Complex Interpolation of Matrices with an application to Multi-Manifold Learning
For researchers in multi-view learning and manifold learning, this work provides theoretical foundations for a method to separate common and distinct structures, though the framework is not yet empirically validated.
The paper studies spectral properties of matrix interpolation A^{1-x}B^x for symmetric positive-definite matrices, showing that exact log-linearity of the operator norm indicates a shared eigenvector, while approximate log-linearity forces alignment of principal singular vectors with leading eigenvectors. This provides theoretical justification for a multi-manifold learning framework to identify common and distinct latent structures in multiview data.
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.