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arXiv · 2604.14118

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

Adi Arbel·Stefan Steinerberger·Ronen Talmon
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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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Adi Arbel, Stefan Steinerberger, Ronen Talmon. 2026-04-15. Complex Interpolation of Matrices with an application to Multi-Manifold Learning. https://arxiv.org/abs/2604.14118

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