soft thresholding by a ReLU proximal projector, with one iteration per threshold, over n ≤ N iterations:

α n = ρ(α n-1 + D t (β -Dα n-1 ) -λ n ) with λ n = λ max λ max λ * -n/N
(5)

By adapting the proof of [START_REF] Jiao | Iterative soft/hard thresholding with homotopy continuation for sparse recovery[END_REF] to positive codes, the next theorem proves in a more general framework that if N is sufficiently large and λ max ≥ D t β ∞ then α n converges exponentially to the optimal positive sparse code.

LISTA algorithm [START_REF] Gregor | Learning fast approximations of sparse coding[END_REF] and its more recent version ALISTA [START_REF] Liu | ALISTA: Analytic weights are as good as learned weights in LISTA[END_REF] accelerate the convergence of proximal algorithms by introducing an auxiliary matrix W , which is adapted to the statistics of the input and to the properties of the dictionary. Such an auxiliary matrix may also improve classification accuracy. We study its influence by replacing D t by an arbitrary matrix W t in (5). Each column W m of W is normalized by |W t m D m | = 1. A generalized ISTC is defined for any dictionary D and any auxiliary W by:

α n = ρ(α n-1 + W t (β -Dα n-1 ) -λ n ) with λ n = λ max λ max λ * -n/N (6)
If W = D then we recover the original ISTC algorithm (5) [START_REF] Jiao | Iterative soft/hard thresholding with homotopy continuation for sparse recovery[END_REF]. Figure 2 The following theorem gives a sufficient condition on this mutual coherence and on the thresholds so that α n converges exponentially to the optimal sparse code. ALISTA [START_REF] Liu | ALISTA: Analytic weights are as good as learned weights in LISTA[END_REF]) is a particular case of generalized ISTC where W is optimized in order to minimize the mutual coherence µ. In Section 4.1 we shall optimize W jointly with D without any analytic mutual coherence minimization like in ALISTA.

Theorem 3.1 Let α 0 be the 0 sparse code of β with error β -Dα 0 ≤ σ. If its support s satisfies s µ < 1/2 (7)

then thresholding iterations (6) with

λ n = λ max γ -n ≥ λ * = W t (β -Dα 0 ) ∞ 1 -2γ µs (8)
define an α n , whose support is included in the support of α 0 if 1 < γ < (2 µs) -1 and λ max ≥ W t β ∞ . The error then decreases exponentially:

α n -α 0 ∞ ≤ 2 λ max γ -n (9)
The proof is in Appendix A of the supplementary material. It adapts the convergence proof of [START_REF] Jiao | Iterative soft/hard thresholding with homotopy continuation for sparse recovery[END_REF] to arbitrary auxiliary matrices W and positive sparse codes. If we set W to minimize the mutual coherence µ then this theorem extends the ALISTA exponential convergence result to the noisy case. It proves exponential convergence by specifying thresholds for a non-zero approximation error σ.

However, one should not get too impressed by this exponential convergence rate because the condition s µ < 1/2 only applies to very sparse codes in highly incoherent dictionaries. Given a dictionary D, it is usually not possible to find W which satisfies this hypothesis. However, this sufficient condition is based on a brutal upper bound calculation in the proof. It is not necessary to get an exponential convergence. Next section studies learned dictionaries for classification on ImageNet and shows that when W = D, the ISTC algorithm converges exponentially although sµ(D) > 1/2. When W is learned independently from D, with no mutual coherence condition, we shall see that the algorithm may not converge.

IMAGE CLASSIFICATION

The goal of this work is to construct a deep neural network model which is sufficiently simple to be analyzed mathematically, while reaching the accuracy of more complex deep convolutional networks on large classification problems. This is why we concentrate on ImageNet as opposed to MNIST or CIFAR. Next section shows that a single 1 sparse code in a learned dictionary improves considerably the classification performance of a scattering representation, and outperforms AlexNet on ImageNet 1 . We analyze the influence of different architecture components. Section 4.2 compares the convergence of homotopy iterated thresholdings with ISTA and FISTA.

IMAGE CLASSIFICATION ON IMAGENET

ImageNet 2012 [START_REF] Russakovsky | ImageNet Large Scale Visual Recognition Challenge[END_REF] is a challenging color image dataset of 1.2 million training images and 50,000 validation images, divided into 1000 classes. Prior to convolutional networks, SIFT representations combined with Fisher vector encoding reached a Top 5 classification accuracy of 74.3% with multiple model averaging [START_REF] Sánchez | High-dimensional signature compression for large-scale image classification[END_REF]. In their PyTorch implementation, the Top 5 accuracy of AlexNet and ResNet-152 is 79.1% and 94.1% respectively2 .

The scattering transform Sx at a scale 2 J = 16 of an ImageNet color image is a spatial array of 14 × 14 of 1539 channels. If we apply to Sx the same MLP classifier as in AlexNet, with 2 hidden layers of size 4096, ReLU and dropout rate of 0.3, the Top 5 accuracy is 65.3%. We shall use the same AlexNet type MLP classifier in all other experiments, or a linear classifier when specified. If we first apply to Sx a 3-layer SLE network of 1x1 convolutions with ReLU and then the same MLP then the accuracy is improved by 14% and it reaches AlexNet performance [START_REF] Oyallon | Scaling the scattering transform: Deep hybrid networks[END_REF]. However, there is no mathematical understanding of the operations performed by these three layers, and the origin of the improvements, which partly motivates this work.

The sparse scattering architecture is described in Figure 3. A 3 × 3 convolutional operator L is applied on a standardized scattering transform to reduce the number of scattering channels from 1539 to 256. It includes 3.5 10 6 learned parameters. The ISTC network illustrated in Figure 2 has N = 12 layers with ReLU and no batch normalization. A smaller network with N = 8 has nearly the same classification accuracy but the ISTC sparse coding does not converge as well, as explained in Section 4.2. Increasing N to 14 or 16 has little impact on accuracy and on the code precision. The sparse code is first calculated with a 1 × 1 convolutional dictionary D having 2048 vectors.

Dictionary columns D m have a spatial support of size 1 and thus do not overlap when translated. It preserves a small dictionary coherence so that the iterative thresholding algorithm converges exponentially. This ISTC network takes as input an array LSx of size 14 × 14 × 256 which has been normalized and outputs a code α 1 of size 14 × 14 × 2048 or a reconstruction Dα 1 of size 14 × 14 × 256. The total number of learned parameters in D is about 5 10 5 . The output α 1 or Dα 1 of the ISTC network is transformed by a batch normalization, and a 5 × 5 average pooling and then provided as input to the MLP classifier. The representation is computed with 4 10 6 parameters in L and D, which is above the 2.5 10 6 parameters of AlexNet. Our goal here is not to reduce the number of parameters but to structure the network into well defined mathematical operators. If we set W = D in the ISTC network, the supervised learning jointly optimizes L, the dictionary D with the Lagrange multiplier λ * and the MLP classifier parameters. It is done with a stochastic gradient descent during 160 epochs using an initial learning rate of 0.01 with a decay of 0.1 at epochs 60 and 120. With a sparse code in input of the MLP, it has a Top 5 accuracy of 81.0%, which outperforms AlexNet.

