58 research outputs found
A typical reconstruction limit of compressed sensing based on Lp-norm minimization
We consider the problem of reconstructing an -dimensional continuous
vector \bx from constraints which are generated by its linear
transformation under the assumption that the number of non-zero elements of
\bx is typically limited to (). Problems of this
type can be solved by minimizing a cost function with respect to the -norm
||\bx||_p=\lim_{\epsilon \to +0}\sum_{i=1}^N |x_i|^{p+\epsilon}, subject to
the constraints under an appropriate condition. For several , we assess a
typical case limit , which represents a critical relation
between and for successfully reconstructing the original
vector by minimization for typical situations in the limit
with keeping finite, utilizing the replica method. For ,
is considerably smaller than its worst case counterpart, which
has been rigorously derived by existing literature of information theory.Comment: 12 pages, 2 figure
Probabilistic Reconstruction in Compressed Sensing: Algorithms, Phase Diagrams, and Threshold Achieving Matrices
Compressed sensing is a signal processing method that acquires data directly
in a compressed form. This allows one to make less measurements than what was
considered necessary to record a signal, enabling faster or more precise
measurement protocols in a wide range of applications. Using an
interdisciplinary approach, we have recently proposed in [arXiv:1109.4424] a
strategy that allows compressed sensing to be performed at acquisition rates
approaching to the theoretical optimal limits. In this paper, we give a more
thorough presentation of our approach, and introduce many new results. We
present the probabilistic approach to reconstruction and discuss its optimality
and robustness. We detail the derivation of the message passing algorithm for
reconstruction and expectation max- imization learning of signal-model
parameters. We further develop the asymptotic analysis of the corresponding
phase diagrams with and without measurement noise, for different distribution
of signals, and discuss the best possible reconstruction performances
regardless of the algorithm. We also present new efficient seeding matrices,
test them on synthetic data and analyze their performance asymptotically.Comment: 42 pages, 37 figures, 3 appendixe
- …