A matrix concentration inequality for products

TL;DR

This paper establishes a non-asymptotic concentration inequality for the product of random positive semidefinite matrices, providing probabilistic bounds on deviations from the expected matrix product.

Contribution

It introduces a novel concentration inequality for products of bounded independent positive semidefinite matrices, extending matrix concentration results to multiplicative processes.

Findings

Provides explicit probability bounds for matrix product deviations.

Shows the inequality holds for small enough step size lpha.

Applicable to high-dimensional matrix analysis and stochastic processes.

Abstract

We present a non-asymptotic concentration inequality for the random matrix product \begin{equation}\label{eq:Zn} Z_n = \left(I_d-\alpha X_n\right)\left(I_d-\alpha X_{n-1}\right)\cdots \left(I_d-\alpha X_1\right), \end{equation} where $\left\{X_k \right\}_{k=1}^{+\infty}$ is a sequence of bounded independent random positive semidefinite matrices with common expectation $\mathbb{E}\left[X_k\right]=\Sigma$. Under these assumptions, we show that, for small enough positive $\alpha$, $Z_n$ satisfies the concentration inequality \begin{equation}\label{eq:CTbound} \mathbb{P}\left(\left\Vert Z_n-\mathbb{E}\left[Z_n\right]\right\Vert \geq t\right) \leq 2d^2\cdot\exp\left(\frac{-t^2}{\alpha \sigma^2} \right) \quad \text{for all } t\geq 0, \end{equation} where $\sigma^2$ denotes a variance parameter.