Events (We may skip the seminar in case it overlaps with some other academic events.)
Abstract: TBA
Abstract: In this talk we study the spectra of matrix-valued contractions of the Gaussian Orthogonal Tensor Ensemble (GOTE). Let $\mathcal{G}$ denote a random tensor of dimension $n$ and order-$r$, drawn from the density \[ f(\mathcal{G}) = \frac{1}{Z_r(n)} \exp\bigg(-\frac{1}{2r}\|\mathcal{G}\|^2_{\mathrm{F}}\bigg).\]
We consider contractions of the form $\mathcal{G} \cdot \mathbf{w}^{\otimes (r - 2)}$ when both $r$ and $n$ go to infinity such that $r / n \to c \in [0, \infty]$. We obtain a Baik-Ben Arous-P\'{e}ch\'{e} phase transition for the largest and the smallest eigenvalues of such contractions at $r = 3$. We also show that the extreme eigenvectors contain non-trivial information about $\mathbf w$. In fact, in the regime $1 \ll r \ll n$, there are two vectors, one of which is perfectly aligned with $w$. We also obtain some results on mixed contractions $\mathcal{G} \cdot \mathbf{u} \otimes \mathbf{v}$ in the case $r = 4$. While the total variation distance between the joint distribution of the entries of $\mathcal{G} \cdot \mathbf{u} \otimes \mathbf{v}$ and that of $\mathcal{G} \cdot \mathbf{u} \otimes \mathbf{u}$ goes to $0$ when $\|\mathbf{u} - \mathbf{v}\| = o(n^{-1})$, the bulk and the largest eigenvalues of these matrices have the same limit profile as long as $\|\mathbf{u} - \mathbf{v}\| = o(1)$. Further, it turns out that there are no outlier eigenvalues when $\langle \mathbf{u}, \mathbf{v}\rangle = o(1)$. This talk is based on a joint work with Soumendu Sundar Mukherjee.
Abstract: TBA