Initialization and training of matrix product state probabilistic models

TL;DR

This paper investigates the challenges in training matrix product state models for quantum-inspired probabilistic modeling, identifies failure modes such as causality traps, and proposes natural gradient descent and warm-start initialization strategies to improve training outcomes.

Contribution

It introduces a natural gradient descent method and a warm-start initialization to overcome training failures in MPS models, enabling more accurate quantum-inspired probabilistic modeling.

Findings

NGD significantly improves training efficiency

NGD with line search converges rapidly to the global minimum

Warm-start initialization prevents causality traps in Born machine training

Abstract

Modeling probability distributions via the wave function of a quantum state is central to quantum-inspired generative modeling and quantum state tomography (QST). We investigate a common failure mode in training randomly initialized matrix product states (MPS) using gradient descent. The results show that the trained MPS models do not accurately predict the strong interactions between boundary sites in periodic spin chain models. In the case of the Born machine algorithm, we further identify a causality trap, where the trained MPS models resemble causal models that ignore the non-local correlations in the true distribution. We propose two complementary strategies to overcome the training failure -- one through optimization and one through initialization. First, we develop a natural gradient descent (NGD) method, which approximately simulates the gradient flow on tensor manifolds and significantly enhances training efficiency. Numerical experiments show that NGD avoids local minima in both Born machines and in general MPS tomography. Remarkably, we show that NGD with line search can converge to the global minimum in only a few iterations. Second, for the BM algorithm, we introduce a warm-start initialization based on the TTNS-Sketch algorithm. We show that gradient descent under a warm initialization does not encounter the causality trap and admits rapid convergence to the ground truth.