G Devineau

W Xi

F Moutarde
email: fabien.moutarde@mines-paristech.fr

J Yang

Convolutional Neural Networks for Multivariate Time Series Classification using both Inter-& Intra-Channel Parallel Convolutions

In this paper, we study a convolutional neural network we recently introduced in [9], intended to recognize 3D hand gestures via multivariate time series classification. The Convolutional Neural Network (CNN) we proposed processes sequences of hand-skeletal joints' positions using parallel convolutions. We justify the model's architecture and investigate its performance on hand gesture sequence classification tasks. Our model only uses hand-skeletal data and no depth image. Experimental results show that our approach achieves a state-of-the-art performance on a challenging dataset (DHG dataset from the SHREC 2017 3D Shape Retrieval Contest).Our model achieves a 91.28% classification accuracy for the 14 gesture classes case and an 84.35% classification accuracy for the 28 gesture classes case.

Introduction

Gesture is a natural way for a user to interact with one's environment. One preferred way to infer the intent of a gesture is to use a taxonomy of gestures and to classify the unknown gesture into one of the existing categories based on the gesture data, e.g. using a neural network to perform the classification. In this paper we present and study a convolutional neural network architecture relying on intraand inter-parallel processing of sequences of hand-skeletal joints' positions to classify complete hand gestures. Where most existing deep learning approaches to gesture recognition use RGB-D image sequences to classify gestures [START_REF] Ye | A survey on human motion analysis from depth data[END_REF], our neural network only uses hand (3D) skeletal data sequences which are quicker to process than image sequences. The rest of this paper is structured as follows. We first review common recognition methods in Section II. We then present the DHG dataset we used to evaluate our network in Section III. We detail our approach in Section IV in terms of motivations, architecture and results. Finally, we conclude in Section VI and discuss how our model can be improved and integrated into a realtime interactive system. Note that the contents of this paper are highly similar to that of [START_REF] Devineau | Deep learning for hand gesture recognition on skeletal data[END_REF], especially sections 1, 2 and 3, as well as the figure illustrating the network, however in this article we fo-cus more on practical tips and on justifying the network architecture whereas the original paper focus was more centered on gesture-related aspects. Readers familiar with [START_REF] Devineau | Deep learning for hand gesture recognition on skeletal data[END_REF] can directly skip to the subsection Architecture Tuning of section IV, in which the network architecture is justified more thoroughly.

Definition & Related Work

We define a 3D skeletal data sequence s as a vector s = (p 1 • • • p n ) T whose components p i are multivariate time sequences. Each component p i = (p i (t)) t∈N represents a multivariate sequence with three (univariate sequences) components p i = (x (i) , y (i) , z (i) ) that alltogether represent a time sequence of the positions p i (t) of the i-th skeletal joint j i . Every skeletal joint j i represents a distinct and precise articulation or part of one's hand in the physical world. In the following subsections, we present a short review of some approaches to gesture recognition. Typical approaches to hand gesture recognition begin with the extraction of spatial and temporal features from raw data. The features are later classified by a Machine Learning algorithm. The feature extraction step can either be explicit, using hand-crafted features known to be useful for classification, or implicit, using (machine) learned features that describe the data without requiring human labor or expert knowledge. Deep Learning algorithms leverage such learned features to obtain hierarchical representations (features) that often describe the data better than hand-crafted features. As we work on skeletal data only, with a deeplearning perspective, this review pays limited attention to non deep-learning based approaches and to depth-based approaches ; a survey on the former approaches can be found in [START_REF] Mitra | Gesture recognition : A survey[END_REF] while several recent surveys on the latter approaches are listed in Neverova's thesis [START_REF] Neverova | Deep learning for human motion analysis[END_REF].

Non-deep-learning methods using handcrafted features

Various hand-crafted representations of skeletal data can be used for classification. These representations often describe physical attributes and constraints, or easily interpretable properties and correlations of the data, with an em-phasis on geometric features and statistical features. Some commonly used features are the positions of the skeletal joints, the orientation of the joints, the distance between joints, the angles between joints, the curvature of the joints' trajectories, the presence of symmetries in the skeletal, and more generally other features that involve a human interpretable metric calculated from the skeletal data [START_REF] Klaser | A spatiotemporal descriptor based on 3d-gradients[END_REF][START_REF] Knopp | Hough transform and 3d surf for robust three dimensional classification[END_REF][START_REF] Truong | Laban descriptors for gesture recognition and emotional analysis[END_REF]. For instance, in [START_REF] Vemulapalli | Human action recognition by representing 3d skeletons as points in a lie group[END_REF],

Deep-Learning based methods

Deep Learning, also known as Hierarchical Learning, is a subclass of Machine Learning where algorithms f use a cascade of non-linear computational units f i (layers), e.g. using convolutions, for feature extraction and transformation : Recurrent Neural Networks (RNN), e.g. networks that use Long Short-Term Memory (LSTM) [START_REF] Hochreiter | Long shortterm memory[END_REF] or Gated Recurrent Units (GRU) [START_REF] Cho | Learning phrase representations using rnn encoderdecoder for statistical machine translation[END_REF], have long been considered as the best way to achieve state-of-the-art results when working with neural networks on sequences like time series. Recently, the emergence of new neural networks architectures that use convolutions or attention mechanisms [START_REF] Van Den | Wavenet : A generative model for raw audio[END_REF][START_REF] Vaswani | Attention is all you need[END_REF] rather than recurrent cells has challenged this assumption, given that RNNs present some significant issues such as being sensitive to the first examples seen, having complex dynamics that can lead to chaotic behavior [START_REF] Laurent | A recurrent neural network without chaos[END_REF] or being models that are intrinsically sequential, which means that their internal state computations are difficult to parallelize, to name only a few of their issues. In [START_REF] Song | An endto-end spatio-temporal attention model for human action recognition from skeleton data[END_REF], Song et al. elegantly combine the use of an LSTM-based neural network for human action recognition from skeleton data with a spatio-temporal attention mechanism. While this approach seems promising, we rather seek to find a convolution-only architecture rather than a recurrent one.

f = f 1 • f 2 • . . . • f n .A traditional
Zheng et al. propose a convolution-based architecture that does not involve recurrent cells in [START_REF] Zheng | Time Series Classification Using Multi-Channels Deep Convolutional Neural Networks[END_REF], although this architecture can easily be extended with recurrent cells : [START_REF] Ordóñez | Deep convolutional and lstm recurrent neural networks for multimodal wearable activity recognition[END_REF].

p i = (x i , y i , z i ) ∈ R 3 , ∀i ∈ 1; 22 in the 3D space.
The dataset is split into 1960 train sequences (70% of the dataset) and 840 test sequences (30% test sequences).

Parallel Convolutions Model 4.1 Motivation

The goals of the original contest where the DHG dataset was introduced were to (1) study the dynamic hand gesture recognition using depth and full hand skeleton, and to (2) evaluate the effectiveness of recognition process in terms of coverage of the hand shape that depend on the number of fingers used. Nevertheless, the goals of this paper are different. Our first goal is to demonstrate that carrying out hand gesture recognition with a sparse representation of the hand (i.e. the 3D hand skeleton) only is competitive with other existing approaches that often focus on RGB-D images. The second goal of this paper is to propose a generic neural network that does not require recurrent cells to achieve this recognition.

Model Architecture

In [START_REF] Devineau | Deep learning for hand gesture recognition on skeletal data[END_REF] we introduce a multi-channel convolutional neural network with two feature extraction modules and a residual branch per channel. The whole model architecture is depicted in figure 1. The architecture is inspired by the highresolution and low-resolution networks from [START_REF] Molchanov | Hand gesture recognition with 3d convolutional neural networks[END_REF]. The use of residual branches in our architecture is inspired from the original Residual Networks paper [START_REF] He | Deep residual learning for image recognition[END_REF]. Residual branches make networks easier to optimize because of a better gradient backpropagation in the training phase ; they empirically improve the accuracy of deep networks.

Our network inputs consist of multiple, fixed-length, 1D sequences (s 1 , s 2 , . . . , s c ) where c ∈ N * is the number of sequences (channels). Each of these sequences s i is directly fed to three parallel branches. The first branch 1 , improperly called residual branch in this paper, is almost an identity function. Instead of outputting exactly its input we perform a pooling on the input in order to reduce the risk of overfitting. The second and third branches both present a similar architecture, detailed below, designed for feature extraction.

In these two branches, the input is processed as follows.

