Stable Manifold Embeddings with Structured Random Matrices

TL;DR

This paper demonstrates that structured random matrices satisfying the RIP can be used to create stable embeddings for manifold-modeled signals by randomizing column signs, enabling efficient projections for complex data.

Contribution

It shows how any RIP-compliant matrix can be adapted for manifold embeddings through sign randomization, extending efficient projection methods to manifold models.

Findings

Structured matrices satisfying RIP can embed manifold signals.

Randomizing column signs enhances embedding stability.

New constructions for manifold embeddings using structured matrices.

Abstract

The fields of compressed sensing (CS) and matrix completion have shown that high-dimensional signals with sparse or low-rank structure can be effectively projected into a low-dimensional space (for efficient acquisition or processing) when the projection operator achieves a stable embedding of the data by satisfying the Restricted Isometry Property (RIP). It has also been shown that such stable embeddings can be achieved for general Riemannian submanifolds when random orthoprojectors are used for dimensionality reduction. Due to computational costs and system constraints, the CS community has recently explored the RIP for structured random matrices (e.g., random convolutions, localized measurements, deterministic constructions). The main contribution of this paper is to show that any matrix satisfying the RIP (i.e., providing a stable embedding for sparse signals) can be used to construct a stable embedding for manifold-modeled signals by randomizing the column signs and paying reasonable additional factors in the number of measurements. We demonstrate this result with several new constructions for stable manifold embeddings using structured matrices. This result allows advances in efficient projection schemes for sparse signals to be immediately applied to manifold signal models.