No-Free-Lunch Theories for Tensor-Network Machine Learning Models

TL;DR

This paper establishes rigorous no-free-lunch theorems for tensor network machine learning models, revealing their fundamental limitations in generalization capabilities across different tensor network architectures.

Contribution

It provides the first formal no-free-lunch theorems for matrix product states and projected entangled-pair states, advancing understanding of tensor network model limitations.

Findings

Proves no-free-lunch theorem for matrix product states.

Extends the theorem to two-dimensional PEPS using combinatorial methods.

Highlights intrinsic limitations of tensor network models in learning tasks.

Abstract

Tensor network machine learning models have shown remarkable versatility in tackling complex data-driven tasks, ranging from quantum many-body problems to classical pattern recognitions. Despite their promising performance, a comprehensive understanding of the underlying assumptions and limitations of these models is still lacking. In this work, we focus on the rigorous formulation of their no-free-lunch theorem -- essential yet notoriously challenging to formalize for specific tensor network machine learning models. In particular, we rigorously analyze the generalization risks of learning target output functions from input data encoded in tensor network states. We first prove a no-free-lunch theorem for machine learning models based on matrix product states, i.e., the one-dimensional tensor network states. Furthermore, we circumvent the challenging issue of calculating the partition function for two-dimensional Ising model, and prove the no-free-lunch theorem for the case of two-dimensional projected entangled-pair state, by introducing the combinatorial method associated to the "puzzle of polyominoes". Our findings reveal the intrinsic limitations of tensor network-based learning models in a rigorous fashion, and open up an avenue for future analytical exploration of both the strengths and limitations of quantum-inspired machine learning frameworks.