The input is passed to a convolution layer, whose output is subsampled using a pooling layer. This process is repeated two more times. For a single branch, the difference between all the three convolutions resides in the number of feature maps used ; the difference between the two branches resides in the size of the convolutions kernels. Having two kernel sizes for the time convolution layers allows the network to directly work at different time resolutions. Formally, let h (l,β ) represent the input of the l-th convolution layer of the β branch, K (l,β ) be the number of feature maps, W the bias shared for the k-th filter map. The output h (l+1,β ) of the l-th convolution layer is calculated as

h (l+1,β ) = σ h (l,β ) * W (l,β ) k + b (l,β ) k
where σ is the activation function. This output h (l+1,β ) serves as the input of the pooling layer that directly follows the convolution layer. For a single channel, the outputs of the three branches are concatenated into a single vector. Finally, a multi layer perceptron "merges" the -concatenated-vectors of all the channels together and acts as a classifier. There are as many MLP outputs as the number of gesture classes. In our experiments, we have two branches (high resolution and low resolution branches) : β ∈ 1; 2 , 3 convolution 1. middle branch in figure 1 . . . FIGURE 1 -Illustration (borrowed from [START_REF] Devineau | Deep learning for hand gesture recognition on skeletal data[END_REF] in which the network architecture is introduced) of the parallel convolutional neural network. Every channel is processed separately before the Multi Layer Perceptron. The parallel feature extraction module presented on the right is not shared between the 66 channels.

and pooling layers : l ∈ 1; 3 , and K (l,β ) = 8 feature maps for l = 1 or l = 2 and K (l,β ) = 4 feature maps for l = 3. The multi layer perceptron has 1 hidden layer with 1996 hidden units. All of the neurons in our network use the ReLU activation function : σ (x) = ReLU(x) = max(0, x), with the exception of the output neurons which use the softmax activation function. All of the 3 [×2 × c] subsampling layers use an average pooling with a temporal pool size of 2. Average pooling computes the average value of features in a neighborhood (of 2 time steps in our case), while max pooling extracts the maximum value of the features in the neighborhood. Empirically, it has been shown that max pooling outperforms average pooling in image recognition problems [START_REF] Boureau | A theoretical analysis of feature pooling in visual recognition[END_REF]. Nevertheless, experiments we conducted on the choice of the pooling method for our model showed that our model exhibits better results with average pooling (we see a 0.88% decrease in validation accuracy for the model with maximum pooling rather than the average one for the model configuration presented in this paper).

Architecture Tuning

To justify the model architecture, we study the performance of -successive improvements of-a MC-DCNN model, created by tweaking its architecture. We iteratively choose the input data format, the activation function used, the number of convolution layers and feature maps, the regularization rate, the pooling method, and finally we introduce a residual and two parallel CNN branches.

Window length. Our model takes a fixed length gesture sequence. In practice, since the hand gestures from the DHG dataset are relatively short, we do not use a sliding window over the sequences as they occur, but rather simply resample them in time to fit in a fixed time window. To determine the length of the window, we start with a simple mono-branch convolution-pooling model (with 4 convolutions instead of 3 in the final model) and evaluate its validation accuracy depending on the input duration after resampling (see table 1). The window length we chose is 100 timesteps.

Input Activation function. While the choice of the rectified linear unit (ReLU) function is a de facto standard in recent deep learning models, each type of activation function has its pros and cons [START_REF] Jiuxiang | Recent advances in convolutional neural networks[END_REF]. We thus decided to investigate whether we should use the exponential linear unit (ELU), the rectified linear unit (ReLU) or the sigmoid activation function as the network activation function. As a reminder, these activations are defined as follows : ReLU(x) = x + = max(0, x), Sigmoid(x) =

pagation issue when x < 0. PReLU(x) = x if x ≥ 0 µx if x < 0 .
The coefficient µ is learnt during the network training. The Swish activation function has empirically been shown to display better performances than the ReLU activation function [START_REF] Ramachandran | Searching for activation functions[END_REF] and is defined by

Swish(x) = x •Sigmoid(β x).
Number of layers. Following a classical convolutionpooling architecture schema, with a single branch and without dropout, we evaluate the performance of the model for a few epochs. The models we evaluate are composed of convolutions with ReLU activations. We denote C ReLU (8, 5) (or simply C(8, 5)) a convolution layer with 8 feature maps and a kernel of width 5, followed by a ReLU activation. We denote P(2) an average pooling layer of size 2. Results are presented in table 3. We chose to use a depth of 3 convolution+pooling layers, since it leads to the best (average) validation accuracy. The depth is likely to be related to the sequence length though, and longer sequences may benefit from being processed by a deeper network, if one wanted to change the input length. Pooling method. Average pooling seems to function better than max pooling on the input data for our model : it outperforms max-pooling by +0.88% in accuracy (result on the 14 classes case). 1D physical sensors data and 1D motion capture data present more regularity than other 1D data such as text, which means that the data is more compressible (in the time domain). With 1D gesture data it is easier to filter outliers (e.g. because of of physical constraints on the gesture), and outliers have less meaning than outliers in the text domain. For specific gesture recognition applications involving a lot of semantics like sign languages, such assumptions may probably not hold. We suppose that averaging values from a 1D channel sequence helps to reduce outliers weight with the smoothing. Average pooling may act as a regularizer. As gestures are smooth, averaging the signal probably leads to more signal removal than noise removal.

Residual and Parallel CNN branches. We duplicated the original CNN branch obtained, and add a pseudoresidual branch constituted of pooling only, whose aim is to better backpropagate the gradient during training. To achieve better performance -i.e. better validation accuracy, recall and F 1 scores-we adopt a multi-(temporal-)resolution approach : we chose to change the kernel width depending on the branch (instead of using a kernel of width 5, we use two kernels of sizes 3 and 7). We highlight the importance of using three parallel branches for the 28 gesture classes case in table 5.

Training

In this section we detail the hyperparameters we used as well as some other information related to the training.

Weights Initialization & Batching. Each training batch contained a set of 32 skeletal gesture sequences of length 100, where a skeletal gesture sequence is a list of 22 × 3 = 66 unidimensional sequences.

For the training, we used the Xavier initialization (also known as GLOROT uniform initialization) [START_REF] Glorot | Understanding the difficulty of training deep feedforward neural networks[END_REF] to set the initial random weights for all the weights of our model.

Implementation. Our model was implemented twice using either PyTorch or Keras as a high level library above Tensorflow. CUDA/CuDNN were used for GPU parallelization of the computations.

Hardware. We trained our model on one machine with a GPU (NVIDIA GeForce GTX 1080 Ti). Using the hyperparameters presented in the paper, each training step took about 12 seconds. We trained the model for 1000 steps.

Loss & Optimizer. We selected negative log-likelihood as the cost function. To train our model, we used the Adam optimization algorithm [START_REF] Kingma | A method for stochastic optimization[END_REF] which calculates an exponential moving average of the gradient and the squared gradient. Two parameters, namely β 1 and β 2 , are used to respectively weight the first moment (i.e. the mean) and the second moment (i.e. the uncentered variance) of the gradient. For the decay rates of the moving averages we used the parameters β 1 = 0.9, β 2 = 0.98. The values of other parameters were α = 10 -3 for the learning rate, and ε = 10 -8 . All these values are either identical (α, β 1 , ε) or close (β 2 β (de f ault) 2

= 0.999) to the default values proposed in the Adam article.

Results on the DHG dataset

We work on the DHG dataset presented on section III. All sequences are preprocessed as described in the previous section. Each resampled skeletal sequence is split into 22 joints' sequences, and then into 3 × 1D sequences of the (x, y, z) positions of the joints. This leads to 66 = 22 × 3 input sequences that are fed to model we introduced. The outputs of the last layer of the MLP represent the labels we will attribute to the gesture corresponding to the 66channel input. Since there are two classification tasks, depending on the number of classes, we use two different neural networks with the same architecture, except that one has 14 outputs and the other has 28 outputs. On the DHG dataset, our model achieves a 91.28% classification accuracy for the 14 gesture classes case and a 84.35% classification accuracy for the 28 gesture classes case. These are the best recognition accuracy scores known for this challenging dataset at this day (see table 6). In [START_REF] Oreifej | Hon4d : Histogram of oriented 4d normals for activity recognition from depth sequences[END_REF][START_REF] Devanne | 3-d human action recognition by shape analysis of motion trajectories on riemannian manifold[END_REF][START_REF] Smedt | Shrec'17 track : 3d hand gesture recognition using a depth and skeletal dataset[END_REF][START_REF] Ohn-Bar | Joint angles similarities and hog2 for action recognition[END_REF][START_REF] Smedt | Skeleton-based dynamic hand gesture recognition[END_REF] more or less complex but handcrafted features are used in the classification pipelines. The main advantage of deep-learning approaches is to automatically discover such (sometimes complex) features. [START_REF] Chen | Motion feature augmented recurrent neural network for skeleton-based dynamic hand gesture recognition[END_REF] is based on deep-learning architecture. It directly applies LSTMsbut without applying CNNs beforehand-to the skeletal data (and to the handcrafted features). Introducing a convolution step before the LSTMs could possibly improve the model in [START_REF] Chen | Motion feature augmented recurrent neural network for skeleton-based dynamic hand gesture recognition[END_REF]. Our model likely uses more efficient representations due to the use of the parallel branches. A comparison of the different approaches is presented in table 6.

