Inverse of a Sum of Random Matrices: A Dynamical Mean-Field Approach

Abstract

We study $(\mathbf{O}^\top \mathbf{A} \mathbf{O} + \mathbf{B})^{-1}$, where $\mathbf{A}, \mathbf{B} \in \mathbb{R}^{N \times N}$ are symmetric positive definite, with limiting eigenvalue distributions ${\rm F}_{\mathbf{A}}, {\rm F}_{\mathbf{B}}$ and spectra bounded and bounded away from zero, and $\mathbf{O}$ is Haar orthogonal. We analyze the deviation of deterministic-equivalent-type approximation $\boldsymbol{\Delta}\doteq (\mathbf{O}^\top \mathbf{A} \mathbf{O} + \mathbf{B})^{-1}- \mathbf{S}^{-1}$ where $\mathbf{S} = \mathbf{B} - z^\star \mathbf{I}$. First, let $\mathbf{O}$ be independent of $(\mathbf{A}, \mathbf{B}, \mathbf{u})$, $\|\mathbf{u}\| = 1$. There is sequence of a standard Gaussian vectors $\mathbf{g}_N$ independent of $(\mathbf{A}, \mathbf{B}, \mathbf{u})$ with $\| \mathbf{S} \boldsymbol{\Delta} \mathbf{S} \mathbf{u} - \sqrt{\frac \tau N}\, \mathbf{g}_N \| \to 0$ a.s. Here $z^\star = -{\rm R}(-\chi)$ and $\tau = {\rm R}'(-\chi)(1 + \eta {\rm R}'(-\chi))$, where ${\rm R}$ is the R-transform of ${\rm F}_{\mathbf{A}}$ and $(\chi, \eta)$ are the first two inverse moments of the free additive convolution of ${\rm F}_{\mathbf{A}}, {\rm F}_{\mathbf{B}}$. Let $\mathbf{K} = \sum_{i \le \lfloor N^q \rfloor} \lambda_i \mathbf{u}_i \mathbf{v}_i^\top$, $q \in [0, C]$, with unit-norm, not necessarily orthogonal $\mathbf{u}_i, \mathbf{v}_i$ and $\limsup_N \max_i |\lambda_i|<\infty$ a.s. If $\mathbf{O}$ is independent of $(\mathbf{A}, \mathbf{B}, \{\mathbf{u}_i, \mathbf{v}_i\})$, then $N^{-q} {\rm tr}(\mathbf{K} \boldsymbol{\Delta}) \to 0$ a.s. The proofs use a dynamical mean-field approach: the generating functions of the free Appell polynomials give a linear dynamics approximating the quantity of interest in the iterated large-$N$, large-time limit, whose fluctuations we analyze.