Learning the nonlinear geometry of high-dimensional data: Models and algorithms

TL;DR

This paper introduces two new nonlinear geometric models, MC-UoS and MC-KUoS, for learning the low-dimensional structures of high-dimensional data, along with efficient algorithms and experimental validation.

Contribution

It formalizes the MC-UoS and MC-KUoS models, develops algorithms for their learning, extends them to handle missing data, and demonstrates their superiority through experiments.

Findings

Proposed models outperform existing methods in experiments.

Algorithms effectively learn geometric structures from synthetic and real data.

Models handle missing data scenarios successfully.

Abstract

Modern information processing relies on the axiom that high-dimensional data lie near low-dimensional geometric structures. This paper revisits the problem of data-driven learning of these geometric structures and puts forth two new nonlinear geometric models for data describing "related" objects/phenomena. The first one of these models straddles the two extremes of the subspace model and the union-of-subspaces model, and is termed the metric-constrained union-of-subspaces (MC-UoS) model. The second one of these models---suited for data drawn from a mixture of nonlinear manifolds---generalizes the kernel subspace model, and is termed the metric-constrained kernel union-of-subspaces (MC-KUoS) model. The main contributions of this paper in this regard include the following. First, it motivates and formalizes the problems of MC-UoS and MC-KUoS learning. Second, it presents algorithms that efficiently learn an MC-UoS or an MC-KUoS underlying data of interest. Third, it extends these algorithms to the case when parts of the data are missing. Last, but not least, it reports the outcomes of a series of numerical experiments involving both synthetic and real data that demonstrate the superiority of the proposed geometric models and learning algorithms over existing approaches in the literature. These experiments also help clarify the connections between this work and the literature on (subspace and kernel k-means) clustering.