Daniel Brooks
email: daniel.brooks@lip6.fr

Olivier Schwander

Frédéric Barbaresco

Jean-Yves Schneider

Matthieu Cord

A Hermitian Positive Definite neural network for micro-Doppler complex covariance processing

Keywords: covariance matrices, complex numbers, micro-Doppler, drone classification, neural networks

In its raw form, micro-Doppler radar data takes the form of a complex time-series, which can be seen as multiple realizations of a Gaussian process. As such, a complex covariance matrix constitutes a viable and synthetic representation of such data. In this paper, we introduce a neural network on Hermitian Positive Definite (HPD) matrices, that is complex-valued Symmetric Positive Definite (SPD) matrices, or complex covariance matrices. We validate this new architecture on synthetic data, comparing against previous similar methods.

I. INTRODUCTION

The usage of deep learning methods has steadily been emerging in the radar community, specifically for micro-Doppler classification [START_REF] Chen | Micro-Doppler effect in radar: phenomenon, model, and simulation study[END_REF]; furthermore, the diversity of possible representations of micro-Doppler signals induce a variety of methods, such as recurrent neural networks (RNNs) in [START_REF] Molchanov | Classification of small UAVs and birds by micro-Doppler signatures[END_REF], or convolutional neural networks (CNNs) in [START_REF] Trommel | Multitarget human gait classification using deep convolutional neural networks on micro-doppler spectrograms[END_REF] and [START_REF] Brooks | Temporal Deep Learning for Drone Micro-Doppler Classification[END_REF]. In parallel, exploiting second-order feature moments, or covariance, is gaining momentum in certain subfields of machine learning, such as in EEG/ECG [START_REF] Barachant | Multiclass BrainComputer Interface Classification by Riemannian Geometry[END_REF], facial recognition [START_REF] Acharya | Covariance Pooling For Facial Expression Recognition[END_REF] or texture classification [START_REF] Harandi | From Manifold to Manifold: Geometry-Aware Dimensionality Reduction for SPD Matrices[END_REF]. The natural representation of micro-Doppler signals as covariance (or self-correlation) led to several classification methods based on real-valued covariances matrices: in [START_REF] Brigant | Geometric barycenters of time/Doppler spectra for the recognition of non-stationary targets[END_REF], authors use Riemannian barycenters on covariance reflexion coefficients; in [START_REF] Arnaudon | Medians and Means in Riemannian Geometry: Existence, Uniqueness and Computation[END_REF], a minimum-distanceto-median scheme on SPD matrices was developed. Of all previously cited methods operating on covariance, the most promising in terms of potential development may be the ones based on neural-like processing: specifically, a first version of networks on SPD matrices, SPDNet, was first introduced in 2017 in [START_REF] Huang | A Riemannian Network for SPD Matrix Learning[END_REF], upon which further works expanded on. We also follow this blooming trend, by extending the theoretical framework of the SPDNet to complex values, thus introducing the HPDNet, or neural network operating on HPD matrices.

In the following section, we review the two theoretical learning frameworks we wish to fuse together. Then we describe our proposed HPDNet architecture and detail its specific mechanics. Finally, we validate the usage of a HPDNet over a SPDNet by studying micro-Doppler drone classification on synthetic radar data.

II. MANIFOLD VALUES IN NEURAL NETWORKS

Both machine learning frameworks we wish to fuse involve manifold values; the first type involves SPD matrices, the second complex values.

A. SPD matrices in neural networks

The fundamental interest of the SPD neural network is to take into account the information geometric structure of the curved Riemannian manifold of SPD matrices, noted S + * . Statistical learning on curved manifolds is part of the field of information geometry [START_REF] Amari | Information Geometry and Its Applications[END_REF]. Here, the learning model involved is a neural network, the layers of which we describe here. As in a standard Euclidean network, the SPDNet builds a hierarchy of activations on linear transformations, ended by a loss function to minimize by gradient descent [START_REF] Huang | A Riemannian Network for SPD Matrix Learning[END_REF]. This linear transformation, a fully-connected layer in perceptrons, a convolution in convolutional networks, becomes a bilinear mappping in the SPDNet, referred to as the BiMap layer, which we illustrate in Fig. 1. The activation function, usually set to the rectified linear unit (ReLU) in Euclidean networks, becomes the ReEig (rectified eigenvalues) in the SPDNet:

P (l) = W (l) T X (l-1) W (l) with W (l) ∈ O(n l-1 , n l ) X (l) = U (l) max(Σ (l) , I n )U (l) T with P (l) = U (l) Σ (l) U (l) T (1)
Thus, the building block of the SPDNet is X (l-1) → P (l) = BiM ap (l) (X (l-1) ) → X (l) = ReEig(P (l) ). In the BiMap layer, O(n l-1 , n l ) is the space of semi-orthogonal rectangular matrices, also called Stiefel manifold, meaning the bilinear mapping is none other than an orthogonal basis change. In order for the parameter W to be non-singular, we must have n l-1 ≤ n l . In the ReEig equation, is a fixed parameter, which thresholds close to zero eigenvlues. Note the activation acts directly on the eigenvalues, thus necessitating a prior eigendecomposition; such a function is sometimes referred to as a structured matrix function [START_REF] Ionescu | Matrix Backpropagation for Deep Networks with Structured Layers[END_REF]. A final layer, of central importance to the SPDNet, is the logarithmic mapping, or Lo-gEig, which maps the final SPD representation to a Euclidean space to perform the actual classification, which we illustrate in Fig. 2; the LogEig is simply the matrix logarithm. The training of an SPDNet entails two main technical difficulties: the optimization constrained to the Stiefel manifold for the update of the BiMap orthogonal parameters on the one hand, and on the other hand the propagation of gradients through the non-linear structured matrix functions found in the ReEig and LogEig layers. Interested readers may refer to [START_REF] Ionescu | Matrix Backpropagation for Deep Networks with Structured Layers[END_REF] and [START_REF] Edelman | The Geometry of Algorithms with Orthogonality Constraints[END_REF] for theoretical background on these matters.

B. Complex numbers in neural networks

Here we explain how to integrate complex values in standard learning frameworks. Real-valued neural networks handle Euclidean data, which can be represented as vectors. A practical approach to handling Hermitian values, i.e. complex values, would simply to concatenate the real and imaginary parts in two independent channels. However, this method fails to take into account the structure of complex numbers. However, given some constraints, it is possible to formally state the equivalence between C and R 2 , which is directly translatable to C n and R 2n . This equivalence was first discovered by Wirtinger in 1927 [START_REF] Wirtinger | Zur formalen Theorie der Funktionen von mehr komplexen Vernderlichen[END_REF], and adapted in [START_REF] Brandwood | A complex gradient operator and its application in adaptive array theory[END_REF] and [START_REF] Bos | Complex gradient and Hessian[END_REF] for electrical engineering purposes; it is sometimes referred to as Wirtinger calculus, or CR calculus, due to the intimacy shared between real and 2D real vectors exposed below. Traditional complex calculus is presented in the context of holomorphic functions; the aforementioned developments aimed to broaden this limited set of functions, nowadays allowing for instance differentiation of complex neural blocks, which are for the most part non-holomorphic, an operation fundamental to statistical learning. The key idea of these developments is to equate the Taylor expansion of a function f : z ∈ C → y ∈ R with a two-dimensional, real-valued counterpart, which by abuse of notation is also noted

f : (u, v) T ∈ R 2 → y ∈ R: f (z) ≈ f (z 0 ) + ∇ z f z0 (z -z0) (2) f (u, v) ≈ f (u 0 , v 0 ) + ∇ u f ∇ v f (u0,v0) u 0 v 0 (3)
We now introduce the 2 × 2 real-to-complex matrix T :

T := 1 j 1 -j such that:            T H T = 2I = T T H T u v = u + jv u -jv = x x * x ( 4
)
As such, we can now write:

∇ u f ∇ v f T (u0,v0) u 0 v 0 = 1 2 ∇ u f ∇ v f T (u0,v0) T H T u 0 v 0 := ∇ x f x 0 x -x 0 (5)
We have isolated a term, ∇ x f , behaving like a gradient, artificially introduced in the CR calculations:

∇ x f x 0 = ∇ x f x0 ∇ x * f x0 T = 1 2 (∇ u f -j∇ v f ) (u0,v0) 1 2 (∇ u f + j∇ v f ) (u0,v0) T (6)
It is now possible to perform optimization on a nonholomorphic complex-valued function f by zeroing ∇ x f . It is now crucial to note that doing so is equivalent to zeroing either ∇ x f or ∇ x * f , which is exactly how [START_REF] Wirtinger | Zur formalen Theorie der Funktionen von mehr komplexen Vernderlichen[END_REF] showed that such a function f can be seen as operating on two independent variables x and x * . We see below how the choice of ∇ x f over ∇ x * f in a neural architecture will lead to a very fluid computational scheme.

III. HPDNET: A HPD-VALUED NEURAL NETWORK

In this section, we present how we adapt the SPDNet layers to complex values, making use of the formal two-channel representation previously exposed. We note H + * the manifold of HPD matrices.

