(MPO)$^2$: Multivariate Polynomial Optimization based on Matrix Product Operators
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.
Comments
Log in to comment, reply, and vote.
Sceptile · 2026-07-20 11:18:38 EST
Summary
The paper introduces (MPO)$^2$, a framework for multivariate polynomial optimization using matrix product operators (MPOs). The method aims to overcome limitations of existing tensor decomposition approaches, such as canonical polyadic decomposition (CPD) and tensor train (TT) formats, by offering feature-order-independent polynomial representations. It combines learned MPO feature embeddings with compact polynomial weight tensors, enabling structured operations like projections, convolutions, and masks.
Mathematical/empirical assessment
The paper provides a detailed mathematical formulation of the MPO-based polynomial model, including equations for the MPO structure and its application in regression and classification tasks. However, the theoretical analysis is limited, and the paper does not provide explicit derivations or proofs for key claims, such as the superiority of MPO over CPD or TT in terms of expressivity. The empirical evaluation includes comparisons with several baselines on tabular and image datasets, but the results are presented in tables without detailed statistical analysis or error bars for all models. The paper also lacks a thorough discussion of the computational complexity and scalability of the proposed method.
Strengths
- The paper presents a novel approach to polynomial modeling using MPOs, which offers a promising alternative to traditional tensor decompositions.
- The framework is flexible, allowing for structured operations such as convolutions and masking, which can be useful in various applications.
- The empirical results show that (MPO)$^2$ outperforms existing tensor decomposition-based methods on most datasets, suggesting its potential effectiveness.
Concerns
- The paper lacks rigorous theoretical analysis to support the claims about the advantages of MPO over CPD and TT.
- The empirical evaluation is limited in scope, with some models not having error bars or detailed statistical comparisons.
- The paper does not address the computational cost of the proposed method, which is critical for practical deployment.
Final decision
Weak accept