If we also jointly optimize W to minimize the classification loss, then the accuracy improves to 83.7%. However, next section shows that in this case, the ISTC network does not compute a sparse 1 code and is therefore not mathematically understood. In the following we thus impose that W = D.

The dimension reduction operator L has a marginal effect in terms of performance. If we eliminate it or if we replace it by an unsupervised PCA dimension reduction, the performance drops by less than 2%, whereas the accuracy drops by almost 16% if we eliminate the sparse coding. The number of learned parameters to compute α 1 then drops from 4 10 6 to 5 10 5 . The considerable improvement brought by the sparse code is further amplified if the MLP classifier is replaced by a much smaller linear classifier. A linear classifier on a scattering vector has a (Top 1, Top 5) accuracy of (26.1%, 44.7%). With a ISTC sparse code with W = D in a learned dictionary the accuracy jumps to (51.6%, 73.7%) and hence improves by nearly 30%.

The optimization learns a relatively large factor λ * which yields a large approximation error LSx -Dα 1 / LSx ≈ 0.5, and a very sparse code α 1 with about 4% non-zero coefficients.

The sparse approximation Dα 1 thus eliminates nearly half of the energy of LS(x) which can be interpreted as non-informative "clutter" removal. The sparse approximation Dα 1 of LSx has a small dimension 14 × 14 × 256 similar to AlexNet last convolutional layer output. If the MLP classifier is applied to Dα 1 as opposed to α 1 then the accuracy drops by less than 2% and it remains slightly above AlexNet. Replacing LSx by Dα 1 thus improves the accuracy by 14%. The sparse coding projection eliminates "noise", which seems to mostly correspond to intra-class variabilities while carrying little discriminative information between classes. Since Dα 1 is a sparse combination of dictionary columns D m , each D m can be interpreted as "discriminative features" in the space of scattering coefficients. They are optimized to preserve discriminative directions between classes.

CONVERGENCE OF HOMOTOPY ALGORITHMS

To guarantee that the network can be analyzed mathematically, we verify numerically that the homotopy ISTC algorithm computes an accurate approximation of the optimal 1 sparse code in (1), with a small number of iterations.

When W = D, Theorem 3.1 guarantees an exponential convergence by imposing a strong incoherence condition s µ(D) < 1/2. In our classification setting, sµ(D) ≈ 60 so the theorem hypothesis is clearly not satisfied. However, this incoherence condition is not necessary. It is derived from a relatively crude upper bound in the proof of Appendix A.1. Figure 4 left shows numerically that the ISTC algorithm for W = D minimizes the Lagrangian L(α) = 1 2 Dα -β 2 + λ * α 1 over α ≥ 0, with an exponential convergence which is faster than ISTA and FISTA. This is tested with a dictionary learned by minimizing the classification loss over ImageNet.

If we jointly optimize W and D to minimize the classification loss then the ImageNet classification accuracy improves from 81.0% to 83.7%. However, Figure 4 right shows that the generalized ISTC network outputs a sparse code which does not minimize the 1 Lagrangian at all. Indeed, the learned matrix W does not have a minimum joint coherence with the dictionary D, as in ALISTA [START_REF] Liu | ALISTA: Analytic weights are as good as learned weights in LISTA[END_REF]. The joint coherence then becomes very large with sμ ≈ 300, which prevents the convergence. Computing W by minimizing the joint coherence would require too many computations.

To further compare the convergence speed of ISTC for W = D versus ISTA and FISTA, we compute the relative mean square error MSE(x, y) = x -y 2 / x 2 between the optimal sparse code α 1 and the sparse code output of 12 iterations of each of these three algorithms. The MSE is 0.23 for FISTA and 0.45 for ISTA but only 0.02 for ISTC. In this case, after 12 iterations, ISTC reduces the error by a factor 10 compared to ISTA and FISTA.

CONCLUSION

This work shows that learning a single dictionary is sufficient to improve the performance of a predefined scattering representation beyond the accuracy of AlexNet on ImageNet. The resulting deep convolutional network is a scattering transform followed by a positive 1 sparse code, which are well defined mathematical operators. Dictionary vectors capture discriminative directions in the scattering space. The dictionary approximations act as a non-linear projector which removes non-informative intra-class variations.

The dictionary learning is implemented with an ISTC network with ReLUs. We prove exponential convergence in a general framework that includes ALISTA. A sparse scattering network reduces the convolutional network learning to a single dictionary learning problem. It opens the possibility to study the network properties by analyzing the resulting dictionary. It also offers a simpler mathematical framework to analyze optimization issues.

A APPENDIX

A.1 PROOF OF THEOREM 3.1 Let α 0 be the optimal 0 sparse code. We denote by S(α) the support of any α. We also write ρ λ (a) = ρ(a -λ). We are going to prove by induction on n that for any n ≥ 0 we have S(α n ) ⊂ S(α 0 ) and α n -α 0 ∞ ≤ 2λ n if λ n ≥ λ * . For n = 0, α 0 = 0 so S(α 0 ) = ∅ is indeed included in the support of α 0 and α 0 -α 0 ∞ = α 0 ∞ . To verify the induction hypothesis for λ 0 = λ max ≥ λ * , we shall prove that α 0 ∞ ≤ 2λ max .

Let

α 0 ∞ (1 -µs) ≤ W t β ∞ + W t w ∞ But given the inequalities W t β ∞ ≤ λ max W t w ∞ ≤ λ max (1 -2γ µs) (1 -γ µs) (1 -µs)
≤ 1 since γ ≥ 1 and (1 -µs) > 0 we get α 0 ∞ ≤ 2λ max = 2λ 0

Let us now suppose that the property is valid for n and let us prove it for n + 1. We denote by D A the restriction of D to vectors indexed by A. We begin by showing that S(α n+1 ) ⊂ S(α 0 ). For any m ∈ S(α n+1 ), since β = Dα 0 + w and W t m D m = 1 we have α n+1 (m) = ρ λn+1 (α n (m) + W t m (β -Dα n )) = ρ λn+1 (α 0 (m) + W t m (D S(α 0 )∪S(αn)-{m} (α 0 -α n ) S(α 0 )∪S(αn)-{m} + w)) For any m not in S(α 0 ), let us prove that α n+1 (m) = 0. The induction hypothesis assumes that S(α n ) ⊂ S(α 0 ) and α 0 -α n ∞ ≤ 2λ n with λ n ≥ λ * so: Because of the thresholding ρ λn+1 , it proves that α n+1 (m) = 0 and hence that S(α n+1 ) ⊂ S(α 0 ).