A. Formal HPD representation in a real-valued computational framework

Because calculus of non-holomorphic functions is to this day not widespread in computational frameworks (for instance as of end 2018, complex tensors are not integrated in the deep learning framework PyTorch [START_REF] Paszke | Automatic differentiation in PyTorch[END_REF]), it remains interesting to provide a custom integration, especially given the simplicity of the resulting implementation. From the results derived above, and taking inspiration from a seminal paper on deep complex networks [START_REF] Trabelsi | Deep Complex Networks[END_REF], we can thus represent an HPD matrix H = H + jH as a two-channel SPD matrix (H , H ). In any neural architecture, the gradient of the loss function l involving (H , H ) is computed as ∇ H l + j∇ H l, which corresponds exactly to ∇ H l. For this reason, it is natural to choose ∇ H l over ∇ H * l, as foreseen above. The backpropagation through the network can therefore be done with no additional pain, using any out-of-the-box backpropagation algorithm in a realvalued computational framework. However, the inference still needs to respect the internal structure of complex numbers. Below we show how to adapt the BiMap layer to adopt complex numbers.

B. Complex bilinear mapping

Here we show how to generalise the BiMap layer described previously to complex values, using the formalism exposed above. First we write the complex expresison of the bilinear mapping, then structure it to fit the CR framework:

P = W XW H = (W + jW )(X + jX )(W H + jW H ) = W X W H -W X W H -W X W H -W X W H + j W X W H -W X W H + W X W H + W X W H := W X W H -W X W H -W X W H -W X W H W X W H -W X W H + W X W H + W X W H (7)
Note that the parameter matrix is now unitary (that is, complex orthogonal), and the transpose becomes a transconjugate.

C. Complex structured non-linearities

We study here the case of the non-linear structured complex functions involved in the HPDNet, the complex ReEig and complex LogEig. As stated previously, both take the form a non-linear function f acting on the an HPD matrix P 's eigenvalues. We assume an eigen-decomposition of P = U ΣU H . Note that although U is unitary, Σ is real because P is Hermitian. This is a very strong result, which greatly simplifies the generalisation of aforementioned functions to the complex setting: in fact, there is nothing in f to actually generalise, since the input eigenvalues are already real. However, one must take care to correctly handle the eigen-decomposition itself, separating the real and imaginary parts in order to respect the CR formalism:

X = f (P ) = U f (Σ)U H = (U + jU )f (Σ)(U -jU ) T = U f (Σ)U T -U f (Σ)U T + j U f (Σ)U T + U f (Σ)U T := U f (Σ)U T -U f (Σ)U T U f (Σ)U T + U f (Σ)U T (8)
Using the equations above, we are now able to perform inference through the complex BiMap, ReEig and LogEig layers. Posterior to the LogEig, a fully-connected layer (or several) handles the hyperplanar separation for classification. At any point in the network, it is possible to go back to R using the C2R transfer function below:

∀H ∈ H + * , 1 2 (H + H ) = S ∈ S + * (9)
This is necessary because, again, we cannot yet perform the classification in the complex manifold. Furthermore, it is not obvious that the best performing model would maximize the use of complex numbers; it remains of interest to study the influence of the C2R's positioning in the networks hierarchy to optimize its performance andd robustness.

IV. EXPERIMENTS

In this section we test the HPDNet architecture against an equivalent SPDNet counterpart on both synthetic and two real micro-Doppler datasets. We also evaluate the robustness to lack of available training data.

A. Description of the synthetic data

The synthetic micro-Doppler data generated from a simulator introduced in [START_REF] Brooks | Temporal Deep Learning for Drone Micro-Doppler Classification[END_REF]. The simulator models three drone classes: a helicopter, a quadrocopter and an octocopter as shown in Fig. 3.

Approximately 4 minutes of signal are synthetisized per class. We set the observation conditions such that the signalto-noise ration (SNR) varies around 5dB before coherent integration. We consider an elementary timeframe to correspond roughly to one or two blade rotations; the fastest propeller having an RPM of 4800 rounds per minute, we set the elementary period of observation to 10ms. We also set a reasonably low period of repetition frequency (PRF) of 2kHz, which results in an elementary signal of 20 discrete timesteps, which translates to setting the input feature dimension to 20.

In other words, one radar signal is modeled as a point z in C n , with n = 20. However, the final objective here is the classification on the underlying Gaussian process; in order to estimate its corresponding covariance, we theoretically need at least 20 independent samples in order for the resulting matrix to be non-singular. In practice, we sample N = 20 successive samples; that is, the strict minimum. We use the unbiased covariance estimator:

X = 1 N -1 i≤N zi zH i ∈ H + * (n) (10)
In the equation above, zi is the centered version of z i . We implement a 2-layer network reducing the matrix dimension from 20 to 16 and 8.

B. Description of the NATO database

To challenge our model to a real-world setting, we make use of a micro-Doppler drone database provided by the NATO organization.

The NATO database consists of recordings of 8 different aiborne subjects, including one class encompassing birds and 7 different drones:

• DJI Phantom 3 (carbon fiber & nylon blades) [quad- copter]; • 3DR X8 (carbon fiber blades) [quadcopter]; • 3DR Iris (carbon fiber & nylon blades) [quadcopter]; • Firefly [hybrid] • Anaconda [fixed wing] • Opterra [fixed wing] • Skywalker [fixed wing]
The Phantom and the Iris come in two versions: carbon fiber or nylon blades, which can consitute child classes. The drones are categorized in three types: quadcopter, fixed wing and hybrid, which can consitute parent classes. The radar signals are accessible in their raw form of time series of complex points of amplitude and phase. The subjects were furthermore recorded in 6 frequency bands (L,S,C,X,Ku and Ka) and with both vertical and horizontal polarization (except for the L band). In our classification setup, we choose to consider all class variation as independent objects; for instance, the Phantom drones with carbon or nylon blades are set as different classes altogether, which we hope to descriminate. All in all, this leads to 10 separate classes.

Each datafile is the continuous recording of one of the above targets; those recordings vary in lenth from 7s to 672s. The P RF is set to 25kHz, meaning a signal of 1min totals 1.5e6 complex numbers. All recordings are split in elementary segments of M = 1024 points (i.e. 40ms) destined for further Fourier processing. Thus, a 1min long signal would yield a file of 1024 * 1465 complex numbers organized in a complex matrix of shape (1024, 1465).

Having a large amount of data sampled at a high P RF allows for customization to worstly-conditioned scenarios, by using only part of the data and downsampling the signals. We set ourselves in a particular scenario, making use of only 3% of the available data, downsampled 3 times (for a P RF of about 8kHz).

We transform the inputs, as we do the synthetic data, to 20 × 20 HPD matrices, and also use the same architecture and learning scheme.

C. Description of the Aveillant database

To further asses the algorithm's sturdiness, we repeat experiments on a dataset provided by the Aveillant company.

The data's main focus is the discrimination of the I1-D drone from other airborne targets. Although three classes are proposed ("bird", "car", "drone"), for which there are respectively 10, 2 and 12 files, we chose to collapse the task to a 2-class problem ("other", "drone"). Each file in the dataset is a series of nearly continuous frames of length 279ms sampled to M = 2048 points, i.e. at a P RF of about 8kHz. The number of frames in the files varies from 72 to 823 with a median value of 207, i.e. the durations of the recordings vary from 20s to 4min with a median duration of 1min. As for the NATO data, classification is performed frame by frame, i.e. one data point consists of 2048 consecutive complex points. A data point is split through a sliding window of length n = 128 with a hop length of 32 (so a 75% overlap); a single covariance matrix of size 128 × 128 is thus sampled from all the resulting sub-frames and passed on to an SPDNet or HPDNet. We implement a 2-layer network reducing the matrix dimension from 20 to 16 and 8. The following paragraph describes the results on both datasets.

D. Comparison of SPDNet and HPDNet

Upon these 20 × 20 Hermitian matrices, we build a 2layer HPDNet with hidden dimensions 16 and 8. After the LogEig Euclidean mapping, the resulting matrix is passed to the real domain via the C2R layer and vectorized to a 8 * 8 = 64-dimensional vector, which is finally mapped to the 3-dimensional space of class distribution. All experiments are run on a 5-fold cross-validation using 25% of all data for validation, and tested on another fixed 25% held-out set.

The remaining 50% is used for training. We compare the HPDNet described above with the equivalent SPDNet. In this experimental configuration, the performance of both models, measured as overall accuracy over the synthetic test set, are as follows:

• SPDNet: 90.2% ± 3.28 • HPDNet: 93.5% ± 1.13 As for the NATO dataset, we get:

• SPDNet: 82.8% ± 0.72 • HPDNet: 86.5% ± 0.53 As for the Aveillant dataset, we get:

• SPDNet: 86.7% ± 1.61 • HPDNet: 89.2% ± 1.42 We see that for all datasets the HPDNet performs slightly better, while also presenting a smaller standard deviation, which indicates higher stability from using complex values. For visual purpose, we plot the training and validation accuracy curves along with the confusion matrix upon convergence for one cross-validation instance of the training on both NATO and Aveillant data using the HPDNet in figures 4 and 5.

