Regularized second-order optimization of tensor-network Born machines

TL;DR

This paper introduces a regularized second-order optimization method for tensor-network Born machines, improving training efficiency and stability in learning complex data distributions with quantum-inspired models.

Contribution

The paper proposes a novel regularized Newton's method on the manifold of normalized states for TNBMs, enhancing convergence and avoiding local minima during training.

Findings

Significantly faster convergence rates.

Improved stability and robustness in training.

Effective on both discrete and continuous datasets.

Abstract

Tensor-network Born machines (TNBMs) are quantum-inspired generative models for learning data distributions. Using tensor-network contraction and optimization techniques, the model learns an efficient representation of the target distribution, capable of capturing complex correlations with a compact parameterization. Despite their promise, the optimization of TNBMs presents several challenges. A key bottleneck of TNBMs is the logarithmic nature of the loss function commonly used for this problem. The single-tensor logarithmic optimization problem cannot be solved analytically, necessitating an iterative approach that slows down convergence and increases the risk of getting trapped in one of many non-optimal local minima. In this paper, we present an improved second-order optimization technique for TNBM training, which significantly enhances convergence rates and the quality of the optimized model. Our method employs a modified Newton's method on the manifold of normalized states, incorporating regularization of the loss landscape to mitigate local minima issues. We demonstrate the effectiveness of our approach by training a one-dimensional matrix product state (MPS) on both discrete and continuous datasets, showcasing its advantages in terms of stability and efficiency, and demonstrating its potential as a robust and scalable approach for optimizing quantum-inspired generative models.