Let us now evaluate α 0 -α n+1 ∞ . For any (α 1 , α 2 , λ), a soft thresholding satisfies

|ρ λ (α 1 + α 2 ) -α 1 | ≤ λ + |α 2 |

Figure 2 :

 2 Figure 2: A generalized ISTC network computes a positive 1 sparse code in a dictionary D by using an auxiliary matrix W . Each layer applies Id -W t D together with a ReLU and a bias λ n to compute α n from α n-1 in (6). The original ISTC algorithm corresponds to W = D.

  illustrates a neural network implementation of this generalized ISTC algorithm over N layers, with side connections. Let us introduce the mutual coherence of W and D

Figure 3 :

 3 Figure 3: Two variants of the image classification architecture: one where the input for the classifier is the sparse code α, and the other where the reconstruction Dα is the input for the classifier.

Figure 4 :

 4 Figure 4: Value of L(α n ) = 1 2 Dα n -β 2 + λ * α n 1 versus the number of iterations n, for ISTC with W = D, ISTA and FISTA on the left, and for ISTC with W = D, ISTA and FISTA on the right.

  us write the error w = β -Dα 0 . For all m α0 (m)W t m D m = W t m β -W t m wm =m α 0 (m ) W t m D m .Since the support of α 0 is smaller than s, W t m D m = 1 and µ = maxm =m |W t m D m | |α 0 (m)| ≤ |W t m β| + |W t m w| + s µ α 0∞ so taking the max on m gives:

I

  = |α 0 (m) + W t m (D S(α 0 )∪S(αn)-{m} (α 0 -α n ) S(α 0 )∪S(αn)-{m} + w)| ≤ |W t m (D S(α 0 ) (α 0 -α n ) S(α 0 ) )| + |W t m w| since S(α n ) ⊂ S(α 0 ) and α 0 (m) = 0 by assumption. ≤ µs α 0 -α n ∞ + W t w ∞Since we assume that λ n+1 ≥ λ * , we haveW t w ∞ ≤ (1 -2γ µs)λ n+1and thusI ≤ µs α 0 -α n ∞ + W t w ∞ ≤ µs2λ n + λ n+1 (1 -2γ µs) ≤ λ n+1 since λ n = γλ n+1 .

Table 1 :

 1 Top 1 and Top 5 accuracy on ImageNet with a same MLP classifier applied to different representations: Fisher Vectors[START_REF] Perronnin | Fisher vectors meet neural networks: A hybrid classification architecture[END_REF], AlexNet(Krizhevsky et al., 2012), Scattering with SLE[START_REF] Oyallon | Scattering networks for hybrid representation learning[END_REF], Scattering alone, Scattering with ISTC for W = D which outputs α 1 , or which outputs Dα 1 , or which outputs α 1 with unconstrained W .

		Fisher AlexNet	Scat.	Scat. Scat.+ ISTC	Scat.+ ISTC	Scat.+ ISTC
	Vectors + SLE alone α Top1 55.6 56.5 57.0 42.0	59.2	56.9	62.8
	Top5	78.4	79.1	79.6	65.3	81.0	79.3	83.7

1 , W = D Dα 1 , W = D α 1 , W = D

Code to reproduce experiments is available at https://github.com/j-zarka/SparseScatNet

Accuracies from https://pytorch.org/docs/master/torchvision/models.html

ACKNOWLEDGMENTS

This work was supported by the ERC InvariantClass 320959, grants from Région Ile-de-France and the PRAIRIE 3IA Institute of the French ANR-19-P3IA-0001 program. We thank the Scientific Computing Core at the Flatiron Institute for the use of their computing resources. We would like to thank Eugene Belilovsky for helpful discussions and comments.


Lakhlifi

Youssef El Jabbar

Rachida El Ouatib

Lahcen Er-Rakho

Sophie Guillemet-Fritsch

Bernard Durand

Structural, morphological and optical properties of cobalt-substituted MgMoO 4 ceramics prepared by pyrolysis of citric acid precursors

Keywords:

Introduction

Nanomaterials have been widely studied in recent years. Their properties vary according to their chemical compositions, sizes, and specific surface areas. These materials of the future have the potential to save energy while protecting the environment; however, achieving these objectives requires excellent mastery of nanomaterial production processes, which must be low cost from an energy point of view [START_REF] Li | Photoluminescence and energy transfer properties of a novel molybdate KBaY(MoO 4 ) 3 : Ln 3+ (Ln 3+ = Tb 3+ , Eu 3+ , Sm 3+ , Tb 3+ /Eu 3+ , Tb 3+ / Sm 3+ ) as a multi-color emitting phosphor for UV w-LEDs[END_REF][START_REF] Panemangalore | Effect of fluoride coatings on the corrosion behavior of Mg-Zn-Er alloys[END_REF]. Molybdates, which also have interesting physical properties [START_REF] Mohar | Dependence of optical properties of Mg-doped ZnO nanorods on Al dopant[END_REF], are widely used in various industrial sectors. Metal molybdates of monoclinic structures have many potential applications involving, for example, photoluminescence, semiconductor lasers, magnets, lithium-ion batteries, microwaves, catalysts, and photoelectric devices [START_REF] Shkir | A facile microwave-assisted synthesis of PbMoO 4 nanoparticles and their key characteristics analysis: a goodcontender for photocatalytic applications[END_REF][START_REF] Karthik | A Study of Electrocatalytic and Photocatalytic Activity of Cerium Molybdate Nanocubes Decorated Graphene Oxide for the Sensing and Degradation of Antibiotic Drug Chloramphenicol[END_REF][START_REF] Howarth | Efficient extraction of sulfate from water using a Zr-metal-organic framework[END_REF][START_REF] Tawiah | Polymer-Encapsulated Colorful Al Pigments with High NIR and UV Reflectance and Their Application in Textiles[END_REF][START_REF] Agustín-Sáenz | Effect of organic precursor in hybrid sol-gel coatings for[END_REF][9][START_REF] Ou | Preparation and Photoelectric Properties of MoSMicrospheres[END_REF].

Magnesium-Cobalt Molybdate (Mg 1-x Co x MoO 4 ) is an important technological material used as a catalyst in many chemical and petrochemical processes such as cracking, dehydrogenation, hydrogenation and hydrodesulfurization (HDS). These molybdates also exhibit interesting optical, catalytic and dielectric properties. They have been used as pigments for decades and in particular as an analytical dosing agent.