E. Model robustness to lack of data

In the context of machine learning on radar data, one is often faced with an unevitable lack of data, due to the underlying cost and sensitivity. To simulate such a scenario, we repeat the experiment using only 10% of the available training data. The performance obtained on the synthetic data are then:

• SPDNet: 83.2% ± 4.18 • HPDNet: 85.1% ± 1.49 As for the NATO data:

• SPDNet: 79.9% ± 3.18 • HPDNet: 81.4% ± 1.95 As for the Aveillant dataset, we get:

• SPDNet: 83.2% ± 1.97 • HPDNet: 85.1% ± 1.56 Although the gap is reduced, the HPDNet still outperforms the SPDNet in both cases; this results was not obvious, as, because it involves complex numbers, the HPDNet intrinsically exhibits twice as many parameters than its real counterpart. Allowing too many paramaters to a neural network can lead to overfitting on the training set, resulting in worsened performance at test time; however, we do not observe this phenomenon, justifying the interest of correctly handling the geometric structure of complex numbers. Again, the HPDNet exhibits a smaller standard deviation, which makes it a suitable generalisation of the SPDNet.

CONCLUSION

Radar signals exhibit a strong structure, which makes way for a variety of meaningful representations; the covariance of the underlying Gaussian process, modeled as a Symmetric Positive Definite (SPD) matrix, is one of them. However, this representation's major drawback is the discarding of phase information; in this work, we introduce a neural network on Hermitian Positive Definite (HPD) matrices, which model the complex covariance of the signal, thus integrating any phase information in the raw signal. The theoretical background of this model is grounded in both complex-valued, and SPDvalued neural networks. We experimentally show, on synthetic micro-Doppler data and larger-scale real-world NATO and Aveillant datasets, that the HPDNet not only increases the performance compared to a real-valued SPDNet counterpart, but also exhibits higher stability and robustness to lack of data, all the while possessing a small amount of parameters compared to standard neural networks. For all these reasons, we believe it to be a suitable and scalable model for radar micro-Doppler classification.

Fig. 1 .

 1 Fig.1. Illustration of the BiMap layer. These are successively set in order to reduce the feature dimension, and more importantly separate the data. Each BiMap is followed by the ReEig activation.

Fig. 2 .

 2 Fig. 2. Illustration of the LogEig. The final feature SPD manifold is logarithmically mapped to a Euclidean space to allow for classification.

Fig. 3 .

 3 Fig. 3. Photographs and corresponding models of the three drones to classify. Scatter points along their surface define the reflected radar wave which in turn constitutes the received micro-Doppler signal.

Fig. 4 .

 4 Fig.4. Training and validation accuracy during the learning of the HPDNet on NATO data, along with the final normalized confusion matrix; true class is written on the left, predicted class underneath. Full class names can be deduced from the letters displayed on the matrix; for instance, class 'B' designates the birds, 'Pc' the Phantom drone with carbon blades. We notice that the Skywalker and X8 drones are never predicted, perhaps due to their challengingly evasive presence within the database.

Fig. 5 .

 5 Fig.5. Training and validation accuracy during the learning of the HPDNet on NATO data, along with the final normalized confusion matrix; true class is written on the left, predicted class underneath. Full class names can be deduced from the letters displayed on the matrix; for instance, class 'B' designates the birds, 'Pc' the Phantom drone with carbon blades. We notice that the Skywalker and X8 drones are never predicted, perhaps due to their challengingly evasive presence within the database.

ACKNOWLEDGMENT

We would like to thank the NATO working group SET245 for providing the drone micro-Doppler database and allowing for publication of classification results.


Daniel Brooks

Olivier Schwander

Frédéric Barbaresco

Jean-Yves Schneider

Matthieu Cord

Riemannian batch normalization for SPD neural networks

Covariance matrices have attracted attention for machine learning applications due to their capacity to capture interesting structure in the data. The main challenge is that one needs to take into account the particular geometry of the Riemannian manifold of symmetric positive definite (SPD) matrices they belong to. In the context of deep networks, several architectures for these matrices have recently been proposed. In our article, we introduce a Riemannian batch normalization (batchnorm) algorithm, which generalizes the one used in Euclidean nets. This novel layer makes use of geometric operations on the manifold, notably the Riemannian barycenter, parallel transport and non-linear structured matrix transformations. We derive a new manifold-constrained gradient descent algorithm working in the space of SPD matrices, allowing to learn the batchnorm layer. We validate our proposed approach with experiments in three different contexts on diverse data types: a drone recognition dataset from radar observations, and on emotion and action recognition datasets from video and motion capture data. Experiments show that the Riemannian batchnorm systematically gives better classification performance compared with leading methods and a remarkable robustness to lack of data.

Introduction and related works

Covariance matrices are ubiquitous in any statistical related field but their direct usage as a representation of the data for machine learning is less common. However, it has proved its usefulness in a variety of applications: object detection in images [START_REF] Tuzel | Region Covariance: A Fast Descriptor for Detection and Classification[END_REF], analysis of Magnetic Resonance Imaging (MRI) data [START_REF] Pennec | A Riemannian Framework for Tensor Computing[END_REF], classification of time-series for Brain-Computer Interfaces [START_REF] Barachant | Classification of covariance matrices using a Riemannian-based kernel for BCI applications[END_REF] (BCI). It is particularly interesting in the case of temporal data since a global covariance matrix is a straightforward way to capture and represent the temporal fluctuations of data points of different lengths. The main difficulty is that these matrices, which are symmetric positive definite (SPD), cannot be seen as points in a Euclidean space: the set of SPD matrices is a curved Riemannian manifold, thus tools from non-Euclidean geometry must be used; see [START_REF] Bhatia | Positive Definite Matrices[END_REF] for a plethora of theoretical justifications and 33rd Conference on Neural Information Processing Systems (NeurIPS 2019), Vancouver, Canada.

properties on the matter. For this reason most of classification methods (which implicitly make the hypothesis of a Euclidean input space) cannot be used successfully.

Interestingly, relatively simple machine learning techniques can produce state-of-art results as soon as the particular Riemannian geometry is taken into account. This is the case for BCI: [START_REF] Barachant | Classification of covariance matrices using a Riemannian-based kernel for BCI applications[END_REF][START_REF] Barachant | Multiclass Brain-Computer Interface Classification by Riemannian Geometry[END_REF] use nearest barycenter (but with Riemannian barycenter) and SVM (but on the tangent space of the barycenter of the data points) to successfully classify covariances matrices computed on electroencephalography multivariate signals (EEG); in the same field, [START_REF] Yger | Supervised LogEuclidean Metric Learning for Symmetric Positive Definite Matrices[END_REF] propose kernel methods for metric learning on the SPD manifold . Another example is in MRI, where [START_REF] Pennec | A Riemannian Framework for Tensor Computing[END_REF][START_REF] Arsigny | Log-Euclidean metrics for fast and simple calculus on diffusion tensors[END_REF] develop a k-nearest neighbors algorithm using a Riemannian distance. Motion recognition from motion skeletal data also benefits from Riemannian geometry, as exposed in [START_REF] Cavazza | When Kernel Methods Meet Feature Learning: Log-Covariance Network for Action Recognition From Skeletal Data[END_REF], [START_REF] Huang | Building Deep Networks on Grassmann Manifolds[END_REF] and [START_REF] Huang | Deep Learning on Lie Groups for Skeleton-based Action Recognition[END_REF]. In the context of neural networks, an architecture (SPDNet) specifically adapted for these matrices has been proposed [START_REF] Huang | A Riemannian Network for SPD Matrix Learning[END_REF]. The overall aspect is similar to a classical (Euclidean) network (transformations, activations and a final stage of classification) but each layer processes a point on the SPD manifold; the final layer transforms the feature manifold to a Euclidean space for further classification. More architectures have followed, proposing alternatives to the basic building blocks: in [START_REF] Dong | Deep Manifold Learning of Symmetric Positive Definite Matrices with Application to Face Recognition[END_REF] and [START_REF] Gao | Learning a Robust Representation via a Deep Network on Symmetric Positive Definite Manifolds[END_REF], a more lightweight transformation layer is proposed; in [START_REF] Zhang | Deep Manifold-to-Manifold Transforming Network[END_REF] and [START_REF] Chakraborty | ManifoldNet: A Deep Network Framework for Manifold-valued Data[END_REF], the authors propose alternate convolutional layers, respectively based on multi-channel SPD representation and Riemannian means; a recurrent model is further proposed in [START_REF] Chakraborty | A Statistical Recurrent Model on the Manifold of Symmetric Positive Definite Matrices[END_REF]; in [START_REF] Mao | COSONet: Compact Second-Order Network for Video Face Recognition[END_REF] and [START_REF] Li | Towards Faster Training of Global Covariance Pooling Networks by Iterative Matrix Square Root Normalization[END_REF], an approximate matrix square-root layer replaces the final Euclidean projection to lighten computational complexity. In [START_REF] Brooks | Exploring Complex Time-series Representations for Riemannian Machine Learning of Radar Data[END_REF], a SPD neural network is appended to a fully-convolutional net to improve on performance and robustness to data scarcity. All in all, most of the developments focus on improving or modifying existing blocks in an effort to converge to their most relevant form, both theoretically and practically; in this work, we propose a new building block for SPD neural networks, inspired by the well-known and well-used batch normalization layer [START_REF] Ioffe | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift[END_REF]. This layer makes use of batch centering and biasing, operations which need to be defined on the SPD manifold. As an additional, independent SPD building block, this novel layer is agnostic to the particular way the other layers are computed, and as such can fit into any of the above architectures. Throughout the paper we choose to focus on the original architecture proposed in [START_REF] Huang | A Riemannian Network for SPD Matrix Learning[END_REF]. Although the overall structure of the original batchnorm is preserved, its generalization to SPD matrices requires geometric tools on the manifold, both for the forward and backward pass. In this study, we further assess the particular interest of batch-normalized SPD nets in the context of learning on scarce data with lightweight models: indeed, many fields are faced with costly, private or evasive data, which strongly motivates the exploration of architectures naturally resilient to such challenging situations. Medical imagery data is well-known to face these issues [START_REF] Pennec | A Riemannian Framework for Tensor Computing[END_REF], as is the field of drone radar classification [START_REF] Brooks | Temporal Deep Learning for Drone Micro-Doppler Classification[END_REF], which we study in this work: indeed, radar signal acquisition is prohibitively expensive, the acquired data is usually of confidential nature, and drone classification in particular is plagued with an ever-changing pool of targets, which we can never reasonably hope to encapsulate in comprehensive datasets. Furthermore, hardware integration limitations further motivate the development of lightweight models based on a powerful representation of the data. As such, our contributions are the following:

• a Riemannian batch normalization layer for SPD neural networks, respecting the manifold's geometry;

• a generalized gradient descent allowing to learn the batchnorm layer;

• extensive experimentations on three datasets from three different fields, (experiments are made reproducible with our open-source PyTorch library, released along with the article).

