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.

翻译：给定两个对称正定矩阵 $A, B \in \mathbb{R}^{n \times n}$，我们研究插值 $A^{1-x} B^x$ 在 $0 \leq x \leq 1$ 上的谱性质。利用这一插值视角，可以探究 $A$ 和 $B$ 中是否存在“共同结构”，即特征向量指向相似方向。一般而言，算子范数 $\|A^{1-x} B^x\|$ 的精确对数线性等价于原始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个矩阵的主特征向量对齐。这些结果催生并提供了一个多流形学习框架的理论依据，该框架用于识别多视图数据中共同与独特的潜在结构。