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

• URL: http://arxiv.org/abs/2604.14118v1
• Date: Wed, 15 Apr 2026 17:40:37 GMT
• Title: Complex Interpolation of Matrices with an application to Multi-Manifold Learning
• Abstract summary: We study the spectral properties of the $A1-x Bx$ for $0 leq x leq 1$.<n>The presence of common structures' in $A$ and $Bx|$ can be investigated using this perspective.<n>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.
• Score: 12.540068560688958
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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