Co-manifold learning with missing data

TL;DR

This paper introduces a co-manifold learning method that leverages the coupled geometry of rows and columns in a data matrix, especially with missing data, to improve data visualization and clustering.

Contribution

The paper presents a novel unsupervised approach for co-manifold learning that estimates complete matrices at multiple scales and uses a new multi-scale metric to capture coupled row and column structures.

Findings

Outperforms existing methods in data visualization

Achieves better clustering results

Effectively handles missing data in matrices

Abstract

Representation learning is typically applied to only one mode of a data matrix, either its rows or columns. Yet in many applications, there is an underlying geometry to both the rows and the columns. We propose utilizing this coupled structure to perform co-manifold learning: uncovering the underlying geometry of both the rows and the columns of a given matrix, where we focus on a missing data setting. Our unsupervised approach consists of three components. We first solve a family of optimization problems to estimate a complete matrix at multiple scales of smoothness. We then use this collection of smooth matrix estimates to compute pairwise distances on the rows and columns based on a new multi-scale metric that implicitly introduces a coupling between the rows and the columns. Finally, we construct row and column representations from these multi-scale metrics. We demonstrate that our approach outperforms competing methods in both data visualization and clustering.