A Manifold Approach to Learning Mutually Orthogonal Subspaces

TL;DR

This paper introduces the partitioned subspace manifold, a novel geometric framework enabling the optimization of matrices representing multiple mutually orthogonal subspaces, facilitating flexible feature learning in machine learning tasks.

Contribution

It formulates a new manifold structure for optimizing matrices constrained to multiple subspaces, extending Riemannian optimization techniques to more general subspace constraints.

Findings

Effective in multiple dataset analysis

Enhances domain adaptation capabilities

Supports learning distinct feature groups

Abstract

Although many machine learning algorithms involve learning subspaces with particular characteristics, optimizing a parameter matrix that is constrained to represent a subspace can be challenging. One solution is to use Riemannian optimization methods that enforce such constraints implicitly, leveraging the fact that the feasible parameter values form a manifold. While Riemannian methods exist for some specific problems, such as learning a single subspace, there are more general subspace constraints that offer additional flexibility when setting up an optimization problem, but have not been formulated as a manifold. We propose the partitioned subspace (PS) manifold for optimizing matrices that are constrained to represent one or more subspaces. Each point on the manifold defines a partitioning of the input space into mutually orthogonal subspaces, where the number of partitions and their sizes are defined by the user. As a result, distinct groups of features can be learned by defining different objective functions for each partition. We illustrate the properties of the manifold through experiments on multiple dataset analysis and domain adaptation.