提出新型多项式模型，实现高效且灵活的复杂关系建模。
(MPO)$^2$: Multivariate Polynomial Optimization based on Matrix Product Operators
- 用矩阵乘积算子学习特征嵌入，结合紧凑权重张量构建多项式
- 在回归与分类任务中性能优于现有张量分解方法
- 支持卷积、投影等结构化操作，适合复杂数据建模
机器学习与信号处理的核心在于从有限观测中学习复杂的输入输出关系。多变量多项式模型通过特征交互自然表达此类关系，但其系数张量随多项式阶数呈指数增长。现有张量化多项式模型虽降低计算成本，但经典帕累托分解表达能力受限，张量列车形式依赖特征顺序。本文提出基于矩阵乘积算子的多变量多项式优化框架（MPO）²，将可学习的MPO特征嵌入与紧凑的多项式权重张量结合，实现特征顺序无关的多项式表示，并可融入投影、卷积、掩码等结构化算子以捕捉权重张量对称性。在回归与分类基准测试中，(MPO)²优于现有基于张量分解的多项式模型，为高效多项式函数逼近提供灵活替代方案。
原文摘要 · Abstract (English)
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.
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