Discussion

Dataset Size, . A common barrier to using deep-learning is small datasets. The DHG dataset has roughly 3000 balanced sequence instances (1960 The input to the network assumes a sequence of poses, which are provided by the Intel RealSense camera. The poses can also be retrieved by using body-worn sensors or estimated by segmenting videos [START_REF] Simon | Hand keypoint detection in single images using multiview bootstrapping[END_REF]. In that specific case, occlusion issues may arise, leading to wrong joint positions. Data augmentation may help reduce this problem. Invariance by rotation is not taken into account yet by the network.

We re-sampled signals to a vector of size 100 due to the nature of the motions that were all both relatively short as well as all being about the same duration in order of magnitude. This may not hold for motion capture data with very variable time spans for which one may prefer encode with a convolution and memorize with an reccurrent cell like an LSTM or a GRU.

Recurrence ; Speed. One of the goals of this paper was to study if a convolution-only network could lead to stateof-the-art results for gesture classification. While this result is established for short gestures with limited semantic meaning, the question remains open for gestures with very variable time span. For those cases, re-sampling the input might not always work and it might probably more efficient to insert recurrent cells in the model, e.g. after the convolutions, in order to benefit from the (long time range) context by keeping track of the processed input in a memory. In that case, one should carefully check if the model does not overfit, as memory cells are often harder to regularize [START_REF] Turkin | Tikhonov regularization for long short-term memory networks[END_REF]. Recurrent cells also tends to significantly increase the training time although there is ongoing work, e.g. [START_REF] Bradbury | Quasi-Recurrent Neural Networks[END_REF], to alleviate this issue. LSTMs and GRUs can warp time through their gating mechanism, but since CNNs can have gating mechanisms too, it would be interesting to see if gestures with very variable time span and limited semantic meaning could be efficiently classified without involving any recurrent or auto-regressive mechanism. One of the main advanges of using sparse (skeletal) input data instead of dense (image) input data lies in inference speed. On a (good) Intel Xeon CPU E5-1630 v4 @ 3.70GHz processor, without any GPU, the inference time is as low as ∼ 10 -5 s for a batch of 32 gestures, which is several orders of magnitude sufficient for real-time applications, even on less efficient processors for embedded systems.

Conclusion

We introduced and studied a new convolutional neural network which classifies (recognizes) hand gestures using skeletal data only. This neural network extends the MC-DCNN framework in several ways. First, it introduces parallel processing branches for each signal. The advantage of two convolutional branches over a single one seems to be that it allows the architecture to access different time resolutions of each signal. Second, the use of residual connection for each signal allows the gradient to better backpropagate in the neural network. Experimentally, it seems to be useful not only regarding the duration of network training, but also in terms of accuracy results. Finally, dropout is also used as a regularization technique. From a neural network perspective, we observe that (intra-and inter-) parallel processing of sequences using convolutional neural networks can be competitive with neural architectures that use cells specifically designed for sequences such as GRU and LSTM cells. We applied our model to perform hand gesture classification on a challenging hand gesture dataset (DHG dataset). Our method outperforms all existing published methods on this dataset. Our model achieves a 91.28% classification accuracy (+3,04% improvement) for the 14 gesture classes case and an 84.35% classification accuracy (+2,45% improvement) for the 28 gesture classes case.

Références

  (l,β ) k the k-th convolution filter of the l-th convolution in the β branch, and b (l,β ) k

  Vemulapalli et al. propose a human skeletal representation within the Lie group SE(3) × . . . × SE(3), based on the idea that rigid body rotations and translations in 3D space are members of the Special Euclidean group SE(3). Human actions are then viewed as curves in this manifold. Recognition (classification) is finally performed in the corresponding Lie algebra. In [8], Devanne et al. represent skeletal joints' sequences as trajectories in a n-dimensional space ; the trajectories of the joints are then interpreted in a Riemannian mani-

	Nearest Neighbors (k-NN) with Dynamic Time Warping
	(DTW) discrepancy.

fold. Similarities between the shape of trajectories in this shape space are then calculated with k-Nearest Neighbor (k-NN) to achieve the sequence classification. In

[START_REF] Smedt | Shrec'17 track : 3d hand gesture recognition using a depth and skeletal dataset[END_REF]

, two approaches for gesture recognition -on the DHG dataset presented in the next section-are presented. The first one, proposed by Guerry et al., is a deep-learning method presented in the next subsection. The second one, proposed by De Smedt et al., uses three hand-crafted descriptors : Shape of Connected Joints (SoCJ), Histogram of Hand Directions (HoHD) and Histogram of Wrist Rotations (HoWR), as well as Fisher Vectors (FV) for the final representation. Regardless of the features used, hand-crafted features are always fed into a classifier to perform the gesture recognition. In

[START_REF] Cippitelli | A human activity recognition system using skeleton data from rgbd sensors[END_REF]

, CIPPITELLI et al. use a multi-class Support Vector Machine (SVM) for the final classification of activity features based on posture features. Other very frequently used classifiers

[START_REF] Xing | A brief survey on sequence classification[END_REF]

are Hidden Markov models (HMM), Conditional Random Fields (CRF), discrete distance-based methods, Naive Bayes, and even simple k-

  DCNN and other deep (skeletal) gesture recognition models lies in the fact that MC-DCNN networks are skeleton-structure agnostic. A naive direct use of the model proposed by that paper does nevertheless not yield to results significantly competitive against other approaches results, but still gives a first glimpse of neural architectures for multi-variate sequences such as hand gesture skeleton data. In[START_REF] Devineau | Deep learning for hand gesture recognition on skeletal data[END_REF] we introduce a new neural network built upon this framework.To evaluate performances of several variations of the proposed neural network model architecture we conducted experiments on the Dynamic Hand Gesture-14/28 (DHG) dataset[START_REF] Smedt | Shrec'17 track : 3d hand gesture recognition using a depth and skeletal dataset[END_REF] created and introduced by DE SMEDT et al. in the SHREC2017 -3D Shape Retrieval Contest. The DHG dataset consists in a total of 2800 labeled hand gesture sequences performed by 28 participants.

	3 Dataset

Zheng et al. introduce a general framework (Multi-Channels Deep Convolution Neural Networks, or MC-DCNN) for multivariate sequences classification. In MC-DCNN, multivariate time series are seen as multiple univariate time series ; as such, the neural network input consist of several 1D time series sequences. The feature learning step is executed on every univariate sequence individually.

The respective learned features are later concatenated and merged using a classic MLP placed at the end of the feature extraction layers to perform classification. The major difference between MC-The sequences are recorded by an Intel RealSense depth camera and have variable lengths. Each labeled sequence consists of the raw data sequence returned by the camera, associated with two labels representing the category of the recorded gesture. For all sequences a depth image of the scene is provided at each timestep, alongside with both a 2D and a 3D skeletal representation of the hand. The hand skeleton returned by the Intel RealSense depth camera is presented in a paragraph below. Each gesture falls into one of 14 categories : Grab (G), Tap (T), Expand (E), Pinch (P), Rotation clockwise (RC), Rotation counter-clockwise (RCC), Swipe right (SR), Swipe left (SL), Swipe up (SU), Swipe down (SD), Swipe x (SX), Swipe + (S+), Swipe v (SV), Shake (Sh). Moreover, each gesture can be performed with either only one finger or with the whole hand. That means that gestures are classified with either 14 labels or 28 labels, depending on the number of fingers used. The gesture recognition method we introduce in the next section only uses the 3D hand skeletal representation returned by the Intel RealSense depth camera. At each time step the 3D hand skeleton consists in an ordered list of 22 joints with their positions

TABLE 1 -

 1 Accuracy after training the model for 50 epochs

TABLE 2 -

 2 epochs and evaluating its performances on the validation set (see table2), we chose to use the ReLU function as the network activation function. ELU are ReLU yield to similar results, although ELU is more computationally expensive. Overall ReLU reveals to be a good choice, one of the reasons being that it introduces sparsity effect on the network. Accuracy after training the model for 50 epochs Other activations worth considering in future works include Leaky ReLUs or Parametric ReLUs, as well as the Swish activation function. Parametric ReLUs are similar to ReLUs but cells do not "die", which can be the case with ReLU due to the backpro-

	1 1+e -x ,

x -1) if x < 0 x if x ≥ 0

, where λ ≥ 0 is a hyper-parameter to be tuned.

