(MPO)$^2$: Multivariate Polynomial Optimization based on Matrix Product Operators

2026-07-17

AI Summary

This work addresses the challenges in multivariate polynomial models arising from the exponential growth of coefficient tensors with model order, the limited expressivity of existing tensor decomposition methods, and their sensitivity to feature ordering. To overcome these limitations, the authors propose an efficient optimization framework based on matrix product operators (MPOs). The approach leverages learnable MPO-based feature embeddings and compact polynomial weight tensors to achieve permutation-invariant function approximation. Furthermore, structured operators—such as projection, convolution, and masking—are incorporated to explicitly model weight symmetries, thereby transcending the expressivity bottlenecks of conventional tensor decompositions. Experimental results demonstrate that the proposed framework significantly outperforms existing tensor-decomposition-based polynomial models on both regression and classification benchmarks, offering a highly expressive, flexible, and computationally efficient solution for polynomial approximation.

Abstract

Central to machine learning and signal processing is the ability to perform universal function approximation and learn complex input-output relationships from limited numbers of observations. Multivariate polynomial models offer a natural way to express such relationships through multiplicative feature interactions, but their coefficient tensors grow exponentially in size with the polynomial degree. Existing tensorized polynomial models reduce this cost, yet canonical polyadic decompositions have rank-limited expressivity, and tensor train formulations are feature order dependent. We introduce Multivariate Polynomial Optimization based on Matrix Product Operators (MPO)$^2$, a framework that combines learned MPO feature embeddings with compact polynomial weight tensors. This yields feature order independent polynomial representations that can incorporate structured operators such as projections, convolutions, and masks for weight tensor symmetries. Across regression and classification benchmarks, (MPO)$^2$ improves over existing tensor decomposition based polynomial models and provides a flexible alternative for efficient polynomial function approximation.

Problem

• multivariate polynomial
• tensor decomposition
• feature order dependence
• coefficient tensor explosion
• function approximation

Innovation

• Matrix Product Operators
• Multivariate Polynomial Optimization
• Tensor Decomposition
• Feature Order Independence
• Structured Operators