Sandrine Péché

Lucas Benigni, Sandrine Péché

This paper is concerned with the asymptotic distribution of the largest eigenvalues for some nonlinear random matrix ensemble stemming from the study of neural networks. More precisely we consider $M= \frac{1}{m} YY^\top$ with $Y=f(WX)$ where $W$ and $X$ are random rectangular matrices with i.i.d. centered entries. This models the data covariance matrix or the Conjugate Kernel of a single layered random Feed-Forward Neural Network. The function $f$ is applied entrywise and can be seen as the activation function of the neural network. We show that the largest eigenvalue has the same limit (in probability) as that of some well-known linear random matrix ensembles. In particular, we relate the asymptotic limit of the largest eigenvalue for the nonlinear model to that of an information-plus-noise random matrix, establishing a possible phase transition depending on the function $f$ and the distribution of $W$ and $X$. This may be of interest for applications to machine learning.

Sandrine Peche, Vianney Perchet

We consider the stochastic block model where connection between vertices is perturbed by some latent (and unobserved) random geometric graph. The objective is to prove that spectral methods are robust to this type of noise, even if they are agnostic to the presence (or not) of the random graph. We provide explicit regimes where the second eigenvector of the adjacency matrix is highly correlated to the true community vector (and therefore when weak/exact recovery is possible). This is possible thanks to a detailed analysis of the spectrum of the latent random graph, of its own interest.

Lucas Benigni, Sandrine Péché

This paper is concerned with the asymptotic empirical eigenvalue distribution of a non linear random matrix ensemble. More precisely we consider $M= \frac{1}{m} YY^*$ with $Y=f(WX)$ where $W$ and $X$ are random rectangular matrices with i.i.d. centered entries. The function $f$ is applied pointwise and can be seen as an activation function in (random) neural networks. We compute the asymptotic empirical distribution of this ensemble in the case where $W$ and $X$ have sub-Gaussian tails and $f$ is real analytic. This extends a previous result where the case of Gaussian matrices $W$ and $X$ is considered. We also investigate the same questions in the multi-layer case, regarding neural network applications.