通过矩阵插值揭示多视角数据中的共性与差异结构
Complex Interpolation of Matrices with an application to Multi-Manifold Learning
- 用A^{1-x}B^x插值分析两矩阵的谱特性
- 近似对数线性对应主奇异向量对齐，暗示共性结构
- 适用于发现多视图数据中的共享与独特潜变量
给定两个对称正定矩阵 $A, B \in \mathbb{R}^{n \times n}$，本文研究插值矩阵 $A^{1-x} B^x$（$0 \leq x \leq 1$）的谱性质。当 $A$ 与 $B$ 存在共同结构（如特征向量方向相近）时，该插值视角可有效揭示其关联。一般情况下，算子范数 $\|A^{1-x} B^x\|$ 的精确对数线性等价于两矩阵存在共享特征向量；稳定性界表明，近似对数线性迫使主奇异向量与两矩阵的主特征向量对齐。这些结果为多流形学习框架提供了理论支撑，可用于识别多视图数据中的共同与差异潜结构。
原文摘要 · Abstract (English)
Given two symmetric positive-definite matrices $A, B \in \mathbb{R}^{n \times n}$, we study the spectral properties of the interpolation $A^{1-x} B^x$ for $0 \leq x \leq 1$. The presence of `common structures' in $A$ and $B$, eigenvectors pointing in a similar direction, can be investigated using this interpolation perspective. Generically, exact log-linearity of the operator norm $\|A^{1-x} B^x\|$ is equivalent to the existence of a shared eigenvector in the original matrices; stability bounds show that approximate log-linearity forces principal singular vectors to align with leading eigenvectors of both matrices. These results give rise to and provide theoretical justification for a multi-manifold learning framework that identifies common and distinct latent structures in multiview data.
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