91 research outputs found
Precise Phase Transition of Total Variation Minimization
Characterizing the phase transitions of convex optimizations in recovering
structured signals or data is of central importance in compressed sensing,
machine learning and statistics. The phase transitions of many convex
optimization signal recovery methods such as minimization and nuclear
norm minimization are well understood through recent years' research. However,
rigorously characterizing the phase transition of total variation (TV)
minimization in recovering sparse-gradient signal is still open. In this paper,
we fully characterize the phase transition curve of the TV minimization. Our
proof builds on Donoho, Johnstone and Montanari's conjectured phase transition
curve for the TV approximate message passing algorithm (AMP), together with the
linkage between the minmax Mean Square Error of a denoising problem and the
high-dimensional convex geometry for TV minimization.Comment: 6 page
Message Passing Algorithms for Compressed Sensing
Compressed sensing aims to undersample certain high-dimensional signals, yet
accurately reconstruct them by exploiting signal characteristics. Accurate
reconstruction is possible when the object to be recovered is sufficiently
sparse in a known basis. Currently, the best known sparsity-undersampling
tradeoff is achieved when reconstructing by convex optimization -- which is
expensive in important large-scale applications. Fast iterative thresholding
algorithms have been intensively studied as alternatives to convex optimization
for large-scale problems. Unfortunately known fast algorithms offer
substantially worse sparsity-undersampling tradeoffs than convex optimization.
We introduce a simple costless modification to iterative thresholding making
the sparsity-undersampling tradeoff of the new algorithms equivalent to that of
the corresponding convex optimization procedures. The new
iterative-thresholding algorithms are inspired by belief propagation in
graphical models. Our empirical measurements of the sparsity-undersampling
tradeoff for the new algorithms agree with theoretical calculations. We show
that a state evolution formalism correctly derives the true
sparsity-undersampling tradeoff. There is a surprising agreement between
earlier calculations based on random convex polytopes and this new, apparently
very different theoretical formalism.Comment: 6 pages paper + 9 pages supplementary information, 13 eps figure.
Submitted to Proc. Natl. Acad. Sci. US
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