Complex Interpolation of Matrices with an application to Multi-Manifold Learning
Adi Arbel, Stefan Steinerberger, Ronen Talmon
TLDR
Analyzes matrix interpolation to detect shared structures, enabling multi-manifold learning for multiview data analysis.
Key contributions
- Studies spectral properties of interpolations A^{1-x} B^x for symmetric positive-definite matrices.
- Shows exact log-linearity of operator norm implies shared eigenvectors between matrices.
- Provides stability bounds linking approximate log-linearity to principal singular vector alignment.
- Develops a multi-manifold learning framework to identify common and distinct latent structures.
Why it matters
This paper reveals how matrix interpolation uncovers shared features in data views, offering a principled method for multi-manifold learning. It bridges spectral theory and machine learning for improved latent structure discovery.
Original 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.
📬 Weekly AI Paper Digest
Get the top 10 AI/ML arXiv papers from the week — summarized, scored, and delivered to your inbox every Monday.