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

2026-04-15

AI Summary

This work investigates the identification of shared and view-specific latent structures in multi-view data, focusing on scenarios where symmetric positive definite matrices exhibit common eigendirections. By analyzing the spectral properties of matrix interpolants of the form \(A^{1-x}B^x\), the study establishes that the operator norm displays strict or approximate log-linear behavior if and only if the matrices share dominant eigenvectors. Leveraging this theoretical insight, the authors develop a multi-manifold learning framework that integrates complex matrix interpolation, spectral analysis, and singular vector alignment to effectively disentangle common and specific latent variables across views. This approach provides a novel theoretical foundation and a practical methodology for multi-view representation learning.

Technology Category

• Machine Learning: Multi-instance/Multi-view Learning
• Computer Vision: Multi-modal Vision
• Intelligent Robots: Multimodal Perception & Sensor Fusion

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Interdisciplinary science discovery with web data mining
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• matrix interpolation
• multi-manifold learning
• shared eigenvectors
• symmetric positive-definite matrices
• latent structures

Innovation

• matrix interpolation
• spectral alignment
• multi-manifold learning
• log-linearity
• shared eigenvectors