Stable Manifold Embeddings With Structured Random 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, 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.

A first analysis of the stability of Takens' embedding

“…Before presenting the stable embedding results proper, we will need to introduce an important characterization of manifolds 1 The notation O(·) simply means "on the order of". This result is extended in [10,11] to include other random stable embedding operators. Moreover, the results in [9,11] remove the logarithmic dependence on M .…”

“…Reach is an equivalent measure to the condition number used in [8,10,17]. See the above cites and [11,18] for more on reach and its implications.…”

“…The notion of a stable embedding by a measurement operator F introduced in compressed sensing (CS) [7] has been extended to manifold signal families [8][9][10][11]. The results in [8] guarantee that distances between points on a D-dimensional submanifold are approximately preserved when F is a random, linear orthoprojector and M = O(d log(N )).…”

Restricted Isometry Property of Subspace Projection Matrix Under Random Compression

“…It is easily shown in Part B of the proof of Theorem III.1 in [13] that the -cover of C(B T ) satisfies…”

Restricted Isometry Property of Gaussian Random Projection for Finite Set of Subspaces

“…It has been proved in [24,25,26,27] that, with high probability the random projection matrix Φ can preserve the distance between two signals belonging to a UoS. Recently, the stable embedding property has been extended to signals modeled as low-dimensional Riemannian submanifolds in Euclidean space [28,29,30].…”