Random projections of signal manifolds

Michael B. Wakin, Richard G. Baraniuk

Research output: Chapter in Book/Report/Conference proceedingConference contribution

13 Scopus citations

Abstract

Random projections have recently found a surprising niche in signal processing. The key revelation is that the relevant structure in a signal can be preserved when that signal is projected onto a small number of random basis functions. Recent work has exploited this fact under the rubric of Compressed Sensing (CS): signals that are sparse in some basis can be recovered from small numbers of random linear projections. In many cases, however, we may have a more specific low-dimensional model for signals in which the signal class forms a nonlinear manifold in ℝN. This paper provides preliminary theoretical and experimental evidence that manifold-based signal structure can be preserved using small numbers of random projections. The key theoretical motivation comes from Whitney's Embedding Theorem, which states that a A'-dimensional manifold can be embedded in ℝ2K+1. We examine the potential applications of this fact. In particular, we consider the task of recovering a manifold-modeled signal from a small number of random projections. Thanks to our more specific model, we can recover certain signals using far fewer measurements than would be required using sparsity-driven CS techniques.

Original languageEnglish (US)
Title of host publication2006 IEEE International Conference on Acoustics, Speech, and Signal Processing - Proceedings
Volume5
StatePublished - Dec 1 2006
Event2006 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 2006 - Toulouse, France
Duration: May 14 2006May 19 2006

Other

Other2006 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 2006
CountryFrance
CityToulouse
Period5/14/065/19/06

ASJC Scopus subject areas

  • Software
  • Signal Processing
  • Electrical and Electronic Engineering

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