Bounded Manifold Completion

TL;DR

This paper introduces a novel matrix completion-based method for detecting low-dimensional manifolds within high-dimensional data, offering theoretical guarantees and robustness over existing nonlinear dimensionality reduction techniques.

Contribution

It presents a new approach leveraging low-rank matrix completion with convex relaxation to identify low-dimensional manifolds, with proven theoretical guarantees and improved robustness.

Findings

The method successfully detects low-dimensional manifolds in synthetic data.

It demonstrates robustness to non-uniform sampling in real-world datasets.

Theoretical analysis confirms the method's effectiveness.

Abstract

Nonlinear dimensionality reduction or, equivalently, the approximation of high-dimensional data using a low-dimensional nonlinear manifold is an active area of research. In this paper, we will present a thematically different approach to detect the existence of a low-dimensional manifold of a given dimension that lies within a set of bounds derived from a given point cloud. A matrix representing the appropriately defined distances on a low-dimensional manifold is low-rank, and our method is based on current techniques for recovering a partially observed matrix from a small set of fully observed entries that can be implemented as a low-rank Matrix Completion (MC) problem. MC methods are currently used to solve challenging real-world problems, such as image inpainting and recommender systems, and we leverage extent efficient optimization techniques that use a nuclear norm convex relaxation as a surrogate for non-convex and discontinuous rank minimization. Our proposed method provides several advantages over current nonlinear dimensionality reduction techniques, with the two most important being theoretical guarantees on the detection of low-dimensional embeddings and robustness to non-uniformity in the sampling of the manifold. We validate the performance of this approach using both a theoretical analysis as well as synthetic and real-world benchmark datasets.