Tensor Extrema Estimation Via Sampling: A New Approach for Determining Minimum/Maximum Elements

Andrei Chertkov, Gleb V. Ryzhakov, Georgii S. Novikov, Ivan V. Oseledets

Abstract: The tensor train (TT) format, widely used in computational mathematics and machine learning, offers a computationally efficient method for handling multidimensional arrays, vectors, matrices, and discretized functions in various applications. In this article, we propose a new algorithm for estimating minimum/maximum elements of TT-tensors, which leads to accurate approximations. The method consists of sequential tensor multiplications of the TT-cores with an intelligent selection of candidates for the optimum. We propose a probabilistic interpretation of the method and estimate its complexity and convergence. We perform extensive numerical experiments with random tensors and various multivariable benchmark functions with the number of input dimensions up to 100. Our approach generates a solution close to the exact optimum for all model problems on a regular laptop.