Multiscale Co-Manifold Learning on Tensors

TMLR Paper8076 Authors

Abstract: Nonlinear manifold learning on tensors typically compute mode-wise embeddings that fail to capture implicit couplings between the geometries of the different modes. We propose a new framework called tensor co-manifold learning (TCML). TCML is designed to recover coupled low-dimensional structures {\em simultaneously} across all modes of multiway data (i.e., tensors or multi-dimensional arrays) and generalizes recent methods for co-manifold learning to higher-order tensors via a tensor-based multiscale approach to co-organizing rows, columns, and higher modes. By imposing smoothness constraints at various levels of granularity, we formulate a family of optimization problems that characterize smoothness across coarse-to-fine scales. We demonstrate that these problems are efficiently solvable and their solutions yield a multiscale distance between tensor slices along a given mode. These distances take into account the structure of the data along the other modes. We demonstrate how to utilize this multiscale distance measure to compute nonlinear embeddings of the data. The resulting embeddings are demonstrably more effective at revealing low-dimensional coupled structure than linear factorizations or nonlinear embeddings obtained by treating each mode independently.

Submission Type: Regular submission (no more than 12 pages of main content)

Assigned Action Editor: ~Mahito_Sugiyama1

Submission Number: 8076