After training the same model for a limited -but sufficient-number of

TABLE 3 -

 3 Accuracy after training the model for 50 epochs the use of dropout in the previous layer with a dropout rate of 20%, which is the rate we finally chose.

	Dropout. To reduce overfitting, we use dropout [31]
	which consists in randomly setting some layer inputs to 0
	at each update during training time only. Results of some

experiments we performed to determine whether to apply dropout to the convolutions layers or not, as well as to determine their rate, are presented in the table 4 below (still on a model without parallel branches). We denote D(0.2)

TABLE 4 -

 4 Acc. after training the models for 100 epochs

TABLE 5 -

 5 Model acc. degradation on branch ablation

  train sequences + 837

	Approaches	Accuracy 14 gestures Accuracy 28 gestures
	OREIFEJ & LIU [26]	78.53	74.03
	DEVANNE et al. [8]	79.61	62.00
	GUERRY et al. [7]	82.90	71.90
	OHN-BAR & TRIVEDI [24]	83.85	76.53
	CHEN et al. [3]	84.68	80.32
	DE SMEDT et al. [6]	88.24	81.90
	Ours	91.28	84.35

TABLE 6 -

 6 Accuracy results on the DHG dataset

			Ours		DE SMEDT et al.	Difference
	Gesture Precision	Recall	F1-score Precision Recall F1-score	F1-score
	G	72.4%	94.8%	82.1%	67.5%	57.0%	61.8%	20.3%
	T	71.2%	77.0%	74.0%	85.2%	87.0%	86.1%	-12.1%
	E	84.7%	90.9%	87.7%	84.8%	87.0%	85.9%	1.8%
	P	90.9%	78.4%	84.2%	52.1%	61.0%	56.2%	28.0%
	RC	69.2%	98.2%	81.2%	80.0%	77.5%	78.8%	2.5%
	RCC	97.8%	77.6%	86.5%	90.9%	85.5%	88.1%	-1.6%
	SR	91.2%	100.0%	95.4%	85.1%	92.5%	88.6%	6.7%
	SL	98.0%	88.9%	93.2%	78.4%	85.5%	81.8%	11.4%
	SU	98.2%	79.4%	87.8%	89.3%	85.5%	87.4%	0.4%
	SD	93.3%	91.8%	92.6%	80.8%	88.0%	84.3%	8.3%
	SX	100.0%	89.9%	94.7%	95.8%	85.0%	90.1%	4.6%
	S+	98.3%	100.0%	99.1%	90.2%	98.5%	94.1%	5.0%
	SV	90.6%	100.0%	95.1%	93.2%	92.0%	92.6%	2.5%
	Sh	96.2%	71.4%	82.0%	88.6%	81.0%	84.7%	-2.7%

TABLE 7 -

 7 Comparison of F 1 score in the 14 gesture classes case

	test sequences) of 100 timesteps with 66 1D-channels. The
	proposed model has 13829454 free parameters in total, or
	13829454/66 ≈ 209537 free parameters by channel. Given
	that each of the 1960 training sequences has 100 times-
	teps, and because of the regularization applied (DropOut
	with p = 0.2), the model likely does not overfit. Qualitati-
	vely speaking, no overfitting was observed experimentally.
	Zero-, One-, or Few-shot learning [39], as well as data aug-
	mentation, transfer learning, model compression and distil-
	lation techniques can help to reduce the minimum size of
	the dataset required for training and validating deep lear-
	ning models.
	Preprocessing, Average Pooling & Data Regularity.

F 1 scores by gesture, for a version of the model without the "residual" branch ; very similar results are obtained for the model with the "residual" branch.


Samir Adly
email: samir.adly@unilim.fr

Jifi V Outrata
email: outrata@utia.cas.cz

Qualitative Stability of a Class of Non-Monotone Variational Inclusions. Application in Electronics

Keywords: Hemivariational înequalities, nonsmooth and variational analysis, Aubin property, isolated calmness, non-regular electrical circuits 1991 Mathematics Subject Classification: 46N10, 47N10, 47J20, 49J40, 34A60

The main concern of this paper is to investigate some stability properties (namely Aubin property and isolated calmness) of a special non-monotone variational inclusion. We provide a characteri zation of these properties in terms of the problem data and show their importance for the design of electrical circuits involving nonsmooth and non-monotone electronic devices like DIAC (Dlode Altematîng Current). Circuits wîth other devices like SCR (Silicon Controlled Rectifiers), Zener diodes, thyristors, va.ractors and transistors can be analyzed in the same way.

Introduction and problem formulation

In this paper, we intend to analyze some qualitative stability properties for a non-monotone variational inclusion of the following form:

p E f(z) + B T [)J(Bz), (1)
where f: R n -+ R n is a given mapping and BE R mxn , p E R n are a given matrix and a given vector with m ::::; n, respectively. Throughout the paper, we assume that B is surjective. The fonction J : !R m -IR, x = ( X1, .... , X m ) 1-+ J ( x) is defined by J(x) = j1 (x 1 ) + j2 (x 2 ) + .. • + jm (X m ), 'v x = (x i, ... ,x m ) E !R m , (2) where j i : lR -lR are supposed to be locally Lipschitz for every i = 1, 2, ... , m.

For a locally Lipschitz fonction <.p : JR N -IR, &<.p denotes its Clarke generalized gradient ( [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF]) defined by 8 <.p (x) := {p E !R N : <.p 0 (x; y ) 2: (p, y), \/y E JR N }, where <.p 0 (x; y) stands for the Clarke generalized directional derivative of <.p at x in the direction y defined by . <.p(z + >.y) -<.p(z) <.p 0 (x; y):= hm sup--'----'----------.

À-+Û+

.,\ Z-+X [START_REF] Addi | Modelling and analysis of a non-regular electronic circuit via a variational inequality formulation[END_REF] Various problems arising in mechanical and electrical engineering can be formu lated in the form [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems[END_REF]. We will show in this paper that the model ( 1) is of particular interest in the theory of electrical circuits containing nonsmooth electronic devices like DIACs (Dlode Alternating Current), Silicon Controlled Rectifiers, thyristors, varactors and transistors.

Modeling of electrical circuits is usually performed by using Kirchhoff 's laws and the Ampere-Volt ( or I-V) characteristics for each involved electronic device which are graphs of mappings (possibly set-valued) relating the voltage V to the current i. Electrical devices like resistors, inductances and capacitors are usually described by a single-valued (1-V) mapping (linear or non-linear) and are called "smooth electrical devices". On the other hand, semiconductors like diodes, DIACs, Silicon Controlled Rectifiers (SÇR) and transistors are defined generally by means of set valued (I-V) characteristics and are called "nonsmooth electrical devices". The mathematical modeling of electronic circuits involving smooth electrical devices leads generally to algebraic/differential equations that can be studied by using classical mathematical analysis. On the other hand, mathematical modeling of electronic circuits involving nonsmooth electrical devices leads generally to varia tional/ differential inclusions that can be handled by using tools from variational and nonsmooth analysis. An efficient approach for the treatment of monotone set-valued maps in the framework of variational inequalities relies on the notion of convex superpotential introduced by J. J. Moreau [START_REF] Moreau | La notion de sur-potentiel et les liaisons unilatérales en élastostatique[END_REF] and generalized by P. D. Panagiotopoulos [START_REF] Panagiotopoulos | Non-convex superpotentials in the sense of F. H. Clarke and applications[END_REF] to the case of non-monotone set-valued maps by using the generalized gradient of F. H. Clarke [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF] for locally Lipschitz fonctions. The usage of tools from variational and nonsmooth analysis for the study of elec trical circuits is a fairly recent and quite promising topic of research [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems[END_REF][START_REF] Addi | A method using the approach of Moreau and Panagiotopoulos for the mathematical formulation of non-regular cir cuits in electronics, Nonlinear Anal[END_REF][START_REF] Addi | A qualitative mathematical analysis of a class of linear variational inequalities via semi-complementarity problems[END_REF]Il].

To the best of our knowledge, this paper is the first work dealing with the quali tative stability of electronic circuits containing nonsmooth electrical devices with non-monotone (I-V)-characteristics. For the monotone case, we can cite e.g. [START_REF] Addi | A sensitivity analysis of a class of semi coercive variational inequalities using recession tools[END_REF],

[4], [START_REF] Addi | Modelling and analysis of a non-regular electronic circuit via a variational inequality formulation[END_REF], [START_REF] Goeleven | Existence and uniqueness for a linear mixed variational inequality arising in electrical circuits with transistors[END_REF].

