Johannes Heiny: Recent advances in large sample correlation matrices and their applications

Many fields of modern sciences are faced with high-dimensional data sets. In this talk, we investigate the spectral properties of large sample correlation matrices. First, we consider a $p$-dimensional population with iid coordinates in the domain of attraction of a stable distribution with index $\alpha\in (0,2)$. Since the variance is infinite, the sample covariance matrix based on a sample of size $n$ from the population is not well behaved and it is of interest to use instead the sample correlation matrix $R$. We find the limiting distributions of the eigenvalues of $R$ when both the dimension $p$ and the sample size n grow to infinity such that $p/n\to \gamma$. The moments of the limiting distributions $H_{\alpha,\gamma}$ are fully identified as the sum of two contributions: the first from the classical Marchenko-Pastur law and a second due to heavy tails. Moreover, the family $\{H_{\alpha,\gamma}\}$ has continuous extensions at the boundaries $\alpha=2$ and $\alpha=0$ leading to the Marchenko-Pastur law and a modified Poisson distribution, respectively. A simulation study on these limiting distributions is also provided for comparison with the Marchenko-Pastur law.

In the second part of this talk, we assume that the coordinates of the $p$-dimensional population are dependent and $p/n \le 1$. Under a finite fourth moment condition on the entries we find that the log determinant of the sample correlation matrix $R$ satisfies a central limit theorem. In the iid case, it turns out the central limit theorem holds as long as the coordinates are in the domain of attraction of a stable distribution with index $\alpha>3$, from which we conjecture a promising and robust test statistic for heavy-tailed high-dimensional data. The findings are applied to independence testing and to the volume of random simplices. Reference: Limiting distributions for eigenvalues of sample correlation matrices from heavy-tailed populations https://arxiv.org/abs/2003.03857

Alexey Onatskiy: Spin-glass to Paramagnetic Transition in Spherical Sherrington-Kirkpatrick Model with Ferromagnetic Interaction

This is joint paper with I.M. Johnstone, Y. Klochkov, and D. Pavlyshyn. This paper studies fluctuations of the free energy of the Spherical Sherrington-Kirkpatrick model with ferromagnetic Curie-Weiss interaction with coupling constant $J \in[0,1)$ and inverse temperature $\beta$. We consider the critical temperature regime $\beta=1+bN^{-1/3}\sqrt{\log N}$, $b \in \mathbb{R} \backslash \{0\}$. For $b<0$, the limiting distribution of the free energy is Gaussian. As $b$ increases from $0$ to $+\infty$, we describe the transition of the limiting distribution from Gaussian to Tracy-Widom.

Sandrine Peche: Non-linear random matrix ensembles
We consider a random matrix model M = YY? where Y = (f(WX)_ij) and W and X are large rectangular matrices with iid entries.The function f is called the activation function in certain neural networks. Pennington and Worah have identified the empirical eigenvalue distribution of such random matrices in the Gaussian case (W and X). We extend their result to a wider class of distributions for a certain class of activation functions. This is joint work with Lucas Benigni