Research
Complex Interpolation of Matrices with an application to Multi-Manifold Learning
arXiv:2604.14118v1 Announce Type: new Abstract: Given two symmetric positive-definite matrices A, B in R^{n imes n}, we study the spectral properties of the interpolation A^{1-x} B^x for 0 leq x leq 1
arXiv:2604.14118v1 Announce Type: new Abstract: Given two symmetric positive-definite matrices A, B in R^{n imes 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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Source: arXiv cs.LG | 2026-04-16