The solid-solid method is often used for the synthesis of Mg 1-x Co x MoO 4 molybdates. This process requires heat treatment at high temperatures, and typically results in powders of poor chemical and morphological homogeneity [START_REF] Cornu | Luminescence properties and pigment properties of A-doped (Zn,Mg)MoO 4 triclinic oxides (with A = Co, Ni, Cu or Mn)[END_REF]. The objective of this work is to develop Mg 1-x Co x MoO 4 (0 ≤ x ≤ 1) molybdates of nanometric size at low temperatures by pyrolysis of citrate precursors. Extremely pure multi-component oxides of submicron grain size and high chemical homogeneity can be obtained rapidly and easily by producing polymerizable complexes [START_REF] Claude | Sol-gel synthesis of Ni/Al 2 O 3 catalysts for toluene reforming: Support modification with alkali, alkaline earth or rare-earth dopant (Ca, K, Mg or Ce)[END_REF]. Typically, for molybdates of the formula A 2+ MoO 4 (A 2+ = Mg 2+ , Co 2+ …) and of the monoclinic type in the space group C2/m in the polymorph α, all the cations are in octahedral coordination. In the β phase, the divalent cation remains in the octahedral site, while the coordination of Mo evolves toward a tetrahedral site [START_REF] Zhao | A flexible all-solid-state asymmetric supercapacitors based on hierarchical carbon cloth@CoMoO 4 @NiCo layered double hydroxide core-shell heterostructures[END_REF][START_REF] Lakhlifi | Synthesis and physicochemical characterization of pigments based on molybdenum[END_REF]. The products thus obtained adopt the α or β type molybdates of divalent 3d element monoclinics depending only on whether they are cooled to ambient temperature with or without grinding. Mg 2+ ions have a strong preference for regular octahedral environments in the β phase rather than the α phase [START_REF] Zagorac | Theoretical and Experimental Study of Structural Phases in CoMoO 4[END_REF][START_REF] Ran | Enhanced energy transfer from Bi 3+ to Eu 3+ ions relying on the criss-cross cluster structure in MgMoO 4 phosphor[END_REF]. When Mg is substituted for Co, the likelihood of affecting the second neighboring Mo atoms increases rapidly, as one Mg atom is surrounded by six Mo atoms [START_REF] Jadhav | Effect of magnesium substitution on the structural, morphological, optical and wettability properties of cobalt ferrite thin films[END_REF].

Methods and materials

The following reagents were used to develop the precursors of the O) (Acros, 99%), and citric acid (Acros, 98%). The citric acid makes it possible to complex the metals along the polymer chains. These reagents were mixed in stoichiometric proportions. After evaporation, the precursors obtained were pre-burned at 300°C for 12 hours under air. The resulting black powders were crushed and then maintained at 700°C for 2 hours.

Depending on the rate of cobalt added, powders of different colours were obtained after grinding. The resulting powders were characterized using an x-ray diffractometer (Bruker D8 Advance equipped with a LynxEye detector. The X-ray generator (40 kV, 40 mA) is a copper anticathode tube that uses the CuKα line. A graphite monochromator eliminates the Kβ lines. The wavelength of the Kα1 / Kα2 lines of copper is 0.15406 / 0.15443 nm. The phases obtained are analyzed using Eva PLUS software by comparing the positions and intensities of the different diffraction lines observed with those available in the PDF-4-2010 Database established by the ICDD (International Center for Diffraction Data). The infrared spectra were taken using a Fourier-transform infrared spectrometer (IR Affinity-1S Shimadzu). The Raman spectra were recorded with a LabRAM HR 800 (Horiba Jobin-Yvon) spectrometer. The morphologies of the powders were examined using scanning electronic microscopy (JEOL JSM 6400), and the specific surface areas were determined using the Brunauer-Emmett-Teller (BET) method (Micrometrics Flowsorb II 2300). The colour parameters (L*a*b*) were measured using the CIE Lab system colorimeter (CR-400/410, KONICA MINOLTA).

Results and discussion

Thermal decomposition (TG-DTA)

The decomposition of the organic part of the xerogels is an essential parameter to be defined in order to optimize the future heat treatments necessary for the preparation of the powders. The thermogravimetric analysis curve of the magnesium-based Mg 0. precursor in air shows that thermal decomposition occurs in several stages. As shown in Fig. 1, this decomposition starts at approximately 20°C and ends before 700°C. The total weight loss is about 60%. The first weight loss of the molybdate (about 8%) is observed at approximately 98°C; it corresponds to the elimination of water adsorbed on the surfaces of the powders. The second series of successive weight losses observed between 350 and 550°C corresponds to a loss of 24%, and can be attributed to the combustion of the precursor. A final weight loss of 28% is associated with three exothermic peaks located at 581, 630 and 692 °C. This process most likely corresponds to the oxidation of the carbon produced during the combustion of the precursor. No phenomenon is observed above 700°C. This temperature was thus used for the rest of our syntheses.

X-ray diffraction (XRD)

In order to check the purity of the phases, X-ray diffraction analyses were carried out. The examination of the X-ray diffractograms of the solid solutions of Mg 1-x Co x MoO 4 (0≤x≤1) present a single continuous system for the compositions corresponding to 0≤x≤1 without grinding and have a single-phase domain related to monoclinic β-MgMoO 4 (JCPDS card No. 072-2153) (Fig. 2). For cobalt contents x ≥ 0.6 in the case of ground powders, the only phase observed is α-CoMoO 4 (JCPDS card No. 073-1331) (Fig. 3).

A simple grinding of the powder (0.6 ≤ x ≤ 1), which is purple in color and which corresponds to the allotropic variety β-CoMoO 4 , leads to powders of green color isotype of the allotropic variety α-CoMoO 4 pure. The change in color under the effect of grinding or pressure in general is called the phenomenon of piezochromism. The molybdates isotype of βCoMoO 4 are known as piezosensitive (sensitive to grinding) compounds. The grinding of the powders obtained (Mg 1-x Co x MoO 4 (0 ≤ x ≤ 1)) was carried out manually with an agate mortar for 5 to 10 • For the compositions 0 ≤ x ≤ 0.4, an iso-type monoclinic phase of β-MgMoO 4 is observed.

• For cobalt contents 0.6 ≤ x ≤ 1, the only phase observed is α- CoMoO 4 with a green color, it is a transition phenomenon of β --> α.

These results are in good agreement to those obtained by solid-state reaction [18].