In electrical engineering, one usually uses a circuit simulator for predicting the behavior of an electronic circuit. Most used software, for the smooth case, are based on various versions of SPI CE (Simulation Program with Integrated Circuits Emphasis). The simulation of nonsmooth electronic circuits are well-known to be a difficult task. Due to the lack of smoothness, classical mathematical analysis is not applicable and requires natural extensions. This paper is devoted to qualitative stability analysis of ( 1) under the structural assumption that

J(x) = Mx -q, Vx E lR n , (4)
where M E lR n x n and q E ]R n are a given matrix and a given vector. Without Joss of generality, we will suppose that q = O.

By stability analysis we mean a study of the local behavior of the solution map

S : ]R n =î ]R n defined by S(p) = {z E ]R n : p E Mz -q + B T [)J(Bz)}, around a reference point (p, z) E Gr(S). ( 5
)
The structure of the paper is as follows: In Section 2 we recall the definitions of several basic notions from nonsmooth analysis and state some principal results, substantial in our development. Section 3 in devoted to characterizations and criteria (sufficient conditions) for the Aubin property of S, whereas in Section 4 we examine in the same way the isolated calmness of S. Section 5 deals then with the application of the derived conditions to concrete electronic devices, namely the DIAC.

The used notation is basically standard. For x, y E JR_ n , (x, y) stands for the Euclidean scalar product on lR n and llxll = ./(x,x) is the corresponding norm. lIB is the unit ball, ô n is the indicatory fonction of a set n and I n denotes the unit matrix in lR n . For a set-valued map (multifonction) <I>: JR. n =î JR. m , Gr(<I>) := {( x, y) E R n X lR.

m IY E <I>( x )} denotes its graph. The Kuratowski-Painlevé upper ( outer) limit of <I> as x ___. x, denoted by Limsup x --+x <I>(x), is defined by Limsup <I> (x) := {x* E lR n : :3xk ___.

Mathematical tools

In this section, we provide definitions and properties of some essential notions from nonsmooth analysis that will be used throughout this paper. For more details we refer to [START_REF] Dontchev | Implicit Functions and Solution Mappings. A View from Variational Analysis[END_REF][START_REF] Mordukhovich | Variational Analysis and Generalized Differentiation. 1: Basic Theory[END_REF][START_REF] Rockafellar | Variational Analysis[END_REF]].

• Aubin Property: Let S : R n � R m be a set-valued mapping and (x, y) E Gr(S ). S is said to have the Aubin property around (x, y) if there exists a constant K 2::: 0 and neighborhoods U C R n of x and V C R m of y such that S(x') n V C S(x) + Kllx' -xlllBI, for all x, x' EU.

(

The Lipschitz modulus of S at x for y, denoted by lip(S; (x, y)), is defined by lip(S; (x, y))= inf {/<i, ER + : ?J U E N (x), VE N (y)

such that condition (8) is satisfied},
where the notation N(x) stands the collection of all neighborhoods of x.

• Isolated calmness: The set-valued mapping S: R n � R m is said to be isolat edly calm at (x, y) if there exists a constant K-2::: 0 and neighborhoods U c ]R n of x and V C R m of y such that S(x) n V c y+ /<i,llx -xlllBI, for all x EU.

(

) 10
We note that the linear mapping associated with every matrix A E R mxn is isolatedly calm at any point . The isolated calmness of its generalized inverse A-1 (-) is equivalent to its injectivity, i.e., A-1 (0) = { O } . More generally, for a fonction f : ]R n -+ lR m (supposed sufficiently smooth around x), the inverse 1-1 (-) is isolatedly calm at (f(x),x) if and only if the derivative mapping v' f(x) is injective.

In [START_REF] Dontchev | Implicit Functions and Solution Mappings. A View from Variational Analysis[END_REF] it is shown that a set-valued mapping S : JR. n � R m , whose graph is the union of finitely many polyhedral convex sets, is isolatedly calm at (x, y) if and only if y is an isolated point of S(x).

• Let A C R n be a nonempty subset of ]R n and i: E A. The Bouligand tangent (contingent) cone to A at x is defined by or equivalently,

A-x

TA(i:) = Limsup --, t-o+ t (11)
• The regular (Fréchet) normal cone to a subset A C ]R n at i:, denoted by NA(x), is defined by

NA(x) = {x* E JR.
• The limiting (Mordukhovich} normal cone to A at x, denoted by NA(x), is the cone

NA(x) = Limsup NA(x). (15)
.,-,i,

xEA
Let us remark that if the set A is closed and convex, then the regular and the limiting normal cones coïncide with the normal cone in the sense of convex analysis defined by

N A (x) := { x * E JR. n : (x *, x -x ) � 0 for all x E A } . • A subset AC JR_ n is called normally regular at x E A provided N A (x) = N A (x).
• Let <I>: JR. n ::::t IR. m be an arbitrary set-valued mapping. For (x , y) E Gr(<I> ), the graphical (contingent) derivative of <I> at (x , y) is the mapping D<I>(x, y) IR. n 4 lR m defined by

D<I>(x,y)(z) = {w E IR. m : (z ,w ) E Tcr(�) (x,y) }, z E IR. n . ( 16
)
• The limiting (Mordukhovich} coderivative of <I> at (x, y) is the multifonction D * <I>(x, y) : IR. m ::::t lR n defined for all y * E IR. m by

D*<I>(x, y)(y * ) := { x * E IR. n : (x *, -y * ) E N cr(�) (x , y) } .
For more details about these notions, we refer the reader to the monographs [START_REF] Mordukhovich | Variational Analysis and Generalized Differentiation. 1: Basic Theory[END_REF] and [START_REF] Rockafellar | Variational Analysis[END_REF].

We recall now the following result for the characterization of the Aubin property, known in the literature as the Mordukhovich criterion.

Theorem 2.1 ([12]

). Let <I> : IR. n :::::::: lR m be a set-valued mapping and (x,y) E Gr(<I>). Suppose that Gr(<I>) is locally closed at (x, y). Then <I> has the Aubin property a round ( x, y) if and only if

D*<I>(x, y)(O) = {O}. ( 17
)
Furthermore, the Lipschitz modulus of <I> at x for y is given by lip ( <I> ;(x , fï)) = IID* <I>( x , '!} ) 11

+ := sup {llx*II : x * E D*<I>(x, y)(y*) , IIY*II � 1} . (18)
To unburden our notation, let us introduce the set-valued mapping Q : JR. n � JR_ n defined by

Q(z) := B77JJ(Bz). ( 19
)
On the basis of Theorem 2.1 one can now obtain the following characterization of the Aubin property of S.

Theorem 2.2 ([14]

). Suppose that Q has a closed graph. Then the set-valued mapping S defined in [START_REF] Addi | A sensitivity analysis of a class of semi coercive variational inequalities using recession tools[END_REF] has the Aubin property around (p, z) if and only if the implication

0 E M T b + D*Q(z,j5-Mz)(b) ==> b = 0, ( 20
) holds true.
The generalized equation on the left-hand sicle of ( 20) is usually called the adjoint generalized equation. For the isolated calmness of S we dispose with a similar characterization on the basis of the graphical derivative.

Theorem 2.3. Suppose that Q has a closed graph. Then the set-valued mapping S defined in (5) has the isolated calmness property at (p, z) if and only if the implication

0 E Mb + DQ(z,p -Mz)(b) � b = 0, (21) holds true.
Proof. The proof of Theorem 2.3 follows easily from [START_REF] Dontchev | Implicit Functions and Solution Mappings. A View from Variational Analysis[END_REF] (Corollary 4 C.2). In fact, by setting

F(z) = Mz + Q(z), one bas DS(p,z)(0) = {0} � ker DF(z ,p) = {O},
and [START_REF] Dontchev | Implicit Functions and Solution Mappings. A View from Variational Analysis[END_REF]. D

DF(z,p)(b) = Mb + DQ(z,p -Mz)(b) for all b E R n by virtue of Proposi tion 4 A.2 in
We conclude this preparatory session with a second order chain rule which enables us to compute the coderivative D*Q of the set-valued mapping Q defined in [START_REF] Panagiotopoulos | Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Fu.nctions[END_REF]. Finally assume that g is twice continuously differentiable, and on a neighborhood W of x E lR n we have 8( ip o g)( x ) = (v'g( x ) f âip(g(x)) for x E W.

( (23) where X E lR m is the unique vec tor satisfying (v'g(x)f>. = ii, i.e., X =

) 22
Let ii E 8(<p o g)(x). If the Jacobian v'g(x) is surjective, then for all y• E ]R n one has D*â(<p o g)(x, v)(y*) � (t, X;V'g,(x)) y•+ ('vg(xW D'ÊJ,p(g(x), X)(Vg(x)y'),
['v'g(x)(v'g(x)f]- 1 v'g(x)v.
Remark 2.5. This statement was established in a slightly less general setting in [START_REF] Mordukhovich | On second-order subdifferentials and their applications[END_REF] (Theorem 3.4) as an inclusion with the remark that the reverse inclusion is easily seen to hold as well. For the reader's convenience, we provide the proof of the above theorem in the Appendix.

