拓展线性回归，用张量捕捉多响应非线性关系
Higher Order Reduced Rank Regression
- 用张量和Tucker分解建模非线性交互关系
- 通过黎曼优化求解，理论与实验验证有效
- 适合处理复杂多输出数据的科研人员
降低秩回归（RRR）是一种广泛用于多响应回归的方法。然而，RRR假设特征与响应之间存在线性关系。尽管线性模型在许多情况下表现良好，但现实世界中的许多问题涉及复杂关系，无法由简单线性交互充分描述。一种建模此类关系的方式是采用多线性变换。本文提出高阶降低秩回归（HORRR），作为RRR的扩展，利用多线性变换，从而能够捕捉多响应回归中的非线性交互。HORRR采用系数的张量表示，并使用Tucker分解施加多线性秩约束，作为正则化手段，类似于RRR中的秩约束。将这些约束编码为流形后，可使用黎曼优化求解该问题。本文从理论和实证角度分析了黎曼优化在求解HORRR问题中的有效性。
原文摘要 · Abstract (English)
Reduced Rank Regression (RRR) is a widely used method for multi-response regression. However, RRR assumes a linear relationship between features and responses. While linear models are useful and often provide a good approximation, many real-world problems involve more complex relationships that cannot be adequately captured by simple linear interactions. One way to model such relationships is via multilinear transformations. This paper introduces Higher Order Reduced Rank Regression (HORRR), an extension of RRR that leverages multi-linear transformations, and as such is capable of capturing nonlinear interactions in multi-response regression. HORRR employs tensor representations for the coefficients and a Tucker decomposition to impose multilinear rank constraints as regularization akin to the rank constraints in RRR. Encoding these constraints as a manifold allows us to use Riemannian optimization to solve this HORRR problems. We theoretically and empirically analyze the use of Riemannian optimization for solving HORRR problems.
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