29 research outputs found
On the Power of Adaptivity in Sparse Recovery
The goal of (stable) sparse recovery is to recover a -sparse approximation
of a vector from linear measurements of . Specifically, the goal is
to recover such that ||x-x*||_p <= C min_{k-sparse x'} ||x-x'||_q for some
constant and norm parameters and . It is known that, for or
, this task can be accomplished using non-adaptive
measurements [CRT06] and that this bound is tight [DIPW10,FPRU10,PW11].
In this paper we show that if one is allowed to perform measurements that are
adaptive, then the number of measurements can be considerably reduced.
Specifically, for and we show - A scheme with measurements that uses
rounds. This is a significant improvement over the best possible non-adaptive
bound. - A scheme with measurements
that uses /two/ rounds. This improves over the best possible non-adaptive
bound. To the best of our knowledge, these are the first results of this type.
As an independent application, we show how to solve the problem of finding a
duplicate in a data stream of items drawn from using
bits of space and passes, improving over the best
possible space complexity achievable using a single pass.Comment: 18 pages; appearing at FOCS 201
Info-Greedy sequential adaptive compressed sensing
We present an information-theoretic framework for sequential adaptive
compressed sensing, Info-Greedy Sensing, where measurements are chosen to
maximize the extracted information conditioned on the previous measurements. We
show that the widely used bisection approach is Info-Greedy for a family of
-sparse signals by connecting compressed sensing and blackbox complexity of
sequential query algorithms, and present Info-Greedy algorithms for Gaussian
and Gaussian Mixture Model (GMM) signals, as well as ways to design sparse
Info-Greedy measurements. Numerical examples demonstrate the good performance
of the proposed algorithms using simulated and real data: Info-Greedy Sensing
shows significant improvement over random projection for signals with sparse
and low-rank covariance matrices, and adaptivity brings robustness when there
is a mismatch between the assumed and the true distributions.Comment: Preliminary results presented at Allerton Conference 2014. To appear
in IEEE Journal Selected Topics on Signal Processin