On a Zero-Finding Problem Involving the Matrix Exponential

An important step in the solution of a matrix nearness problem that arises in certain machine learning 3 applications is finding the zero of f(α) = z exp(logX + αzz )z − b. The matrix valued exponential and logarithm 4 in f(α) arises from the use of the von Neumann matrix divergence tr(X logX − X log Y − X + Y ) to measure the 5 nearness between the positive definite matrices X and Y . A key step of an iterative algorithm used to solve the 6 underlying matrix nearness problem requires the zero of f(α) to be repeatedly computed. In this paper we propose 7 zero-finding algorithms that gain their advantage by exploiting the special structure of the objective function. We 8 show how to efficiently compute the derivative of f , thereby allowing the use of Newton-type methods. In numerical 9 experiments we establish the advantage of our algorithms. 10