Our article is organized as follows: we first recall the essential required tools of manifold geometry; we then proceed to describe our proposed Riemannian batchnorm algorithm; next, we devise the projected gradient descent algorithm for learning the batchnorm; finally, we validate experimentally our proposed architecture.

Riemannian metrics on SPD matrices

In a general setting, a Riemannian distance δ R (P 1 , P 2 ) between two points P 1 and P 2 on a manifold is defined as the length of the geodesic γ P1→P2 , i.e. the shortest parameterized curve ξ(t), linking them:

δ R (P 1 , P 2 ) = inf ξ | (ξ(0)=P1,ξ(1)=P2) 1 0 ds(t)dt ds(t) 2 = ξ(t) T F ξ(t) ξ(t) (1)
In the equation above, ds is the infinitesimal distance between two close points and F is the metric tensor, which defines a local metric at each point on the manifold. ξ is the velocity of the curve, sometimes noted dξ. For manifolds of exponential family distributions, F is none other than the Fisher information matrix (FIM) (the inverse of which defines well-known Cramer-Rao bound), which is the Hessian matrix of the entropy. This connection between entropy and differential metrics was first made in 1945 by C.R. Rao [START_REF] Rao | Information and the Accuracy Attainable in the Estimation of Statistical Parameters[END_REF] and in 1943 by M. Fréchet [START_REF] Fréchet | Sur l'extension de certaines evaluations statistiques au cas de petits echantillons[END_REF], and further axiomatized in 1965 by N.N. Chentsov [START_REF] Cencov | Statistical Decision Rules and Optimal Inference[END_REF]. Then, in a 1976 confidential report cited in [START_REF] Atkinson | Rao's Distance Measure[END_REF], S.T. Jensen derived the infinitesimal distance between two centered multivariate distributions ds(ξ) 2 = 1 2 tr(ξ -1 ξξ -1 ξ). Such distributions being defined entirely by the covariance matrix, they are isomorphic to the SPD manifold, so the integration of ds along the geodesic leads to the globally-defined natural distance on S + * [START_REF] Marceau-Caron | Natural Langevin Dynamics for Neural Networks[END_REF], also called affine-invariant Riemannian metric (AIRM) [START_REF] Pennec | A Riemannian Framework for Tensor Computing[END_REF], which can be expressed using the standard Frobenius norm || • || F :

δ R (P 1 , P 2 ) = 1 2 ||log(P -1 2 1 P 2 P -1 2 1 )|| F (2)
The interested reader may note that while the above metric is the correct one from the information geometric viewpoint, it is notoriously computation-heavy. Other metrics or divergences, either closely approximate it or provide an alternate theoretical apporach, while contributing the highly desirable property of lightweight computational complexity, especially in the modern context of machine learning. Notable examples may include the usage of the Fisher-Bures metric [START_REF] Sun | Fisher-Bures Adversary Graph Convolutional Networks[END_REF], the Bregman divergence [START_REF] Boissonnat | Bregman Voronoi diagrams[END_REF][START_REF] Siahkamari | Learning Bregman Divergences[END_REF][START_REF] Banerjee | Clustering with Bregman Divergences[END_REF], and optimal transport [START_REF] Arjovsky | Wasserstein Generative Adversarial Networks[END_REF].

Another matter of importance is the definition of the natural mappings to and from the manifold and its tangent bundle, which groups the tangent Euclidean spaces at each point in the manifold. At any given reference point P 0 ∈ S + * , we call logarithmic mapping Log P0 of another point P ∈ S + * at P 0 the corresponding vector S in the tangent space T P0 at P 0 . The inverse operation is the exponential mapping Exp P0 . In S + * , both mappings (not to be confused with the matrix log and exp functions) are known in closed form [START_REF] Amari | Information Geometry and Its Applications[END_REF]:

∀S ∈ T P0 , Exp P0 (S) = P 1 2 0 exp(P -1 2 0 SP -1 2 0 )P 1 2 0 ∈ S + * (3a) ∀P ∈ S + * , Log P0 (P ) = P 1 2 0 log(P -1 2 0 P P -1 2 0 )P 1 2 0 ∈ T P0 (3b)

Riemannian barycenter

The first step of the batchnorm algorithm is the computation of batch means; it may be possible to use the arithmetic mean 1 N i≤N P i of a batch B of N SPD matrices {P i } i≤N , we will rather use the more geometrically appropriate Riemannian barycenter G , also known as the Fréchet mean [START_REF] Yang | Riemannian median, geometry of covariance matrices and radar target detection[END_REF] , which we note Bar({P i } i≤N ) or Bar(B). The Riemannian barycenter has shown strong theoretical and practical interest in Riemannian data analysis [START_REF] Pennec | A Riemannian Framework for Tensor Computing[END_REF], which justifies its usage in this context. By definition, G is the point on the manifold that minimizes inertia in terms of the Riemannian metric defined in equation 2. The definition is trivially extensible to a weighted Riemannian barycenter, noted Bar w ({P i } i≤N ) or Bar w (B), where the weights w := {w i } i≤N respect the convexity constraint:

G = Bar w ({P i } i≤N ) := arg min G∈S + * N i=1 w i δ 2 R (G, P i ) , with w i ≥ 0 i≤N w i = 1 (4)
Figure 1: Illustration of one iteration of the Karcher flow [START_REF] Karcher | Riemannian center of mass and mollifier smoothing[END_REF].

When N = 2, i.e. when w = {w, 1 -w}, a closed-form solution exists, which exactly corresponds to the geodesic between two points P 1 and P 2 , parameterized by w ∈ [0, 1] [12]:

Bar (w,1-w) (P 1 , P 2 ) = P 1 2 2 P -1 2 2 P 1 P -1 2 2 w P 1 2
2 , with w ≥ 0

Unfortunately, when N > 2, the solution to the minimization problem is not known in closed-form: thus G is usually computed using the so-called Karcher flow algorithm [START_REF] Karcher | Riemannian center of mass and mollifier smoothing[END_REF][START_REF] Yang | Riemannian median, geometry of covariance matrices and radar target detection[END_REF] , which we illustrate in Figure 1 . In short, the Karcher flow is an iterative process in which data points projected using the logarithmic mapping (equation 3b) are averaged in tangent space and mapped back to the manifold using the exponential mappings (equation 3a) , with a guaranteed convergence on a manifold with constant negative curvature, which is the case for S + * [START_REF] Karcher | Riemannian center of mass and mollifier smoothing[END_REF]. The initialization of G is arbitrary, but a reasonable choice is the arithmetic mean. A key point is that convergence is guaranteed on a manifold with constant negative curvature, which is the case for the SPD manifold S + * [START_REF] Karcher | Riemannian center of mass and mollifier smoothing[END_REF]. Another point of interest is that selecting K = 1 (that is, only one iteration of the flow) and α = 1 (unit step size) in the Karcher algorithm , corresponds exactly to the barycenter from the Log-Euclidean metric viewpoint [START_REF] Pennec | A Riemannian Framework for Tensor Computing[END_REF]. We actually use this setting in the layer: as the batch barycenter is but a noisy estimation of the true barycenter, a lax approximation is sufficient, and also allows for much faster inference.

Centering SPD matrices using parallel transport

The Euclidean batchnorm involves centering and biasing the batch B, which is done via subtraction and addition. However on a curved manifold, there is no such group structure in general, so these seemingly basic operations are ill-defined. To shift SPD matrices around their mean G or towards a bias parameter G, we propose to rather use parallel transport on the manifold [START_REF] Amari | Information Geometry and Its Applications[END_REF]. In short, the parallel transport (PT) operator Γ P1→P2 (S) of a vector S ∈ T P1 in the tangent plane at P 1 , between P 1 , P 2 ∈ S + * defines the path from P 1 to P 2 such that S remains parallel to itself in the tangent planes along the path. The geodesic γ P1→P2 is itself a special case of the PT, when S is chosen to be the direction vector γ P1→P2 (0) from P 1 to P 2 . The expression for PT is known on S + * :

∀S ∈ T P1 , Γ P1→P2 (S) = (P 2 P -1 1 ) 1 2 S (P 2 P -1 1 ) 1 2 ∈ T P2 (6)
The equation above defines PT for tangent vectors, while we wish to transport points on the manifold.

