86,911 research outputs found
"Compressed" Compressed Sensing
The field of compressed sensing has shown that a sparse but otherwise
arbitrary vector can be recovered exactly from a small number of randomly
constructed linear projections (or samples). The question addressed in this
paper is whether an even smaller number of samples is sufficient when there
exists prior knowledge about the distribution of the unknown vector, or when
only partial recovery is needed. An information-theoretic lower bound with
connections to free probability theory and an upper bound corresponding to a
computationally simple thresholding estimator are derived. It is shown that in
certain cases (e.g. discrete valued vectors or large distortions) the number of
samples can be decreased. Interestingly though, it is also shown that in many
cases no reduction is possible
Information Theoretic Limits for Standard and One-Bit Compressed Sensing with Graph-Structured Sparsity
In this paper, we analyze the information theoretic lower bound on the
necessary number of samples needed for recovering a sparse signal under
different compressed sensing settings. We focus on the weighted graph model, a
model-based framework proposed by Hegde et al. (2015), for standard compressed
sensing as well as for one-bit compressed sensing. We study both the noisy and
noiseless regimes. Our analysis is general in the sense that it applies to any
algorithm used to recover the signal. We carefully construct restricted
ensembles for different settings and then apply Fano's inequality to establish
the lower bound on the necessary number of samples. Furthermore, we show that
our bound is tight for one-bit compressed sensing, while for standard
compressed sensing, our bound is tight up to a logarithmic factor of the number
of non-zero entries in the signal
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