Clearly, the above statement applies, for instance, if 8 is the Clarke subdifferential and tp is locally Lipschitz.

Aubin property of the solution map

In this section we give a characterization of the Aubin property of the map S defined in (5) around the reference pair (p, z). Without any lack of generality we will assume that q = 0 and start with the general case without any additional assumptions on the problem data M, B and J. Thereafter we will try to specialize these conditions under additional assumptions.

In what follows TE !R mxn is the matrix defined by [START_REF] Yakubovich | Solution of certain matrix inequalities in the stability theory of nonlinear control systems[END_REF] Note that the product BB T is non-singular by virtue of the surjectivity of B. (

) 27
Using the specific structure of J given in (2), we have for any x E IR "'

m 8J (x) = IJ 8j i(xi) and i=l Gr(tJJ) = { (x, y) E !R m x IR "' : (xi, Yi) E Gr(8ji), Vi = 1, 2, ... , m }. ( 28
)
The second equality in (28) enables us to employ [START_REF] Rockafellar | Variational Analysis[END_REF] (Proposition 6.41 ), which leads for every b E !R m to the equivalences

a E D•BJ ( x, u)(b) {=::> ai E D•aj i( xi, ui)(bi) Vi = 1,
) 33
Then S has the Aubin property around (p, z).

Proof. Suppose by contradiction the existence of a nonzero vector c E lR m \ {0} such that the implication (31) is not satisfied. By virtue of (33), we have (c ;, (Rë)ï) ::.; 0, Vi = 1, 2, ... , m.

(34)

However, since Ris a P-matrix, it follows that for each vector c E lR m , there must be an index i E {1, 2, ... , m} such that (ci, (Rc)i) > 0, which contradicts (34) and ensures the validity of (31). D Remark 3.4. In this way, we have established the fact that the Aubin property holds whenever M is nonsingular, Ris a P-matrix and all fonctions j i are convex for i = 1, 2, ... , m. Observe that R can be a P-matrix even if M T is not. To see this consider e.g. the matrices

M T = (-1 2) 1 -1 '
In some situations the Aubin property of S around (p, z ) may be ensured via the (more restrictive) notion of strong metric regularity of the inversé map s-1 defined by s-1 (z) := Mz + B T 8J(Bz) at (z,p), cf. [ 9, Proposition 3G.l ] . This amounts to the statement that the mapping S has a single-valued Lipschitz localization around (p, z). Such a situation arises if, e.g., B = I n , Mi s symmetric positive definite and J is convex, cf. [ 23, Proposition 12.54 ] .

We conclude this section with an attempt to combine the positive semi-definitness of M with condition (25). and consider the reference point (p, z ) with j5 = (-1, -1 f and z = (0, Of. Here, one can easily show that, due to the small quadratic term in j 2 , condition (35) is fulfilled. Since M is positive semi-definite and condition (33) holds true by the convexity of j 1 and h, Theorem 3.5 applies and so the respective S has the Aubin property around (p, z ).

In some applications it might be useful to know the value of lip(S; (p, z)). By virtue of ( 18) and ( 27 If problem (38) possesses a solution, say (b, w), then it can be numerically solved by standard techniques and K = llbll. If (38) does not possess a solution and the suprémal value of its objective is finite, we get typically only a lower estimate of K. Finally, if there is a sequence of feasible points (b(k ) , w( k ) ) in (38) with ll b(k ) I l -+ +oo, then the respective S does not have the Aubin property around (p, z).

In Section 5 we will illustrate the nature of problem (38) by means of a simple academic example.

Isolated calmness of the solution map

In this section, we give a characterization of the isolated calmness of the map S defined in [START_REF] Addi | A sensitivity analysis of a class of semi coercive variational inequalities using recession tools[END_REF] in terms of the data (M, B, J) involved in problem [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems[END_REF].

We start with an auxiliary lemma which might be helpful also in other situati ons.

Lemma 4.1. Let 2 = A-1 (r) where r c JR. n is closed and A E JR. nXm is a surjec tive matrix. Let A = G( 2), where G E JR. lxm is an injective matrix, u E A and ïi be uniquely given by Gïi = u. Then one has T A (u) = {Gw : Aw E Tr(Av)}.

(39)

Proof. By virtue of the surjectivity of A, we have, cf. [START_REF] Rockafellar | Variational Analysis[END_REF] (Exercise 6.7) that Ts(v) = {w : Aw E Tr(Aïi)}.

We have, {Gw : w E Ts(ïi)} C T A (u).

(40) Indeed, let w E Ts (v). By definition of the tangent cone, there are sequences v i --+ v in 2 and ti L o + as i --+ +oo such that Vi -ÏÏ ----+ w as i --+ +oo. We : how now the boundedness of the sequence (h i ) defined by

W• -ïi hi = T•
Assume by contradiction that ( h i ) is unbounded. Hence there exists a subsequence (still denoted by (hi)) such that Jlhill --+ +oo as i --+ +oo. By passing to a subsequence if necessary, we have It follows that ll�:II --+ s with 11s11 = 1.

Gs=O, which contradicts the injectivity of G. Thus, the sequence (h i ) possesses a conver gent subsequence with the limit in Ts(ïi). We conclude that k E GTs(ïi), which com;:,letes the proof.

D

Gr(8Ji) nui = LJ cyl, ( 42
)
v=l where l i are given integers and the sets cS i ), v = 1, ... , l i , are closed and normally regular at ((Bz) i , ïi i) . Then the isolate.d calmness property of S at (p, z) holds true if and only if one has the implication 1,

((Bb)i, -(Wb)i) E LJ Td, i((Bz)i, v i) Vi = 1, 2, ... , m ==> b = 0, ( 4 3)
v=l where ii is given by (26).

Proof. Our starting point is the implication (21) in Theorem 2.3. To compute the graphical derivative of Q, we observe that

Gr(Q) = { G [ ! ] : A[ ! ] E Gr(8J)},
where is injective and is surjective. By virtue of (28) one has generally only the inclusion

m Ta r �J)(Bz,v) C fITar&j; ((Bz) i ,v i ). ( 44
)
i=l However, thanks to the structural assurnption ( 42), inclusion ( 44) becornes equality (cf. {23], Proposition 6.41). This enables us to invoke Lemma 4.1, according to which we have i.e.,

{ 1 ; } DQ(z,p -Mz)(b) = B T C : ((Bb)i, ci) E � Tdi) ((Bz)i, vi) ' Vi .
The implication [START_REF] Robinson | Sorne continuity properties of polyhedral multifonctions[END_REF] can thus be rewritten to the form

Mb+B T c=0 } ((Bb) i , Ci) E LJ�=l Tdil (( Bz ) i , 1Yï), Vi ===? b = O,
from which condition ( 43) follows because c = -Wb. This completes the proof of Theorem 4.3. D

Illustration in non-regular electrical circuits

The airn of this section is to develop a mathematical model of electronic circuits involving devices like diodes, Zener diodes, DIACs, Silicon controlled rectifiers that are characterized by set-valued monotone or non-monotone ampere-volt character istics.

Electrical devices like diodes are described in terms of Ampere-Volt characteristic (I, V) which is a multifonction expressing the difference of potential V across the de vice as a fonction of current i going through the device ( for more details we refer to [START_REF] Addi | A method using the approach of Moreau and Panagiotopoulos for the mathematical formulation of non-regular cir cuits in electronics, Nonlinear Anal[END_REF][START_REF] Addi | A qualitative mathematical analysis of a class of linear variational inequalities via semi-complementarity problems[END_REF][START_REF] Adly | A stability theory for second-order nonsmooth dynamical systems with applications to friction problems[END_REF] and references therein).

An electronic circuit is formed by the interconnection of electronic components like generators, resistors, capacitors, inductors, diodes, transistors, etc. The behavior of a circuit is usually described in terms of currents and voltages that can be speci fied for each involved electrical device. An approach to state a mathematical model that can be used to determine these currents and voltages requires to formulate the ampere-volt characteristic of each electrical device, to write clown the Kirchoff ' s voltage law expressing that the algebraic sum of the voltages between successive nodes in all loops of the circuit are zero and to write down the Kirchhoff 's current law stating that the algebraic sum of the currents in all branches which flow to a common node equals zero.

Let A E JR. nX n' D E JR. nXm ' C E JR. mXn and E E JR. nX p be given matrices. Let J : JR. m -lR. be a given fonction. It is assumed that x f--+ J(x) is locally Lipschitz. Suppose that the state-space equations attain the form where u E IR." is a given vector (usually u is a control vector that drives the system).

