Abstract
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.
