1 research outputs found
Unknown sparsity in compressed sensing: Denoising and inference
The theory of Compressed Sensing (CS) asserts that an unknown signal
can be accurately recovered from an underdetermined set of
linear measurements with , provided that is sufficiently
sparse. However, in applications, the degree of sparsity is typically
unknown, and the problem of directly estimating has been a
longstanding gap between theory and practice. A closely related issue is that
is a highly idealized measure of sparsity, and for real signals with
entries not equal to 0, the value is not a useful description of
compressibility. In our previous conference paper [Lop13] that examined these
problems, we considered an alternative measure of "soft" sparsity,
, and designed a procedure to estimate
that does not rely on sparsity assumptions.
The present work offers a new deconvolution-based method for estimating
unknown sparsity, which has wider applicability and sharper theoretical
guarantees. In particular, we introduce a family of entropy-based sparsity
measures
parameterized by . This family interpolates between
and as ranges over .
For any , we propose an estimator
whose relative error converges at the dimension-free rate of , even
when . Our main results also describe the limiting distribution
of , as well as some connections to Basis Pursuit Denosing, the
Lasso, deterministic measurement matrices, and inference problems in CS.Comment: The title of the previous tech report has been updated so that it
matches the published version. The published version contains additional
materia