To do so, we simply project the data points to the tangent space using the logarithmic mapping , parallel transport the resulting vector from Eq. 6 which we then map back to the manifold using exponential mapping . It can be shown (see [START_REF] Yair | Parallel Transport on the Cone Manifold of SPD Matrices for Domain Adaptation[END_REF], appendix C for a full proof) that the resulting operation, which we call SPD transport, turns out to be exactly the same as the formula above, which is not an obvious result in itself. By abuse of notation, we also use Γ P1→P2 to denote the SPD transport. Therefore, we can now define the centering of a batch of matrices {P i } i≤N with Riemannian barycenter G as the PT from G to the identity I d , and the biasing of the batch towards a parametric SPD matrix G as the PT from I d to G.

Batch centering and biasing

We now have the tools to define the batch centering and biasing:

Centering from G := Bar(B): ∀i ≤ N, Pi = Γ G→I d (P i ) = G -1 2 P i G -1 2 (7a) Biasing towards parameter G: ∀i ≤ N, Pi = Γ I d →G ( Pi ) = G 1 2 Pi G 1 2 (7b)
3 Batchnorm for SPD data

In this section we introduce the Riemannian batch normalization (Riemannian BN, or RBN) algorithm for SPD matrices. We first briefly recall the basic architecture of an SPD neural network.

Basic layers for SPD neural network

The SPDNet architecture mimics that of classical neural networks with a first stage devoted to compute a pertinent representation of the input data points and a second stage which allows to perform the final classification. The particular structure of S + * , the manifold of SPD matrices, is taken into account by layers crafted to respect and exploit this geometry. The layers introduced in [START_REF] Huang | A Riemannian Network for SPD Matrix Learning[END_REF] are threefold:

The BiMap (bilinear transformation) layer, analogous to the usual dense layer; the induced dimension reduction eases the computational burden often found in learning algorithms on SPD data:

X (l) = W (l) T P (l-1) W (l) with W (l) semi-orthogonal (8)
The ReEig (rectified eigenvalues activation) layer, analogous to the ReLU activation; it can also be seen as a eigen-regularization, protecting the matrices from degeneracy:

X (l) = U (l) max(Σ (l) , I n )U (l) T , with P (l) U (l) Σ (l) U (l) T (9)
The LogEig (log eigenvalues Euclidean projection) layer:

X (l) = vec( U (l) log(Σ (l)
)U (l) T ) , with again U (l) the eigenspace of P (l) [START_REF] Bhatia | Positive Definite Matrices[END_REF] This final layer has no Eucidean counterpart: its purpose is the projection and vectorization of the output feature manifold to a Euclidean space, which allows for further classification with a traditional dense layer. As stated previously, it is possible to envision different formulations for each of the layers defined above (see [START_REF] Dong | Deep Manifold Learning of Symmetric Positive Definite Matrices with Application to Face Recognition[END_REF][START_REF] Zhang | Deep Manifold-to-Manifold Transforming Network[END_REF][START_REF] Mao | COSONet: Compact Second-Order Network for Video Face Recognition[END_REF] for varied examples). Our following definition of the batchnorm can fit any formulation as it remains an independent layer.

Statistical distribution on SPD matrices

In traditional neural nets, batch normalization is defined as the centering and standardization of the data within one batch, followed by the multiplication and addition by parameterized variance and bias, to emulate the data sampling from a learnt Gaussian distribution. In order to generalize to batches of SPD matrices, we must first define the notion of Gausian density on S + * . Although this definition has not yet been settled for good, several approaches have been proposed. In [START_REF] Jaquier | Gaussian mixture regression on symmetric positive definite matrices manifolds: Application to wrist motion estimation with sEMG[END_REF], the authors proceed by introducing mean and variance as second-and fourth-order tensors. On the other hand, [START_REF] Said | Riemannian Gaussian Distributions on the Space of Symmetric Positive Definite Matrices[END_REF] derive a scalar variance. In another line of work synthesized in [START_REF] Barbaresco | Jean-Louis Koszul and the Elementary Structures of Information Geometry[END_REF], which we adopt in this work, the Gaussian density is derived from the definition of maximum entropy on exponential families using information geometry on the cone of SPD matrices. In this setting, the natural parameter of the resulting exponential family is simply the Riemannian mean; in other words, this means the notion of variance, which appears in the Eucidean setting, takes no part in this definition of a Gaussian density on S + * . Specifically, such a density p on SPD matrices P of dimension n writes:

p(P ) ∝ det(α G -1 )e -tr(α G -1 P ) , with α = n + 1 2 (11)
In the equation above, G is the Riemannian mean of the distribution. Again, there is no notion of variance: the main consequence is that a Riemannian BN on SPD matrices will only involve centering and biasing of the batch.

Final batchnorm algorithm

While the normalization is done on the current batch during training time, the statistics used in inference are computed as running estimations. For instance, the running mean over the training set, noted G S , is iteratively updated at each batch. In a Euclidean setting, this would amount to a weighted average between the batch mean and the current running mean, the weight being a momentum typically set to 0.9. The same concept holds for SPD matrices, but the running mean should be a Riemannian mean weighted by η, i.e. Bar (η,1-η) (G S , G B ), which amounts to transporting the running mean towards current batch mean by an amount (1 -η) along the geodesic. We can now write the full RBN algorithm 1. In practice, Riemannian BN is appended after each BiMap layer in the network.

Algorithm Pi ← Γ G S →I d (P i ) // center batch using set statistics 3:

Pi ← Γ I d →G ( Pi ) // bias batch using learnt parameter 4: end for return normalized batch { Pi } i≤N

Learning the batchnorm

The specificities of a the proposed batchnorm algorithm are the non-linear manipulation of manifold values in both inputs and parameters and the use of a Riemannian barycenter. Here we present the two results necessary to correctly fit the learning of the RBN in a standard back-propagation framework.

Learning with SPD constraint

The bias parameter matrix G of the RBN is by construction constrained to the SPD manifold. However, noting L the network's loss function, the usual Euclidean gradient ∂L ∂G , which we note ∂G eucl , has no particular reason to respect this constraint. To enforce it, ∂G eucl is projected to the tangent space of the manifold at G using the manifold's tangential projection operator ΠT G , resulting in the tangential gradient ∂G riem . The update is then obtained by computing the geodesic on the SPD manifold emanating from G in the direction ∂G riem , using the exponential mapping defined in equation 3a. Both operators are known in S + * [START_REF] Yger | A review of kernels on covariance matrices for BCI applications[END_REF]:

∀P, ΠT G (P ) = G P + P T 2 G ∈ T G ⊂ S + (12)
We illustrate this two-step process in Figure 2, explained in detail in [START_REF] Edelman | The Geometry of Algorithms with Orthogonality Constraints[END_REF], which allows to learn the parameter in a manifold-constrained fashion. However, this is still not enough for the optimization of the layer, as the BN involves not simply G and G, but G 1 2 and G -1 2 , which are structured matrix functions of G, i.e. which act non-linearly on the matrices' eigenvalues without affecting its associated eigenspace. The next subsection deals with the backpropagation through such functions.

Figure 2: Illustration of manifold-constrained gradient update. The Euclidean gradient is projected to the tangent space, then mapped to the manifold.

Structured matrix backpropagation

Classically, the functions involved in the chain rule are vector functions in R n [START_REF] Lecun | Gradient-based learning applied to document recognition[END_REF], whereas we deal here with structured (symmetric) matrix functions in the S + * , specifically the square root (•)

1 2 for the bias and the inverse square root (•) -1 2 for the barycenter (in equations 7b 7a). A generalization of the chain rule to S * + is thus required for the backpropagation through the RBN layer to be correct. Note that a similar requirement applies to the ReEig and LogEig layers, respectively with a threshold and log function. We generically note f a monotonous non-linear function; both (•) 1 2 and (•) -1 2 check out this hypothesis. A general formula for the gradient of f , applied on a SPD matrix' eigenvalues (σ i ) i≤n grouped in Σ's diagonal, was independently developed by [START_REF] Ionescu | Matrix Backpropagation for Deep Networks with Structured Layers[END_REF] and [START_REF] Brodskiȋ | Thirteen Papers on Functional Analysis and Partial Differential Equations[END_REF]. In short: given the function P -→ X := f (P ) and the succeeding gradient ∂L (l+1) ∂X , the output gradient ∂L (l) ∂P is:

∂L (l) ∂P = U L (U T ( ∂L (l+1) ∂X )U ) U T (13)
The equation above, also decribed in [START_REF] Nielsen | Matrix Information Geometry[END_REF], is called the Daleck ȋi-Kre ȋn formula and dates back to 1956, (but was translated from Russian 9 years later), predating the other formulation by 60 years. It involves the eigenspace U of the input matrix P , and the Loewner matrix L, or finite difference matrix defined by:

L ij = f (σi)-f (σj ) σi-σj if σ i = σ j f (σ i ) otherwise (14)
In the case at hand, (•)

-1 2 = -1 2 (•) -3 2 and (•) 1 2 = 1 2 (•) -1 2 .
We credit [START_REF] Engin | DeepKSPD: Learning Kernel-matrix-based SPD Representation for Fine-grained Image Recognition[END_REF] for first showing the equivalence between the two cited formulations, of which we expose the most concise.

