AI 中文总结
该研究针对多元多项式模型系数张量随次数指数增长问题，提出基于矩阵乘积算子的多元多项式优化框架（MPO）$^2$，结合特征嵌入与权重张量，提升表现力且与特征顺序无关，在回归和分类基准测试中优于现有模型。
AI 中文摘要
机器学习和信号处理的核心是执行通用函数逼近并从有限数量的观测中学习复杂输入输出关系的能力。多元多项式模型通过乘法特征交互提供了表达此类关系的自然方式，但其系数张量大小随多项式次数呈指数增长。现有张量多项式模型降低了此成本，但规范多adic分解具有秩限制的表现力，张量序列公式依赖于特征顺序。我们引入基于矩阵乘积算子（MPO）$^2$的多元多项式优化，它将学习到的MPO特征嵌入与紧凑的多项式权重张量相结合。这产生了与特征顺序无关的多项式表示，可纳入结构化算子，如投影、卷积和用于权重张量对称性的掩码。在回归和分类基准测试中，（MPO）$^2$优于现有的基于张量分解的多项式模型，并为高效多项式函数逼近提供了灵活选择。
英文摘要
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.
Comments13 pages, 1 figure, 2 tables
