Complex Interpolation of Matrices with an application to Multi-Manifold Learning

TL;DR

This paper analyzes the spectral properties of matrix interpolations to identify shared structures in multiview data, providing theoretical insights and stability bounds relevant for multi-manifold learning.

Contribution

It introduces a spectral analysis framework for matrix interpolation that reveals conditions for shared eigenvectors and supports multi-manifold learning applications.

Findings

Exact log-linearity of the operator norm indicates shared eigenvectors.

Approximate log-linearity implies alignment of principal singular vectors with leading eigenvectors.

The results justify a multi-manifold learning approach for multiview data.

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.