In summary, the Riemannian barycenter (approximation via the Karcher flow for a batch of matrices, or exact formulation for two matrices), the parallel transport and its extension on the SPD manifold, the SPD-constrained gradient descent and the derivation of a non-linear SPD-valued structured function's gradient allow for training and inference of the proposed Riemannian batchnorm algorithm. 72.6% ± 0.61 82.3% ± 0.80 88.7 88.7 88.7% ± 0.83 73.4% ± 3.66 69.7% ± 1.12 Acc. (10% data) 69.1% ± 0.97 77.7 77.7 77.7% ± 0.95 65.6% ± 2.74 61.1% ± 3.50 67.1% ± 2.17

Experiments

Here we evaluate the gain in performance of the RBN against the baseline SPDNet on different tasks: radar data classification, emotion recognition from video, and action recognition from motion capture data. We call the depth L of an SPDNet the number of BiMap layers in the network, and denote the dimensions as {n 0 , • • • , n L }. The vectorized input to the final classification layer is thus of length n 2

L . All networks are trained for 200 epochs using SGD with momentum set to 0.9 with a batch size of 30 and learning rate 5e -3 , 1e -2 or 5e -2 . We provide the data in a pre-processed form alongside the PyTorch [START_REF] Paszke | Automatic differentiation in PyTorch[END_REF] code for reproducibility purposes. We call SPDNetBN an SPDNet using RBN after each BiMap layer. Finally, we also report performances of shallow learning method on SPD data, namely a minimum Riemannian distance to Riemannian mean scheme (MRDRM), described in [START_REF] Barachant | Multiclass Brain-Computer Interface Classification by Riemannian Geometry[END_REF], in order to bring elements of comparison between shallow and deep learning on SPD data.

Drones recognition

Our first experimental target focuses on drone micro-Doppler [START_REF] Chen | Micro-Doppler effect in radar: phenomenon, model, and simulation study[END_REF] radar classification. First we validate the usage of our proposed method over a baseline SPDNet, and also compare to state-ofthe-art deep learning methods. Then, we study the models' robustness to lack of data, a challenge which, as stated previously, plagues the task of radar classification and also a lot of different tasks. Experiments are conducted on a confidential dataset of real recordings issued from the NATO organization1 . To spur reproducibility, we also experiment on synthetic, publicly available data.

Radar data description A radar signal is the result of an emitted wave reflected on a target; as such, one data point is a time-series of N values, which can be considered as multiple realizations of a locally stationary centered Gaussian process, as done in [START_REF] Charon | A new approach for target detection in radar images based on geometric properties of covariance matrices' spaces[END_REF]. The signal is split in windows of length n = 20, the series of which a single covariance matrix of size 20 * 20 is sampled from, which represents one radar data point. The NATO data features 10 classes of drones, whereas the synthetic data is generated by a realistic simulator of 3 different classes of drones following the protocol described in [START_REF] Brooks | Temporal Deep Learning for Drone Micro-Doppler Classification[END_REF]. We chose here to mimick the real dataset's configuration, i.e. we consider a couple of minutes of continuous recordings per class, which correspond to 500 data points per class.

Comparison of SPDNetBN against SPDNet and radar state-of-the-art We test the two SPD-based models in a {20, 16, 8}, 2-layer configuration for the synthetic data, and in a {20, 16, 14, 12, 10, 8}, 5-layer configuration for the NATO data, over a 5-fold cross-validation, split in a train-test of 75% -25%. We also wish to compare the Riemannian models to the common Euclidean ones, which currently consitute the state-of-the-art in micro-Doppler classification. We compare two fully convolutional networks (FCN): the first one is used as given in [START_REF] Brooks | Temporal Deep Learning for Drone Micro-Doppler Classification[END_REF]; for the second one, the number of parameters is set to approximately the same number as for the SPD neural networks, which amounts to an unusually small deep net. All in all, the SPDNet, SPDNetBN and small FCN on the one hand, and the full-size FCN on the other hand respectively have approximately 500 and 10000 parameters. Table 1 reports the average accuracies and variances on the NATO data. We observe a strong gain in performance on the SPDNetBN over the SPDNet and over the small FCN, which validates the usage of the batchnorm along with the exploitation of the geometric structure underlying the data. All in all, we reach better performance with much fewer parameters.

Finally, in the interest of convergence analysis, we also report learning curves for the model's accuracy with and without Riemannian batchnorm in figure 3. Robustness to lack of data As stated previously, it is of great interest to consider the robustness of learning algorithms when faced with a critically low amount of data. The last line in table 1 shows that when given only 10% of available training data, the SPD-based models remain highly robust to the lack of data while the FCNs plummet. Further, we study robustness on synthetic data, artificially varying the amount of training data while comparing performance over the same test set. As the simulator is unbounded on potential training data, we also increase the initial training set up to double its original size. Results are reported in Figure 4. We can conclude from these that the SPDNetBN both exhibits higher robustness to lack of data and performs much better than the state-of-the-art deep method with much fewer parameters. When the available training data allowed skyrockets, we do observe that the FCN comes back to par with the SPDNetBN to the point of outperforming it by a small margin in the extremal scenario; in the meantime, the SPDNet lags behind by a large margin to the SPDNetBN, which thus seems to benefit strongly from the normalization. In any case, the manifold framework seems well suited in a scarce data learning context, especially considering the introduced normalization layers, which again pinpoints the interest of taking into account the geometric structure of the data, all the while without introducing prior knowledge during training.

Other experiments

Here we validate the use of the RBN on a broader set of tasks. We first clarify we do not necessarily seek state-of-the-art in the general sense for the following tasks, but rather in the specific case of the family of SPD-based methods. Our own implementation (as an open PyTorch library) of the SPDNet's performances match that in [START_REF] Huang | A Riemannian Network for SPD Matrix Learning[END_REF], ensuring a fair comparison. Emotion recognition In this section we experiment on the AFEW dataset [START_REF] Dhall | Static facial expression analysis in tough conditions: Data, evaluation protocol and benchmark[END_REF], which consists of videos depicting 7 classes of emotions; we follow the setup and protocol in [START_REF] Huang | A Riemannian Network for SPD Matrix Learning[END_REF]. Results for 4 architectures are summarized in table 2. In comparison, the MRDRM yields a 20.5% accuracy. We observe a consistent improvement using our normalization scheme. This dataset being our largest-scale experiment, we also report the increase in computation time using the RBN, specifically for the deepest net: one training lasted on average 81s for SPDNet, and 88s (+8.6%) for SPDNetBN.

Action recognition

In this section we experiment on the HDM05 motion capture dataset. We use the same experimental setup as in [START_REF] Huang | A Riemannian Network for SPD Matrix Learning[END_REF] results are shown in table 3. Note that all tested models exhibit noticeable variance depending on the weights initialization and the initial random split of the dataset; the results displayed were obtained by setting a fixed seed of 0 for both. In comparison, the MRDRM yields a 27.3% ± 1.06 accuracy. Again, we validate a better performance using the batchnorm.

Conclusion

We proposed a batch normalization algorithm for SPD neural networks, mimicking the orginal batchnorm in Euclidean neural networks. The algorithm makes use of the SPD Riemannian manifold's geometric structure, namely the Riemannian barycenter, parallel transport, and manifold-constrained backpropagation through non-linear structured functions on SPD matrices. We demonstrate a systematic, and in some cases considerable, performance increase across a diverse range of data types. An additional observation is the better robustness to lack of data compared to the baseline SPD neural network and to a state-of-the-art convolutional network, as well as better performance than a well-used, more traditional Riemannian learning method (the closest-barycenter scheme). The overall performances of our proposed SPDNetBN makes it a suitable candidate in learning scenarios where data is structured, scarce, and where model size is a relevant issue.

Figure 3 :Figure 4 :

 34 Figure3: Test accuracy on the NATO dataset of the SPDNet with and without RBN, measured in hours. The RBN exhibits a steeper learning curve. For the same number of epochs, it does take more time overall, but reaches better accuracy much faster, allowing to reduce the number of epochs.

1

  Riemannian batch normalization on S + * , training and testing phase TRAINING PHASE Require: batch of N SPD matrices {P i } i≤N , running mean G S , bias G, momentum η 1: G B ← Bar({P i } i≤N ) // compute batch mean 2: G S ← Bar η (G S , G B ) SPD matrices {P i } i≤N , final running mean G S , learnt bias G

		// update running mean
	3: for i ≤ N do 4: Pi ← Γ G B →I d (P i ) 5: Pi ← Γ I d →G ( Pi )	// center batch // bias batch
	6: end for return normalized batch { Pi } i≤N
	TESTING PHASE
	Require: batch of N

1: for i ≤ N do 2:

Table 1 :

 1 Accuracy comparison of SPDNet, SPDNetBN and FCNs on NATO radar data, in function of amount of training data.

	Model	SPDNet	SPDNetBN	FCN		MRDRM
	# Parameters	∼ 500	∼ 500	∼ 10000	∼ 500	-
	Acc. (all data)

Table 2 :

 2 Accuracy comparison of SPDNet with and without Riemannian BN on the AFEW dataset. Model architecture {400, 50} {400, 100, 50} {400, 200, 100, 50} {400, 300, 200, 100, 50}

	SPDNet	29.9%	31.2%	34.5%	33.7%
	SPDNetBN (ours)	34.9 34.9 34.9%	35.2 35.2 35.2%	36.2 36.2 36.2%	37.1 37.1 37.1%

Table 3 :

 3 Accuracy comparison of SPDNet with and without Riemannian BN on the HDM05 dataset.

	Model architecture	SPDNet SPDNetBN (ours)
	{93, 30}	61.6%±1.35	65.2 65.2 65.2% ± 1.15

Geometry on the manifold of SPD matricesWe start by recalling some useful geometric notions on the SPD manifold, noted S + * in the following.

