The Dynamics of Message Passing on Dense Graphs, with Applications to Compressed Sensing

"Approximate message passing" (AMP) algorithms have proved to be effective in reconstructing sparse signals from a small number of incoherent linear measurements. Extensive numerical experiments further showed that their dynamics is accurately tracked by a simple one-dimensional iteration...

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Veröffentlicht in:IEEE transactions on information theory Jg. 57; H. 2; S. 764 - 785
Hauptverfasser: Bayati, M, Montanari, A
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York IEEE 01.02.2011
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:
Algorithm design and analysis
Algorithms
Approximation algorithms
Compressed
Compressed sensing
Context
Convergence
Density
density evolution
Detection
Dynamics
Evolution
Graphs
Message passing
message passing algorithms
random matrix theory
Sensors
state evolution
ISSN:0018-9448, 1557-9654
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Zusammenfassung:"Approximate message passing" (AMP) algorithms have proved to be effective in reconstructing sparse signals from a small number of incoherent linear measurements. Extensive numerical experiments further showed that their dynamics is accurately tracked by a simple one-dimensional iteration termed state evolution. In this paper, we provide rigorous foundation to state evolution. We prove that indeed it holds asymptotically in the large system limit for sensing matrices with independent and identically distributed Gaussian entries. While our focus is on message passing algorithms for compressed sensing, the analysis extends beyond this setting, to a general class of algorithms on dense graphs. In this context, state evolution plays the role that density evolution has for sparse graphs. The proof technique is fundamentally different from the standard approach to density evolution, in that it copes with a large number of short cycles in the underlying factor graph. It relies instead on a conditioning technique recently developed by Erwin Bolthausen in the context of spin glass theory.
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2010.2094817