Infrared spectroscopy (FT-IR)

Spectroscopy analyses (FTIR) were carried out on the Mg 1-x Co x MoO 4 (x=0.3 and x=0.6) powders prepared by sol gel route and treated at 700°C (Fig. 4). Infrared spectra show the presence of the Mo-O-Mo stretching vibration bands observed at 950 and 911 cm -1 [START_REF] Mobeen Amanulla | magnetic Antibacterial, optical and humidity sensor studies of β-CoMoO 4 -Co 3 O 4 nanocomposites and its synthesis and characterization[END_REF][START_REF] Lakhlifi | Synthesis of molybdates Zn 1-x Co x MoO 4 (0 ≤ x ≤ 1), by decomposition of the precursors developed by the glycine-nitrate process (GNP), and their characterization[END_REF].The band located at 856 cm -1 is associated with the vibration mode of the Mo-O band [START_REF] Uchagawkar | Highly dispersed molybdenum containing mesoporous silicate (Mo-TUD-1) for olefin metathesis[END_REF]. Thus, the bands observed at 744 and 710 cm -1 are associated with the vibration of the Mg-O band [START_REF] Aboul-Enein | Impact of Co/Mo ratio on the activity of CoMo/ MgO catalyst for production of high-quality multi-walled carbon nanotubes from polyethylene waste[END_REF], the band at 495 cm -1 is specific to the vibrations of the Co-O-Mo band [START_REF] Jeseentharani | Co-precipitation synthesis, humidity sensing and photoluminescence properties of nanocrystalline Co 2+ substituted zinc (II) molybdate (Zn 1-x Co x MoO 4 ; x = 0, 0.3, 0.5, 0.7, 1[END_REF]. While the peak located at 436 cm -1 is due to the bending mode of Mo-O [START_REF] Ateyya | Impact of Co/Mo ratio on the activity of CoMo/MgO catalyst for production of high-quality multi-walled carbon nanotubes from polyethylene waste[END_REF].

Raman spectroscopy

In order to confirm the results obtained previously, analyses using Raman spectroscopy were carried out on the Mg 1-x Co x MoO 4 powders (x = 0.3 and x = 0.6) produced by sol-gel route (Fig. 5). Generally, Raman bands corresponding to the symmetrical stretching (ʋ 1 ) and asymmetrical stretching (ʋ 3 ) modes were observed in the 700-1000 cm -1 region, whereas those corresponding to the symmetrical modes (ʋ 2 ) and asymmetrical modes (ʋ 4 ) were observed in the 50-520 cm -1 region [START_REF] Mafa | Visible Light Driven ZnMoO 4 /BiFeWO 6 /rGO Z-Scheme Photocatalyst for the Degradation of Anthraquinonic Dye[END_REF]. With regards to the solid solutions of Mg 1-x Co x MoO 4 (x = 0.3 and x = 0.6) prepared through the sol-gel route, the most intense vibration observed at 934 cm -1 is assigned to the mode d symmetric stretch ʋ 1 (Ag) of the MoO 4 2tetrahedron [START_REF] Jiang | Thermal annealing effects on the luminescence and scintillation properties of CaMoO4 single crystal grown by Bridgman method[END_REF] of the C 2h point group. Those located at 892 and 834 cm -1 correspond to the anti-symmetric stretching ʋ3 (Bg/D 2 h) and ʋ3 (Eg) vibration modes of the Mo-O-Mo of German entartet (e.g.), respectively [START_REF] Uchagawkar | Highly dispersed molybdenum containing mesoporous silicate (Mo-TUD-1) for olefin metathesis[END_REF]; the band observed at 711 cm -1 corresponds to the antisymmetric stretching modes of oxygen in the O-Mo-O band [START_REF] Rodriguez | Electrocatalytic performance of different cobalt molybdate structures for water oxidation in alkaline media[END_REF]; The peaks at 360 and 330 cm -1 correspond to the anti-symmetric ʋ 4 (Bg/D 2 h) and ʋ 2 (Ag/C 2 h) bending modes, respectively [START_REF] Meng | Facile preparation of CoMoO 4 nanorods at macroporous carbon hybrid electrocatalyst for non-enzymatic glucose detection[END_REF][START_REF] Liu | Phosphide-oxide honeycomblike heterostructure CoP@CoMoO 4 /CC for enhanced hydrogen evolution reaction in alkaline solution[END_REF][START_REF] Padmanathan | Hydrothermal synthesis of carbonand reduced graphene oxide-supported CoMoO 4 nanorods for supercapacitor[END_REF][START_REF] Yang | Enhanced electrochemical performance of CoMoO 4 nanorods/reduced graphene oxide as anode material for lithium ion batteries[END_REF]. Thus, the free rotation mode was observed at 276 cm -1 [START_REF] Luz-Lima | Temperature-dependent Raman spectroscopy studies of phase transformations in the K 2 WO 4 and the MgMoO 4 crystals[END_REF]. These results were consistent with those obtained by diffraction analysis techniques X-rays and infrared spectroscopy (FTIR). The Raman bands of the compositions x = 0.3 and x = 0.6 are summarised in Table 1.

Specific surface area measurements (BET)

Specific surface area measurements were carried out on the 2 presents these measurements and the average sizes of the particles which were calculated, taking into consideration that the particles are spherical and individualised. It should be noted that the powders obtained via the sol-gel route have low specific surfaces that vary from 0.4 to 0.8 m 2 /g which depending on the rate of incorporation of the cobalt.

Scanning electron microscopy (SEM)

The SEM micrographs for Mg 1-x Co x MoO 4 (x = 0.3 and x = 0.6) prepared at 700°C via sol-gel route show a powders formed by compact porous agglomerates forming cages (Fig. 6a andb).The porosity comes from the rapid dissipation of the gaseous products (NO 2 , CO 2 and H 2 O) formed during the combustion. The particles forming the agglomerates range in size from 1 to 5 μm). These images show advanced sintering, which confirms the low values of the specific BET surfaces obtained. sol-gel method were measured in the CIE L*a*b* system (Fig. 7). We noted a maximum value for the composition x = 0.6 with purple colour. Fig. 8 shows the different shades of purple obtained essentially depend on the content of cobalt x, such that 0 ≤ x ≤ 1.

Conclusion

Cobalt-substituted MgMoO 4 compounds were prepared through the sol-gel route. The X-ray diffraction result shows the formation of continuous systems of the solid solution Mg 1-x Co x MoO 4 (0 ≤ x ≤ 1) and the grinding of these powders seems to have an influence on the phase obtained. In fact, for the compositions 0 ≤ x ≤ 1:

• Without grinding, they present a single-phase domain of monoclinic symmetry isotype at β-MgMoO 4 .

• With grinding, the fractions x ≥ 0.6 show, a single-phase isotype with α-CoMoO 4 .

The ceramic pigments obtained have an advanced pre-sintering state, which explains the low values of specific surface area between 0.41 and 0.77 m 2 /g. The measurements of the L*a*b* colorimetric coordinates show that the component (-b*), which is characterised by purple colour, has a maximum value in the case of Mg 0.4 Co 0.6 MoO 4 .

7 Fig. 1 .

 71 Fig. 1. TG-DTA curves of the Mg 0.7 Co 0.3 MoO 4 precursor obtained by pyrolysis of citrate precursors.

Fig. 2 .

 2 Fig. 2. XRD patterns of Mg 1-x Co x MoO 4 (0 ≤ x ≤ 1) compounds prepared by sol gel route without grinding

Fig. 3 .

