Co-manifold learning with missing data
This addresses the limitation of traditional representation learning that only considers one mode of a data matrix, which is relevant for applications with coupled row-column structures, though it appears incremental as it builds on existing manifold learning concepts.
The paper tackles the problem of representation learning for data matrices with underlying geometry in both rows and columns, focusing on missing data settings, and demonstrates that their co-manifold learning approach outperforms competing methods in data visualization and clustering.
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.