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Matrices de covariance : diffusions, probabilités libres, et apprentissage profond

Abstract : This thesis is motivated by the study of covariance matrices, and is naturally structured in three parts. In the first part, we study dynamic models related to covariance matrices. More precisely, we study the systems of stochastic differential equations inherited from the dynamics of the eigenvalues of matrix valued processes named the Wishart process and the Jacobi process. The solutions to these systems are respectively the [dollar]beta[dollar]-Wishart process and the [dollar]beta[dollar]-Jacobi process. We extend the known results on the existence and uniqueness of solutions to these equations and we characterize their long term behaviour. In the second part, in the light of modern results from the free probability theory, especially about the rectangular additive free convolution, we study the behaviour of the empirical measure of the particules of the [dollar]beta[dollar]-Wishart process in the large dimension limit, and establish the commutativity between the long time and the high dimensional limits for this sequence of measure valued process. The third part is related to the study of stability of the backpropagation procedure in the learning phase of a feed-forward neural network with a variable width profile. This work focuses on the Jacobian matrix of the network, which can be seen as a long product of matrices, and whose spectral properties are crucial for the stability of the gradient descent. This appeals for the definition of a free rectangular multiplicative convolution. We suggest an efficient algorithm to compute the empirical measure of the square of the singular values of this matrix. The conclusion of this work give insights allowing to assess the stability of a feed-forward neural network, aiming to assist practitioners in the choice of the features in the design of neural networks.
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Submitted on : Monday, January 24, 2022 - 6:36:19 PM
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Ezechiel Kahn. Matrices de covariance : diffusions, probabilités libres, et apprentissage profond. Probabilités [math.PR]. École des Ponts ParisTech, 2021. Français. ⟨NNT : 2021ENPC0016⟩. ⟨tel-03541689⟩



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