Dimensionality reduction

Quantum mechanics

Algorithm

Engineering

Pure mathematics

Statistics

Materials science

Riemannian manifold

Random projection

Curse of dimensionality

Rank (graph theory)

Eigenvalues and eigenvectors

Computer science

Projection (relational algebra)

Embedding

Random matrix

Mechanical engineering

Matrix (chemical analysis)

Physics

Mathematics

Artificial intelligence

Composite material

Combinatorics

Manifold (fluid mechanics)

Compressed sensing

Isometry (Riemannian geometry)

Restricted isometry property

IEEE Journal of Selected Topics in Signal Processing

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, thereby generalizing previous stable manifold embedding results beyond unstructured random matrices. 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.

Stable Manifold Embeddings With Structured Random Matrices