MPO-Based Polynomial Optimization Boosts Function Approximation

In machine learning and signal processing, universal function approximation and learning complex input-output relationships from limited data are central challenges. Multivariate polynomial models naturally express these relationships through multiplicative feature interactions, but their coefficient tensors grow exponentially with polynomial degree, posing a significant computational hurdle. While existing tensorized polynomial models aim to reduce this cost, canonical polyadic decompositions have limited expressivity, and tensor train formulations are dependent on feature order. Researchers have developed Multivariate Polynomial Optimization based on Matrix Product Operators, or (MPO)$^2$. This innovative framework combines learned MPO feature embeddings with compact polynomial weight tensors. The result is a feature-order independent polynomial representation that can also integrate structured operators like projections, convolutions, and masks to handle weight tensor symmetries. Across various regression and classification benchmarks, (MPO)$^2$ consistently outperforms existing polynomial models based on tensor decomposition. It provides a flexible and efficient alternative for approximating complex polynomial functions, addressing key limitations of previous approaches in terms of expressivity and computational scalability.