On a Zero-Finding Problem Involving the Matrix Exponential

Abstract

An important step in the solution of a matrix nearness problem that arises in certain machine learning applications is finding the zero of $f(\alpha) = \bm{z}^T \exp(\log X + \alpha\bm{z}\bm{z}^T)\bm{z} - b$. The matrix valued exponential and logarithm in $f(\alpha)$ arises from the use of the von Neumann matrix divergence $\operatorname{tr}(X \log X - X \log Y - X + Y)$ to measure the nearness between the positive definite matrices $X$ and $Y$. A key step of an iterative algorithm used to solve the underlying matrix nearness problem requires the zero of $f(\alpha)$ to be repeatedly computed. In this paper we propose zero-finding algorithms that gain their advantage by exploiting the special structure of the objective function. We show how to efficiently compute the derivative of $f$, thereby allowing the use of Newton-type methods. In numerical experiments we establish the advantage of our algorithms.