We would like to thank the NATO working group SET245 for providing the drone micro-Doppler database and allowing for publication of classification results.


Andreas Ebbelind
email: andreas.ebbelind@lnu.se

Understanding student teachers' professional development by looking beyond mathematics teacher education

Keywords: Professional development, teacher education, social practices

The empirical material in this paper is from a multiple case study exploring the role, if any, that social practices related to teacher education and beyond teacher education play in primary student teachers' tales of themselves as teachers-to-be. The case of Lisa is used as an example to illustrates how different past and present social practices influence how she talks about competitive teaching during different phases of her teacher education. Of particular influence is her past school-related experiences as they in turn influence how she interprets other social practices. Thus, the case of Lisa illustrates the importance of widening our research interest beyond teacher education if we want to understand the process of becoming mathematics teachers.

Introduction

Student teachers' professional development is by many researchers considered as a rather complex process (Skott, Mosvold & Sakonidis, 2018). One concern is how student teachers' school-related experiences influence how they perceive teacher education and how they view their future profession. According to [START_REF] Forgasz | Beliefs about mathematics and mathematics teaching[END_REF] prior experience impact student teachers becoming as teachers more than undergoing teacher education. To this, [START_REF] Scott | Pre-service teachers' experiences and the influences on their intentions for teaching primary school mathematics[END_REF] concludes that student teachers are less sceptical to advise from others, "outside" teacher education, than from teacher educators in mathematics. Student teachers use educated teachers at the internship or from other settings or family and friends instead of teacher educators to conclude their vision of future teaching. For example, internship, which is often part of teacher education, is said to frame how student teachers perceive the content in mathematics education courses. Thus, the teaching of the internship supervisor during internship becomes important in student teachers' professional development [START_REF] Mosvold | Positioning in identifying narratives of/about pre-service mathematics teachers in field practice[END_REF].

To widen the perspective of the "outside" this paper reports parts of a multiple case study with the aim of contributing with insights about how experience from teacher education and other past and present social practices play for generalist student teachers' tales of themselves as primary mathematics teachers-to-be. This is in line with [START_REF] Morgan | Studying discourse implies studying equity[END_REF] and [START_REF] Skott | Re-centring the individual in participatory accounts of professional identity[END_REF] that mentions that we need to look outside the immediate social practice [teacher education in this case] if we want to view what the [student] teachers bring into the situation. To be interested in any data "that point to any practice […] that appear to orient the [student] teachers' action or meaning making as they relate to the profession (Skott, 2018, p. 615)".

Thus, this paper aims to illustrate how different past and present social practices play for a primary generalist student teacher, tales as primary mathematics teacher-to-be. The empirical material used relate to how a student teacher, whom I will refer to as Lisa, talks about competitive teaching during different phases of her teacher education.

The process of becoming a mathematics teacher

To enter teacher education and become a teacher is in this paper regarded as an emerging learning process. To follow the process of developing into a teacher is partly to follow how student teachers negotiate and re-negotiate their mathematical and pedagogical knowledge. In this process [START_REF] Hodges | Unpacking personal identities for teaching mathematics within the context of prospective teacher education[END_REF] recognise student teachers bringing together different perspectives on teaching and learning mathematics, often conflicting while creating their idea of teaching mathematics. While bringing different perspectives together student teachers can, for example, position themselves in relation to future pupils, relate their participation to different educational models, expand educational models through new experience and develop a critical positioning to either the teaching at internship or the pedagogy taught and used at teacher education [START_REF] Bjuland | Professional mathematics teacher identity: Analysis of reflective narratives from discourses and activities[END_REF].

Jong (2016), as an example, illustrate that teacher education in different ways impact student teachers' professional development. However, attention is on to the characteristics of their participation that draw on ideas from past teaching experience and their cultural, socioeconomically and linguistically background. Jong concludes that becoming a teacher starts the day when children enter school for the first time and meet different educational models. These models influence the idea of teaching that student teachers have. The main result in Jong's study is that any social practices related to teaching and learning mathematics might be critical in professional development, this because social practices, past and present, are used to negotiate and re-negotiate the meaning of teaching mathematics.

The conceptual framework Patterns of Participation

The overall study and this paper start with an interest in professional development, a construct that elaborates within the conceptual framework of Patterns of Participation (PoP). Professional development can be viewed as "a process of flexibility, autonomy, and adaptation to the teaching context" (Hošpesová, Carrillo & Santos, 2018, p. 181). Learning and thus becoming a teacher is therefore considered from a situated perspective. Thus a perspective, that considers learning as participation in social practices take social, cultural and historical systems as well as prior experiences and present participation into consideration when studying professional development. A social practice as a collective way of being with a common endeavour where entities interact and reconcile/ mould together.

PoP "seeks to understand how a [student] teacher's interpretations of and contributions to immediate social interaction relate dynamically to her prior engagement in a range of other social practices" (Skott, 2013, p. 549). Skott makes an important move, according to [START_REF] Lerman | Theories in practice: Mathematics teaching and mathematics teacher education[END_REF], when focusing on the emergence of the situation, the process. "This dynamic process is studied through eliciting the teacher's interpretation of what she does in classrooms and how that relates to her prior engagement in other social practices (p. 625)". The intention with PoP is to disentangle shifts of participation in different social practices, and by doing so shed light on the role of the student teacher in the practices that emerge.

Through the conceptual framework Patterns of Participation, Skott tries to accomplish three things. First, he tries to reduce the emphasis on objectification in research about student teachers and teachers. Secondly, he tries to re-centre the student teacher and teacher in research. Finally, he tries to re-conceptualise what is known as beliefs, knowledge, and identity in participatory terms. In this quest [START_REF] Skott | Re-centring the individual in participatory accounts of professional identity[END_REF] point out that the conceptual framework PoP focuses on the pre-reified processes that precede, give rise to, what others term beliefs, knowledge, and identity.

Patterns of Participation draw on two main theoretical sources, symbolic interactionism and social practice theory. Symbolic interactionism views humans as actors and reactors in situations, and position meaning as something that one engages in when experiencing things in the situation, on the spot. Humans respond to the situation by interacting with others and with the self, and by taking [interpreting] the role of others [START_REF] Skott | Re-centring the individual in participatory accounts of professional identity[END_REF]. Social practice theory views social practices as ordered across time and space. They are linked historically to other social practices (Wenger, 1998). Social practices are stratified. Most important is that social practice theory stress that it is through the engagement with different or using different social practices within communication that individuals understand the world around them when they, for example, emerge into the teaching profession.

By considering that social practices are ordered, stratified, in time and space allow researching complex patterns in multiple social practices. To make visible and describe this plurality of relations lies in the core of PoP but also to make visible how these pluralities of relations change over time.

The overall study

The overall study adopts a multi-sited ethnographic approach. Two primary student teachers were followed for two and a half years as cases during their teacher education with a focus on their education in mathematics. Often student teachers at primary school level are generalists with no specialisation or interest to the teaching of mathematics. However, that was not the case in this study. The student teachers in this study related strongly to the teaching of mathematics in the interview before entering teacher education and they were selected as critical cases [START_REF] Flyvbjerg | Five misunderstandings about case study research[END_REF]) based on their commitment, their mathematical knowledge and interest. The choice of critical cases was based on the indicated research gap related to the lack of research related to primary student teachers that are interested in mathematics, mathematics teaching, mathematics learning and regard themselves as knowledgeable/proficient in mathematics.

In Sweden, the combined course in mathematics and mathematics education for primary school student teachers is a 30 ECTS point course in the four-year teacher education programme (240 ECTS). ECTS means European Credit Transfer System, where one year of studies is 60 ECTS (40 weeks). Student teachers have four internship periods, five weeks each, during their teacher education. Teacher education in Sweden does not have any pre-requisite of university studies before entering the education.

In this paper, only one of the student teachers is used as an example, Lisa. Lisa, who is in her early 20s, started teacher education directly after high school. Lisa's positive experiences of participating in a competitive mathematics classroom are expressed as her primary reason for entering teacher education. Lisa perceived herself as "good" at mathematics and had the highest possible grade in upper secondary school. It is also worth noticing and most important concerning this paper that Lisa described herself as extremely competitive both as a private person and as an athlete, playing soccer at a high national level.

The role of social practices in different phases

The empirical material in this paper is interview transcripts, five in total, and field-notes from observations made during the internship and the mathematics education course. During the mathematics education course, I attended 31 lectures, seminars, examinations or study-group sessions. Based on the multi-sited ethnographic approach, considering social practices as ordered in time and space and considering the "outside", persons related to Lisa were also interviewed, in this paper the internship supervisor whom I will refer to as Mr Higgins.