 3 Fig. 3. XRD patterns of Mg 1-x Co x MoO 4 (0≤x≤ 1) compounds prepared by sol gel route with grinding.

Fig. 4 .

 4 Fig. 4. FT-IR spectra of Mg 1-x Co x MoO 4 (x=0.3 and x=0.6) powders obtained at 700°C.

Fig. 5 .

 5 Fig. 5. Raman spectra of the Mg 1-x Co x MoO 4 (x = 0.3, 0.6) powders obtained at 700°C.

Frequencies

  Ag/C 2 h) MoO 4 : Symmetrical stretch 892-834 ʋ 3 (Ag/C 2 h and Eg/(e.g)) Mo-O-Mo: Ant-isymmetric stretch 711 ʋ 4 (Bg/D 2 h) O-Mo-O: Anti-symmetric oxygen stretching vibration 360 ʋ 4 (Bg/D 2 h): anti-symmetric vibration 330 ʋ 2 (Ag/C 2 h): Bending modes 276 Free rotation mode

3. 7 .Fig. 6 .

 76 Fig. 6. SEM micrographs of Mg 1-x Co x MoO 4 powders (a): x = 0.3 and (b): x = 0.6 prepared by sol-gel route.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Evolution of the colour parameters of Mg 1-x Co x MoO 4 (0 ≤x≤ 1) powders obtained by sol-gel route.

  Mg 1-x Co x MoO 4 molybdates: 0.2M magnesium nitrate (Mg(NO 3 ) 2 •6H 2 O) (Aldrich, 98%), 0.2M cobalt nitrate (Co(NO 3 ) 2 •6H 2 O) (Aldrich, 98%),0.2M ammonium heptamolybdate ((NH 4 ) 6 Mo 7 O 24• 4H

	T
	E-mail address: lakhlifihind21@gmail.com (H. Lakhlifi).
	https://doi.org/10.1016/j.surfin.2020.100718

⁎ Corresponding author.

  [START_REF] Panemangalore | Effect of fluoride coatings on the corrosion behavior of Mg-Zn-Er alloys[END_REF]

Table 1

 1 Raman bands, modes and attributions of the Mg 1-x Co x MoO 4 (x=0.3 and x=0.6).

Table 2

 2 Elementary grain size solid solutions Mg 1-x Co x MoO 4 (x = 0.3 and x = 0.6). Co x MoO 4 (x = 0.3, 0.6) powdered samples. Table

	Synthesis route	The Co 2+ amount	Specific surface	Average particle
		incorporated (x)	area BET (m 2 /g)	size BET (nm)
	Sol-gel (700°C)	0.3	0.41	3622
		0.6	0.77	1824

CRediT authorship contribution statement

Hind Lakhlifi: Conceptualization, Software, Formal analysis, Writing -original draft. Youssef El Jabbar: Conceptualization, Software, Formal analysis, Conceptualization, Software, Formal analysis. Rachida El Ouatib: Supervision, Project administration, Validation. Lahcen Er-Rakho: Methodology, Data curation. Sophie Guillemet-Fritsch: Resources, Visualization. Bernard Durand: Investigation, Visualization, Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Y El Jabbar"'

H Lakhlifi

R El

L Er-Rakho

S Guillemet-Fritsch\

B Durand

the work of Toulouse researchers and makes it freely available over the web where possible Structure, microstructure, optical and magnetic properties of cobalt aluminate nanopowders obtained by sol-gel process

Keywords:

The aim of this study was to detennine the effect of different complexing agents and of the annealing tem perature on the structural, morphological and optical properties of the synthesized precursors. Thus, cobalt aluminate nanoparticles were prepared by the sol-gel method using polyacrylic acid, glycine or citric acid as complexing agents. The synthesis performed at 500 °C for 5 hours led to dark green powders composed of solid solutions denoted Co 2+ [ Agci-x) Co�;i]O4 and of amorphous alumina (39.8wt%). Calcination at temperatures above 900 °C caused the powder colour to change from dark green to blue. A direct spinel structure Co 2+ [ Al� + ] 04 is all the more achieved as the annealing temperature is high. The powder obtained using polyacrylic acid as the complexing agent at 900 °C for 5 hours revealed the best morphology; it consisted of agglomerates of primary particles with a quasi-spherical shape and a size in the range 20-40 nm.

Introduction

Transition-metal aluminates witb general formula

Cobalt aluminate spinel(CoAl 2 O 4 ) is known for its blue colour and it is widely used as pigment.s for ceramics, paint.s, fibres and so on. Among its advantages, this compound presents high thermal and excellent chemical stability [START_REF] Deraz | a, structural Synthesis, morphological properties of cobalt aluminum nano-composite[END_REF][START_REF] Gilabert | Effect of secondary thermal treatment on crystallinity of spinel-type Co(Cr,Alh 04 pigments synthesized by solution combustion route[END_REF][START_REF] Song | Fabrication and stability of CoA12O4 ceramic pigment for 3D printing[END_REF][START_REF] Wang | Ultrafine CoA12O4 ceramic pigment prepared by Pechini-sacrificial agent method[END_REF]. The spinel oxides powders are generally prepared by solid state reactions [ll,12]. This metbod has tbe ad vantage of being simple and inexpensive, but it requires long tbermal treatments at high temperatures (T~l300 °C) [START_REF] Ouahdi | Investigation of the reactivity of A1Cl3 and CoCl2 toward molten alkali-metal ni trates in order to synthesize CoA12O4[END_REF][START_REF] Yu | Preparation of nanosized CoAl2O4 powders by sol-gel and sol-gel-hydrothermal methods[END_REF]. Therefore, tbe resulting powders exhibit a low specific surface area, witb morpholo gies tbat are most often heterogeneous. Recently, several research studies have focussed on tbe preparation of ultrafine CoAl,O 4 powder for its use in the decoration of ceramics by ink-jet printers, which re quires nanometric sizes to avoid clogging of tbe nozzles [START_REF] Pan | Recent development on pre paration of ceramic inks in ink-jet printing[END_REF].