This framework is particularly usefol for the study of non-regular circuits involving nonsmooth electrical devices. In this case, the matrices A, C, D and E are u.sed to state the Kirchhoff's voltage and current laws in matrix form. In general, A contains some electrical parameters like resistance, capacitance or inductance. The state x denotes a current vector and YL is a voltage vector corresponding to the electrical devices involved in the circuit.

Suppose that the following key assumption is satisfied:

there exists a symmetric and invertible matrix RE IR." xn such that: R 2 D = c r (45)

For the connection of assumption (45) with the positive realness (which may be seen as a generalization of the positive definiteness of a matrix to the case of a dynamical system) of the transfer fonction H(s) = C(sl n -A)-1 D, s E C asso ciated to problem (P), and the Kalman-Yakubovich-Popov lemma, see Lemma 1 in [START_REF] Addi | A qualitative mathematical analysis of a class of linear variational inequalities via semi-complementarity problems[END_REF] and references therein. The Kalman-Yakubovich-Popov lemma has been a cornerstone in control and system theory due to its wide range of applications. It relates the frequency domain conditions for positive realness to a set of algebraic equations (Linear Matrix Inequalities describing the state-space representation of the system) and to the dissipativity of the storage fonction. For more details see [START_REF] Brogliato | Dissipative Systems Analysis and Control[END_REF].

Problem (P) is equivalent to the following variational inclusion: 0 E Ax + D8 J(Cx) + Eu.

Setting z = Rx, we have

0EAx+D8J(Cx)+Eu # 0ERAR-1 Rx+RD8J(CR-1 Rx)+REu # 0 E RAR-1 z + R-1 R 2 D8J(CR-1 z) + REu # 0 E RAR-1 z + R-1 crlJJ(CR-1 z) + REu.
This allows us to consider the problem

Q { Find z E IR." such that ( ) 0 E RAR-1 z + REu + R-1 craJ(CR-1 z).
We note that problem (Q) is of the form (1) with f(z) = RAR-1 z + REu and B = CR- Proof. We have seen above that if (x, YL) is a solution of Problem (P) then z = Rx is a solution of Problem Q. Suppose now that z is a solution of Problem Q. Then setting x = R-1 z, we see as above that: breakover voltage while / 1 (resp. / 2 = -/ 1 ) is the forward (resp. reverse) breakover current. For example, for a practical trigger DIAC, V 1 = 30 volts and /1 = 25 µA.

0 E Ax + D8J(Cx) + Eu.
It is clear that there exists a locally Lipschitz fonction j : R ---+ R such that the set-valued map in Figure 5.1 can be written as

VE 8j(i).

A DIAC clipper circuit can be used to limit the output voltage signal to a certain level. Let us consider the circuit from Figure 5.2 involving a load resistance R > 0, an input-signal source u and the corresponding instantaneous current i, a DIAC as a shunt element and a supply voltage E. Using Kirchho:ff's laws, we have

U= V R + V +E, ( 46
)
where V R = Ri and V E 8j(i) is the voltage across the resistor and the diode respectively. Hence, 0ERi+E-u+8j(i), (p,z) = (-1,0)

1] UR [ a � 1 ] U cone { [ �1] , [ a � 1]} } (b 1, b 2) ER [6 ] UR [ -\ -a] U cone { [6 ] , [ -\-a]} (
It is easy to show that if a > 0, then (60) 1s satisfied. Therefore, S satisfies the isolated calmness property at (p,z) with z = (0,0) and p= (-1 , -1).

• Suppose now that the reference point is (fi, z) with z = (z 1 , z 2 ), z 1 < 0 and . z 2 < 0, see Figure 5 In this case we conclude that since a > 0, the implication in (61) holds true.

Consequently, the map S has the Aubin property around (p, z).

By the way of conclusion

In this paper, we studied the Aubin property and the isolated calmness of the solution map to a non-monotone variational inclusion with respect to canonical perturbations. If, instead of canonical perturbations, we perturb the matrix M (in a sufficiently smooth way) then, as explained in the theory developed in [START_REF] Dontchev | Implicit Functions and Solution Mappings. A View from Variational Analysis[END_REF], the conditions in Theo rems 3.1, 4.1 will become only sufficient ( for the respective stability properties ) . We showed that our theoretical results are applicable to the mathematical analysis of nonregular circuits involving nonsmooth electrical devices like DIAC (whose I-V characteristic is nonmonotone) which is a topic of major importance in electrical engineering. Some other electronic devices like SCR.s, Zener diodes or transistors can be treated in the same way. In order to use the chain rules in subdifferential calculus, we have assumed that the matrix B is surjective; it would be interesting to relax this condition. Another interesting question is to study the (non-isolated) calmness of S. This is, however, out of the scope of this work and will probably be the subject of a forthcoming paper.

Acknowledgements. The authors would like to express their gratitude to the hand ling editor and to an anonymous reviewer for their important comments and suggestions.

Appendix

Proof of Theorem 2.4. By the assumption [START_REF] Rockafellar | Convex Analysis[END_REF], for all x around x one has

8(cp o g)(x) = (f o G)(x),

where G(x) := (x, G(x)) and the mappings f: ]R n x Ilr ----. R n , G: ]R n =t JR_ m in the latter representation are defined by f(x, >. ) := (v'g(x)f >., G(x) := Ô cp(g(x)).

Next we invoke [START_REF] Mordukhovich | Variational Analysis and Generalized Differentiation. 1: Basic Theory[END_REF] (Lemma 1.126, Statement (1.64)), the assumptions of which are clearly fulfilled. One arrives for all y• E ]R n at the equality D*(f o G)(x, v)(y*) = (v' xf(x, X)f y*+ D*G(x, X)(v'g(x)y*) = ( t ,\; V 2 g,(X)) y• + D'G(x, À)(V g(X)y'),

  n 1 (x*,x} � 0 for all x E TA(x)} { ,. m,n I 1 . (x*, x -x} } = x E li'\. 11;1 �� P llx -xll � O • xEA[START_REF] Mordukhovich | Variational Analysis and Generalized Differentiation. 1: Basic Theory[END_REF]

Theorem 2 . 4 .

 24 Consider the composition tp o g, with tp : lR m -lR and g : ]R n -lR m . Let 8 be an arbitrary subdiff erential such that 8 <p has a closed graph.

Theorem 3 . 1 .

 31 The mapping S in (5) has the Aubin property around (p, z) if and only if one has the implirotion((Tb)i, (Bb)i) E -Nar ( é j ,) ((Bz)i, vi) Vi = 1, 2, ... , m � b = 0, ( �'. 5 )where v is the unique solution of the linear system ('. :6) Proof. Our starting point is the implication (20) in Theorem 2.2. So the ta.sk is to compute the coderivative of the multifonction Q given by[START_REF] Panagiotopoulos | Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Fu.nctions[END_REF]. To this aim, we invoke the second-order chain rule (23), whose assumptions are evidently satisfied because Bis surjective and Gr ( 8J) is closed (due to the assumed Lipschitz continuity of the fonction J). In this case, the adjoint general equation in[START_REF] Panagiotopoulos | Hernivariational lnequalities[END_REF] can be rewritten as o E M T b + B T D*8J ( B z, v)( Bb).

□Remark 3 . 2 .Theorem 3 . 3 .

 3233 2, ... , m {=::> (ai, -bi) E Nar(8j,)( xi, ui) \:/i = 1, 2, ... , m. The generalized equation (27) attains now the form of the system 0 = M T b+B T c ( e; ,( -Bb)i ) E Nar ( EJ j ,)(( B z)i,vi), Vi= 1,2, ... ,m. (29) It remains to express the vector c = (c1,c2, ... ,c,.,,,f E !R m in the form c = -Tb and the proof of Theorem 3.1 is thereby completed. (i) If n = m and B = I n (the identity matrix), then T = M7'. In this case, condition (25) takes the following form (ii) If M is nonsingular, we may use the equation in (29) in the reverse way to arrive at the condition ( Ci, ( Rc) i) E -Nac ( 8j;) ((Bz)i, vi) Vi = 1, 2, ... , m => c = 0, (31) where R := B(M T )-1 B T . Indeed, in this case b = -(M7')-1 B T c and the result follows easily from ( 29) because b=0�c = 0 due to the injectivity of B T .If, in addition, B = I n , then (31) reduces to the form Quite often in applications Mor Ris a P-matrix or a positive semi-definite matrix and then we can sometimes exploit these additional properties in connection with condition (30) and (31). We recall that a matrix ME R nxn is a P-matrix if all its principal minors are positive or equivalently (Vx E lR n ,x =/ 0)(3i E {1, ... ,n}): xi( Mx) i > O. Assume that M is nonsingular, R is a P-matrix and for all i = 1,2, ... ,m Nac ( /Ji,) ((B z)i, i\) C {( a ,b) E lR 2 : ab::_; 0}. (

Theorem 3 . 5 .

