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

arXiv: https://arxiv.org/abs/2604.14118 · HTML: https://mlsanity.com/paper/2604.14118/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 25 references that are in the map.
- First posted on arXiv 15 Apr 2026.

## Builds on (most cited first, 6 of 6)

- [Nonparametric Canonical Correlation Analysis](https://mlsanity.com/paper/1511.04839/) (2015)
- [Latent common manifold learning with alternating diffusion: analysis and applications](https://mlsanity.com/paper/1602.00078/) (2016)
- [Recovering Hidden Components in Multimodal Data with Composite Diffusion Operators](https://mlsanity.com/paper/1808.07312/) (2018)
- [Spatiotemporal Analysis Using Riemannian Composition of Diffusion Operators](https://mlsanity.com/paper/2201.08530/) (2022)
- [Spectral Discovery of Jointly Smooth Features for Multimodal Data](https://mlsanity.com/paper/2004.04386/) (2020)
- [A common variable minimax theorem for graphs](https://mlsanity.com/paper/2107.14747/) (2021)

## Abstract (arXiv)

Given two symmetric positive-definite matrices A, B ∈ ℝn × n, we study the spectral properties of the interpolation A1-x Bx for 0 ≤ x ≤ 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 |A1-x Bx| 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.