For tbe reasons described above, several soft-chemistry metbods have been studied to prepare tbese materials, namely, tbe hydrotbermal metbod [START_REF] Kim | Characteriza.tion of blue CoA12O4 nana-pigment synthesized by ultrasonic hydro thennal method[END_REF], tbe co-precipitation route, the molten sait syntbesis [START_REF] Ouahdi | Synthesis of CoA12O4 by double decomposition reaction between LlA1O2 and molten KCoC13[END_REF], tbe sol-gel process [START_REF] Bayal | Synthesis of metal aluminate nanoparticles by sol-gel method and studies on their reactivity[END_REF] and recently, tbe polymerisable complex metbod [START_REF] Jafari | ajaf, Characterization of Ba 2+ -added alumina/cobalt nanoceramic pigment prepared by polyacrylamide gel method[END_REF][START_REF] Jafari | Preparation of CoA12O4 nanoblue pigment via polyacrylamide gel method[END_REF].This last metbod provides nano-sized oxide powders • Corresponding author. E-mai l aMress: ysf.eljabbar@grnail.com (Y.E. Jabbar).

with high chemical and morphological homogeneities. This process uses a carboxylic acid as a chelating agent towards metal cations, which allows a mixed precursor to be obtained. The calcination of tbis latter leads to tbe formation of tbe mixed mdde at a relatively low tempera ture [START_REF] Gama | Synthesis and characterization of the NiAl2O4, CoA12O4 and ZnA12O4 spinels by the Journal of Non-Ctysta[END_REF]. Severa! authors have shown that the morphological char acteristics of powders syntbesized by tbis process depend on tbe che mical nature of tbe chelating agent, tbe reaction pH, tbe calcination temperatureand so on [START_REF] Jabbar | Influence of temperature and pH on the morphology and the color of the CoAI2O 4 prepared by sol gel method[END_REF][START_REF] Gomes | Optimizing the synthesis of cobalt aluminate pigment using fractional factorial design[END_REF][START_REF] Merino | Combustion syntheses of CoA12O4 powders using different fuels[END_REF][START_REF] Salem | Effect of calcination temperature on colorant behavior of cobalt-aluminate nano-particles synthesized by combustion technique[END_REF][START_REF] Torkian | Effects of �-alanine on morphology and optical properties of CoAl2O4 nanopowders as a blue pigment[END_REF]. The aim of this work is to prepare nanosized CoA1 2 O 4 powders using sol-gel syntbesis witb different complexing agents (polyacrylic acid, citric acid and glycine) at pH 7 and with different calcination temperatures. Then, the effect of the synthesis parameters on the structural, morphological, optical and magnetic properties of CoAl2O 4 nanoparticles is studied. The use of tbis powder as a blue pigment is discussed.

Materials and methods

Synthesis

Cobalt nitrate Co(NO 3 h•6H2O (99.0%, Aldrich) and aluminum ni trate Al(NO 3 )s-9H2O (99.0%, Aldrich) were dissolved in water at a molar ratio 1/2. Tuen an excess of one of tbe following complexing and gelling agents was added: polyacrylic acid (PM) at a concentration of https:/ /doi.org/10.1016/j.jnoncrysol.2020.120115 100 g L -,, citric acid (CA) with molar ratio CA/cations=3 or glycine (Gly) with a molar ratio Gly /nitrate= 2/3. After adjusting the pH to 7 by adding an ammonium hydroxide solution under constant stirring, the solution was evaporated at 80"C until obtaining a viscous gel, that was dried at 120'C for 24 hours (precursor noted PRO). Then the or ganic malter was removed by a precalcination at 300'C for 12 hours under air and cobalt aluminate samples noted PRl, PR2 and PR3 were obtained by annealing for 5 hours at respectively 500, 900 and 11 00'C.

Characterization

The decomposition in air atmosphere of the pre-calcined powders (PRO) was investigated by differential thermal analysis and thermo gravimetric analysis using a simultaneous DTA/TGA analyzer (DTA-TG-60H Shimadzu); the heating rate was set at 2.5'C/min. The X-ray dif fraction patterns of the powders were recorded by means of a dif fractometer equipped with a detector Lynx Eye (Bruker D8 Advance, Â.cuKa = 1.5406 Â). The infrared spectra were taken using FfIR spec trometer (IR affinity-1S Shimadzu). The morphology and size of the powders were examined with a field emission scanning electronic mi croscope (JEOL JSM 6700F) and the specific surface area were de terrnined by the BET method (Micrometrics Flowsorb II 2300). The magnetic measurements were performed using a superconducting quantum interference magnetometer (SQUID, MPMS-XI). The absor bance spectra were determined by UV-Vis spectrophotometry (Perkin Elmer Lambda 35) and the colorimetric parameters (L*, a* and b*) were measured in the system (CIE) using a colorimeter (Chroma Meler CR-400/410, Konica Minolta).

Results and discussion

Thermal analysis

The thermal analysis curves of the PRO precursors prepared with different complexing and gelling agents and obtained after pyrolysis at 300'C for 12 hours are illustrated in Fig. 1

Powder X-ray diffraction analysis

The XRD patterns of the samples prepared with PAA, CA and Gly as complexing agents and heat treated at different temperatures are given in Fig. 2 2) are interrnediate between those 0 ---50

•,

•- ....

•- o_ 15% 1 ,-.. •- � -._ <( r 10 •- ,_ •-• ••••-... -50 O 1 Exo -100 20 (a)
Temperature ( °C) -- ' 12% � •., 0� r •-... 10 23 <( •••-�--,--,. , •••-•-•••••••---- ,_ F-- -50 ° 15
Co 2+ [Al' + 2c1-xJC0 3+ 2xJO4 + 3xAl20, ➔ (1 + 2x)Co 2+ [A/ 3+ 2JO4 + 1/202 (1)
The transformation is ail the more complete as the annealing tem perature is high. The mechanism of amorphous alumina insertion during the calcination of the gels is illustrated in Fig. 4.

FI'IR spectroscopy analysis

The inlrared spectra of the prepared samples are given in Fig. 5. The PR0PAA and PR0 G ly spectra show two weak bands in the spectral range 500-700 cm-1. While in the PR0 CA spectrum, no IR band is observed.

Magnetic measurement

The mixed and direct spinel structures were characterized using magnetic measurements. The PRlpAA and PR2 PAA powders were ana lysed using a SQUID magnetometer in the temperature range 50-300K. lsothermal magnetization measurements ( o) as a function of the a pp lied field (Hl were performed at 50 and 300K. The evolutions of o = f (Hl curves illustrated in Fig. 6a andb showed that the magnetization o is a linear fonction of the applied field, according to the following equation:

r;=x•H (2)
x: the magnetic susceptibility of the compound.

H: the applied field (Oe). Therefore, the PRlpAA and PR2PAA compounds exhibited a para magnetic behaviour at 50 and 300K. A small deviation of the slope of o=f(H) was observed between PRlpAA and PR2PAA; this indicated a significant influence of the calcination temperature on the magnetic properties of the samples studied.

Magnetic susceptibility measurements x were made under a con stant field of 10,000 Oe in the temperature range 50-300K, as presented in Fig. 7. The obtained curves evolutions were in good agreement with the Curie-Weiss law and could therefore be given by the following equation:

Gel

Colour characterization

The heat treatment (500 ,:; T,:; 11 OO"C) under air of the precursors ,------------------,

  MAl 2 O 4 (M: Ni, Co, Fe) have attracted significant attention in recent decades because of tbeir fundamental and applied interest. The excellent physical proper ties of tbese materials offer many possibilities for use in different fields, such as coatings, pigmentation and photocatalysis [1--6].