 35 Assume that M is symmetric and positive semi-definite, B = I n , condition (33) is fulfilled and Vb E ker M, :3i E { 1, 2, ... ,n} such that ( O,bi) <t -Nar(lJj;) ( zi,ïïi) .(35)Then S has the Aubin property around (p, z). Proof. Let us analyze a possible violaticn of condition (30) by a vector b. Assume first that b E ker ( M) \ { O}. Due to assumption (35) such a vector evidently cannot violate this condition. So let now b ft ke:r(M) so that ( b, Mb ) > O. However, the relations on the left-hand side of implication (30) imply that n (b, Mb ) = L bi ( \.1b ) i :S 0, i=l by virtue of (33). Consequently, this b cannot violate condition (30) as well , and thus the statement has been established. D We note that all the above conditions becc me really workable only in the case when we are given the functions ji and the reference point (p, z). A possible usage of Theorem 3.5 is now illustrated by the fol10wing academic example. Example 3.6. Let M= G �), B=h, i2 ( x2 ) = Jx2l + �x� with some E: > 0,

3 (

 3 ) we obtain that lip(S; (p, z)) = sup{llbll : M r b + B r D*8J(Bz, v)(Bb) n r& =I= 0}.To simplify the notation, let us denote lip(S; (p, z)) by K. Clearly, K equals the supremal value of the objective in the optimization problem maximize llbllsubject to M T b + B T w E B (w, -Bb) E N Gr ( 8J ) (Bz,v ) .Due to (2) the last constraint in (36) amounts to (w., (-Bb).) E N Gr ( 8i; ) (( Bz )i , v i ), i = 1, 2, ... , m.Since the fonctions j i have typically the form (36) (37) where r. is a closed interval and v. is piecewise C 2 in the sense defined in[START_REF] Mordukhovich | On second-order subdifferentials and their applications[END_REF], one can employ the results from [15, Section 4] and conclude that the normal cones in (37) can be expressed as unions of at most three convex polyhedral cones in IR 2 . So, if N Gr (8j; ) ( (Bz)i, vi) = u;= l A{ for all i, we can replace (36) by the disjunctive optimization problem maximize ll b ll subject to M7'b + B T w E B w i ,( -Bb ) i)E LJAi , i =l,2, ... ,m.

ti

  Clearly we have Gv i --+ Gv = u as i --+ +oo. Therefore, Hence, Gv--u • --+ Gw as i --+ +oo. ti It remains to prove that T A (ü) C {Gw : w E Ts(ïi)}, which can be conducted by a similar reasoning. Let k E T A (u), i.e., there are sequences k i --+ k, Àï L O such that u + Àiki = Gw i for some sequence (w i ) C 2. One has thus k-= G w iïi • Ài .

Remark 4 . 2 .Theorem 4 . 3 .

 4243 The surjectivity of A can be replaced by the weaker qualification condition A r z= 0 } z E Nr(Aïi) ==> z = O, provided r is normally regular at Av. On the basis of Theorem 2.3 and equality (39) we are now in a position to state the following characterization of the isolated calmness of S in terms of the problem data. Thereby we set (41) Assume that for each i = l, 2, ... , m there are neighborhoods Ui of the points ((Bz);,v i ) such that l;

(

  'P) {Ax + DyL + Eu = _ 0 y= Cx and YL E âJ(y),

  lt follows that there exists a vector y L E 8J ( Cx) such that:0 = Ax + DyL + Eu.By setting y= Cx, we have 0 = Ax+ DyL + Eu, y= Cx and YL E 8J(y).

Figure 5 . 1 :

 51 Figure 5.1: 1-V characteristic of a DIAC

1 K

 1 Its solution can be readily found in Figure 5.3, according to which . Using (49) and (43), it is easy to check that for the DIAC, S satisfies the isolated calmness property at (p, z) = (-1, 0) if and only if a=/= 1. The results about the Aubin property around and the isolated calmness at (p, z) = (-1, 0) for the DIAC circuit, depicted in Figure 5. 2, are summarized in Table 5. 1.

  59) => (b1, b 2) = (0, 0). with a = a 2 > 0 and /3 = a + l > 0, we observe that the Aubin property of the map S does not hold around (p, z) with z = (0, 0) and p = (l, -1 ). The isolated calmness property of S at ('\ z) is satisfied if and only if the following implication holds true }-(b1, b,) = (0, 0).

  .

  7.

Figure 5 . 7 :

 57 Figure 5.7: A double DIAC Clipping circuit: Characteristics of the DIACs D1 and D2

a2, ... a,) -{ t À;a; À;� 0, i-1,2, ... ,p} • (7)

	x�x
	For more details see [23] (Definitions 4.1). Let a 1 ,a 2 , ... a P E JR. n some given vectors. The conical hull of {a 1 ,a 2 , ... a p } is
	defined by
	cone{o1,

x, xk ___. x* with xk E <I>(x1o) Vk EN}.

[START_REF] Adly | A stability theory for second-order nonsmooth dynamical systems with applications to friction problems[END_REF]

1 •

 1 Proposition 5.1. Suppose that assumption (45) is satisfied. If (x,yL) is a solu tion of Problem (P) then z = Rx is a solution of Problem ( Q). Conversely, if z is a solution of Problem (Q) then there exists a function YL such that (R-1 z,YL) is a solution of Problem (P).

which is an inclusion of the form [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems[END_REF] with the variable z := i. For simplicity, we assume that R = 1. We will describe all situations concerning the Aubin property and the isolated calmness of the DIAC by using Theorem 3.1 and Theorem 4.3. In this case the operators T and W defined in [START_REF] Yakubovich | Solution of certain matrix inequalities in the stability theory of nonlinear control systems[END_REF] and (41) are equal to 1 and the matrix B = 1 . Therefore, the mapping S defined in [START_REF] Addi | A sensitivity analysis of a class of semi coercive variational inequalities using recession tools[END_REF] and associated to (47) has the Aubin property around a point (p, z) if and only if the following condition holds true

(48) Since the graph in Figure 5.1 is symmetric, we have to discuss only the following three situations:

• Suppose that z = 0 and j5 = -1. Using the (I-V)-characteristic of the DIAC in Figure 5.1, it is clear that the Aubin property will depend on the slope of the curve at x = o-. Let us suppose that J"(o-) = -a and we restrict ourselves to the nonmonotone case i.e. a > O. By definitions of the contingent and the limiting normal cones (11)- [START_REF] Mordukhovich | On second-order subdifferentials and their applications[END_REF], we have

Therefore, condition ( 48) is satisfied for (z, p) = (0, -1) if and only if 0 < a < 1.

Consequently, for the DIAC, the map S satisfies the Aubin property around (-1, 0)

if and only if O < a < 1.

In this case problem (38) attains the simple form Let us consider the circuit depicted in Figure 5.4 with a resistor R > 0, two DIACs, an input-signal source u(t) and two supply voltages E1 < E 2 • The current through the resistor is denoted by i = i 1 +i2 . Using Kirchhoff's laws, we have the following system

where ½ E â j 1(-i 1 ) and½ E âj2(i2) are the differences of potential across the DIACs D 1 and D 2 respectively. We suppose that the graphs of âj 1 and âj 2 are of the form depicted in Figure 5.1. In this case, it is easy to see that âj 1 ( -i 1 ) = -âj 1 (i 1 ). Setting M= ( � � ), p = (�=�:), z= ( !: ) (52)

and J(z) := j1 (i i ) -j2 (i 2 ), we see that the system in (51) is equivalent to the variational inclusion

which is of the form (1) with B = 1 2 . We note that here the matrix Mis symmetric and positive semidefinite. We will thus apply Theorem 3.1 and Remark 3.2 (i).

Let j�'(o-) = -a and j; ( o-) = a with a > 0 and assume that R = l .

• Suppose that the reference point (p, z) is given by z = (0, 0) and j5 = (l , 1), see Figure 5.5. which is equivalent to

A simple computation shows us that the limiting normal cones are given by and

In this case , equation ( 54) is equivalent to

=? ( b1, b2) = (0, 0).

A simple computation shows us that {57) does not hold. In fact, with a = a + l > 0 and /3 = l.

Consequently, the Aubin property of S around the point (p, z) with z = (0, 0) and p = (-1, 1) does not hold.

By Theorem 4.3, the isolated calmness property of S at (p, z) is satisfied if and only if the following implication holds true

which is equivalent to x* = (Vg(x)f a with a E D* âcp (g(x) ,>, )(Vg(x)y*),

and we are done.