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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.14118 |
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| _version_ | 1866913035760697344 |
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| author | Arbel, Adi Steinerberger, Stefan Talmon, Ronen |
| author_facet | Arbel, Adi Steinerberger, Stefan Talmon, Ronen |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_14118 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Complex Interpolation of Matrices with an application to Multi-Manifold Learning Arbel, Adi Steinerberger, Stefan Talmon, Ronen Machine Learning Spectral Theory 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. |
| title | Complex Interpolation of Matrices with an application to Multi-Manifold Learning |
| topic | Machine Learning Spectral Theory |
| url | https://arxiv.org/abs/2604.14118 |