  . The TG curves of the PR0PAA and PR0my precursors (Figs.la and lb) showed a total weight Joss of about 15% and 12%, respectively, due to the elimination of the ad sorbed water and the organic residues. The thermogram of the PR0 CA precursor presented a weight Joss of about 23%, and the OTA curve showed a strong exothermic peak around 415'C (Fig. le). This could be attributed to the combustion of organic matter [27]. An additional slight endothermic inflection at 920'C was observed for the three pre cursors decomposition; this phenomenon might be due to the reduction of trivalent cobalt.

  . Those of PR0 PAA and PR0my showed some small peaks, in dicating the beginning of crystallization, while the one of PR0CA showed an amorphous phase. The annealing at 500'C/5h of PRO pre cursors with different complexing and gelling agents led to dark green powders PRl. The XRD patterns of these powders showed several peaks at 26 values of 31.1 ', 36.8', 44.7', 55.T, 59.4', 65.3', 74.2'and 77.4'. These peaks could be indexed with space group Fd3m to (220), (311), (400), (422), (511), (440), (620) and (533) planes of a cubic unit cell. The Rietveld refinement pattern of the PRl sample obtained with PAA as complexing agent is given in Fig. 3. The corresponding data (Table 1) showed that cobalt ions were located both in the tetrahedral Ba and octaltedral 16d sites, whereas the aluminium ions predominantly oc cupied the octaltedral 16d sites. This could indicate a partial oxidation of Co 2+ in Co 3 + , giving the sample a greenish hue. Moreover, the structural properties (ce!! parameter, cell volume and density) deduced from the Rietveld refinement (Table

Fig. 1 .Fig. 2 .Fig. 3 .

 123 Fig. 1. Thermal analysis (TGA-DTA) curves of dried gel precalcined at 300 °C for 12h PROPAA (a), PRO G !y (b) and PROAc (c). of the direct spinel phases Co 2+ [Co� + ]O 4 and Co 2+ [AJ� + ]O 4 • It is then concluded that the PRl dark green powders are solid solutions between the two spinel phases, denoted Co 2+ [Al�éi-,J Co:l:]O4• In these PRl powders, the unreacted aluminium is present, beside the crystallized spinel phase, under the forrn of amorphous alumina. In agreement with Yan et al [28], the internai standard method was in volved to quantify the amorphous phase in the samples. This method consists in using an exact quantity of well-crystallized a-Al 2 O 3 powder to be mixed and milled with the studied sample. The calculation of the percentage of amorphous phase present in a powder was done using the Ful!Prof software by Rietveld refinement of diagram patterns. The percentages of the phases obtained by refinement were corrected with those of the prepared mixture. This allowed the rate of amorphous phase to be deduced. The content in amorphous alumina of the PR PAA sample is about 39.8 wt%. Whatever the nature of the complexing agent, blue powders PR2 and PR3 were obtained by annealing the PRl powders at respectively 900'C/5h and 1100'C/5h. Their XRD patterns (Fig. 2) showed the same system of lines as the PRl samples but they were narrower and became

The

  PRl spectrum shows two bands in the spectral range 500-700 cm-1 whatever the nature of the complexing agent. These bands could be attributed to the Co 2+ [ AIM.-xl Co�:]0 4 solid solution. The width of the peaks indicates the presence of amorphous phases in the sample [13]. When the thermal treatment raised-up 900"C (PR2 and PR3), a sup plementary band appeared at 501 cm -l and the two other bands be came more visible at 563 and 669 cm -l. These three bands are attri butable to the vibration modes of the CoAl204 phase [31]. These results are in good agreement with XRD analysis.

  Based on the linear evolution of the inverse molar magnetic sus ceptibility, the effective magnetic moment could be extracted for dif ferent samples. The value obtained for PR2 PAA was close to 4.65 µ8 • This value characterizes Co 2 + in a tetrahedral site with a spin-orbit inter action (Lande g-factor [g] = 2.4 and a spin [S] = 3/2) [12]. This con firmed the Rietveld refinements, which presented a direct spinel structure for the PR2 PAA sample. However, the observed value for PRl was 3.74 µ 8 far from that of isolated Co 2+ (µ.," = 4.65 µ8 ). This sug gested that a mixture of Co 2 + and Co 3 + existed in this compound [32,33]. Table 3 shows the comparison of the experimental values of the effective magnetic moments with those theoretically obtained by applying the following equation [34]: (4) n: percentage of the crystalline phase obtained by the internai standard method. In Eq. (4), x and y are the fraction of cobalt Co 2+ in tetrahedral sites (Td) and the fraction of Co 3+ in octahedral sites (Oh), respectively. The theoretical values of µ,ff(Co};t) = 4.65µ 8 and µ,ff(Cobtl = 0µ 8 (Co 3+ was considered in low spin). The effective magnetic moments calcu lated by Eq. ( 4) were close to those obtained experimentally; and confirmed the results obtained by Rietveld refinement analysis.

1Fig. 4 .Fig. 5 .- 4 Fig. 6 .Fig. 7 . 3 . 6 .Fig. 9 .

 45467369 Fig. 4. Mechanism of alumina insertion.

Fig. 8 .

 8 Fig. 8. L * a * b * Colourimetric parameters of powders prepared by P AA as complexing agent at pH = 7.

Table 1 Rietveld refinement for PRlPAA, PR2PAA and PR3PAA samples.

 1

	Sample	Atom Wyckoff	Occupancy Bisa	Refinement
			Positions			quality
			X=y=z			parameters
	PRl (500 °C/5h)	Co	0.125	0.78	0.114(5) Rp-0.87%
		Al	0.125	0.08	0.114(5) Rwp=l.09%
		Co	0.5	0.51	0.207

(3) :x: 2 =1.03

		Al	0.5	0.43	0.207(3)
		0	0.262(4)	1	0.502(6)
	PR2 (900'C/5h)	Co	0.125	0.88	0.899(3) Rp=0.87%
		Al	0.125	0.02	0.899(3) Rwp=l.08%
		Al	0.5	0.84	0.642(7) X 2 =1.05%
		Co	0.5	0.10	0.642(7)
		0	0.263(3)	1	0.744(5)
	PR3 (ll00 °C/5h) Co	0.125	0.90	0.468(4) Rp-0.95%
		Al	0.125	0.10	0.468(4) Rwp=l.20%
		Al	0.5	0.92	0.314(3) X 2 =1.18%
		Co	0.5	0.06	0.314(3)
		0	0.264(6)		0.268(5)

Table 2 Structural properties and amorphous phase ratio for PRl, PR2 and PR3 sample obtained by Rietveld method refinements.

 2

Declaration of Competing Interest

Ail authors declare no conflict of interests.