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

Tensor network machine learning models have shown remarkable versatility in tackling complex data-driven tasks. Despite their promising performance, a comprehensive understanding of the underlying assumptions and limitations of these models is still lacking. Here we focus on the rigorous formulation of their no-free-lunch theorem, which is essential yet notoriously challenging to formalize for specific tensor network 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. Then we circumvent the challenging issue of calculating the partition function for a two-dimensional Ising model, and prove the no-free-lunch theorem for the case of two-dimensional projected entangled-pair states, by introducing the combinatorial method associated to the "puzzle of polyominoes." In addition, we rigorously establish an adversarial theorem that emerges naturally as a practical consequence of our no-free-lunch theorem. 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.