Lisa's professional development is described, in this paper, in three different main phases related to how she re-negotiates the use of competitions in teaching mathematics. Each phase will start with excerpts in time spread over the actual phase. Then there will be a summary accentuating the role of social practices within the phase. This section set out to offer the reader a "virtual reality" to explore [START_REF] Flyvbjerg | Five misunderstandings about case study research[END_REF], to provide the reader with situated aspects (Bjurland et al., 2012).

Phase one -The internship experience strengthens her experience of learning mathematics

In phase one, Lisa re-engages in her prior school-related experience, the social practice of her upper secondary mathematics classroom. This experience is later in this phase re-negotiated with the social practice where her internship supervisor Mr Higgins is a part.

In the first interview, conducted before entering teacher education, Lisa talks a lot about how she, from an early age, developed a passion for mathematics as a subject. She credits this passion to the fact that she found mathematics easy and she recalls positive memories with the subject itself. During primary school, Lisa perceives herself as extremely good in mathematics. "Secondary school was okay, it became harder, but it was not hard […] I chose a direction with as much mathematics as possible, so I thought it was fun." Lisa especially remembers two teachers. "Both were extremely interested… they taught playfully…" Lisa describes these teachers as knowledgeable and dedicated to their commitment as teachers. She also describes how they frequently used "competitive teaching" which increased Lisa's motivation: "this to get it a little more challenging and you got interested."

The first internship is conducted approximately eight months after Lisa enters teacher education. Lisa's supervisor Mr Higgins "admits" in an interview two months before that he uses competitions when teaching mathematics, all the time. However, he emphasises that the main idea is for students to compete with themselves, but that there might be some competition among the pupils as well. One example is how the class "do multiplication every third week […] we do it both on Mondays and on Fridays… on time… it might seem a little stressful and inconvenient, but they like it very much." At the internship, Lisa is stunned after her initial participation in Higgins classroom.

I think he explains in a way that is very close to pupils thoughts… it is very close to pupils, and it is… it is at their level of understanding ... and it is very playful all the time […] they [pupils] are involved in everything he does... then they have tests in mathematics where the competitive instinct comes into play... when he says something it becomes fascinating… Lisa accentuates her admiration for the teaching that she participates in and relate the teaching to the use of competitions, "he has it all the time." Lisa concludes that "I am very impressed with him as a teacher and would I… I would be like him."

Two relevant social practices are visible in Lisa's tales. Lisa re-engages in the past social practice, upper secondary mathematics classroom, as a hugely successful pupil. The success was related to the competitive structure which increased her motivation. The common endeavour of participating in competitive teaching practices that she shared with some others is central in her positive experience of mathematics education. It is also central to her view of teaching. Lisa do not question her first internship. Her prior participation in the upper secondary mathematics classroom is instead reinforced as valid in the present school-related social practice.

These two social practices are closely related, there are many similarities. Lisa brings two different social practices together, her past and present participation moulds together while re-negotiating the role of competitions. By participating in a present educational model, the role of competitions is actualised from past participation to present experience.

Phase two -Questioning competitive teaching when teaching

The second phase links to the 30 ECTS course in mathematics education and the common endeavours of the teacher educators. This paper stresses two different ways of participating, ways of being, within the mathematics education course. There are also other social practices of importance, for example, the study group containing three other student teachers.

During the mathematics education course, it became apparent, to Lisa, that most students did not enjoy mathematics at school and competitive teaching is filled with anxiety. She immediately starts to question her experience in mathematics classrooms and begins to align with the teaching in mathematics education. To understand this shift in her participation, the re-negotiation of her past participation, we need to enter lecture halls and seminar sessions that the mathematics education course provided during the first week.

Several different teacher educators indicate that the student teachers' bad experience of mathematics teaching might relate to competitive teaching. During the first week, it becomes clear to Lisa that her positive experience of competitive teaching is uncommon and she starts to question her past participation in mathematics classrooms. "Everyone is more or less competitive, and it is a huge strong feeling in all of us… or in me… it is […] it suits them who are good and not those who have difficulties ... so I do not know how much one will use it ... you will have to think about the pros and cons... and then usually… well, the disadvantages take over ..." There are positive things with competitive teaching "everyone concentrates more and makes every effort to get ahead [of others] or to win… there will be more ... a bit more fight and quality ... sort of ..." However, the negative side takes over "it ruins the confidence and quality deteriorates because it usually goes on time and that it should go as quickly as possible ... instead of as good as possible. I think it is important that they compete against themselves, but I think it is difficult for students to grasp that ..." However, when engaging in informal discussions in other social practices, she talks differently. For example, in her study-group, she several times during this course highlight that "competitions are excellent" and "to get ahead in the textbook is a great feeling."

During the course in mathematics education, Lisa starts to contrast her prior experience with the mathematics education course developing a critical stance against the kind of teaching that is promoted by the teachers. "You do not agree with everything they say."

At the beginning of the mathematics course, Lisa makes a shift concerning competitive teaching. By participating in the social practice set up by mathematics education, she tries to balance her past participation while re-negotiating the role of competitive teaching. The manifestation of the conflict, Lisa participates in, can be viewed between the prior experience of competitive teaching and the present experience of mathematics teacher education. Also relevant is the experience of the "other", those with negative experience from the past. The point is that Lisa re-engages and engage in two distinctly different themes that relate to different social practices, that do not relate to each other, trying to balance the role of competitive teaching during the mathematics education course. The teacher education course in mathematics education has a significant role in the shift describes, how Lisa use past and present experience. While trying to bring these different perspectives together, Lisa starts to develop a critical positioning to the promoted teaching presented at the mathematics education course.

Phase three -Competitions is positive for all learners

The last phase starts before the end of the mathematics education course. She engages in an imagined, present or future, social practice and relates that social practice against the content promoted at the teacher education course.

Even though Lisa think that this has been the best course, so far in her teacher education experience, she does not align with the general promoted teaching of the teacher educators. Lisa concludes that she is still positive to competitions even though it is described as an insufficient teaching strategy by the teacher educators. She admits that she may "sound mean" but "everyone is competitive more or less… everyone wants to win, and no one likes to lose". I do believe that competitions engage pupils… then one might not do it so outspoken that the one who wins gets a candy box … but competitions are there… not that anyone gets knocked down because they never win… however, still there... something that engages them and makes them want to make an effort... and that…well… in that way, I do even believe that one can involve them who have it easy and those who find mathematics hard … that they can applaud and encourage each other in this way… but then I do not know if competitions are the perfect teaching way, but I do think that it can be of importance to put it into teaching… and mathematics feels like a subject where it can be easily done.

Lisa also conducts her second internship with Mr Higgins as a supervisor. He still works in the same way, and Lisa is very comfortable in that classroom. She highlights that Higgins still uses competitions as a teaching strategy and in the final interview, two years and five months after the first one Lisa concludes that "Higgins is an excellent role-model at the moment… he is in my thoughts a splendid teacher… he is like the teachers I liked when I was young".

There is an apparent shift in Lisa's PoP. She is no longer obliged to participate in the collective way of being and share the common endeavour set by the teacher educators. She need not consider the role of the mathematics education course and the way of participating in this social practice. While bringing different perspectives together Lisa positions herself concerning future pupils, expand her educational model related to competitive teaching and develop a critical positioning to the teaching promoted by teacher education.

Discussion

This paper aimed to illustrate how different past and present social practices play for Lisa's, a generalist student teacher, tales of herself as primary mathematics teacher-to-be. In the result section, it has been illustrated what experiences from teacher education and other relevant social practices that are visible. PoP has offered a way to follow the re-negotiation of the meaning of teaching mathematics, how Lisa is using different social practices within communication to understand the world around her when emerging into the teaching profession.

Lisa's pathway through teacher education is viewed as "a process of flexibility, autonomy, and adaptation to the teaching context" (Hošpesová, Carrillo & Santos, 2018, p. 181). In her professional development, she has for example negotiated and re-negotiated her experience of competitions, upper secondary, internship, teacher education, and study-group work. She has first of all re-engaged positively in prior experience and in teachers teaching rather than teacher educators promoted teaching. The teaching of the teacher educators has been used as a discursive counterpart. Most important has been her own experience of being a pupil at school and how she re-negotiated that experience with Mr Higgins teaching practice. It can also be interpreted that her background as an athlete has contributed to her professional development.

Lisa draws on competitive classroom when entering teacher education and she still draws on competitive classrooms at the end of this study. In this sense, there is no shift in her participation. However, the third phase is no regression to the first phase, as Lisa brings in different perspectives and social practices in the different phases. Lisa uses other information in her argumentation and justification during different parts of her teacher education. Also notable is that teacher education promotes another teaching than the one Lisa draw on, and this evolves into the fact that Lisa questions some of her assumptions during her education. In this way, the mathematics education course has played a role, but in the end, not the role the teacher education community expects or wants.

The different phases in this paper are critical because we can interpret how social practices play a role in her tales of herself as teacher-to-be. We can interpret how past and present social practices influence how Lisa talks about competitive teaching during different parts in her teacher education. The point is, by looking closer to the process one can understand how and what students change or not. Many things happen during an education, both inside and "outside". While most others, for example, [START_REF] Hodges | Unpacking personal identities for teaching mathematics within the context of prospective teacher education[END_REF] focuses on social practices related to teacher education I suggest that there are other social practices of importance when student teachers become educated teachers. There is a need to be interested in any social practice when trying to understand student teachers as teachers-to-be.