15 research outputs found
On Finding a Subset of Healthy Individuals from a Large Population
In this paper, we derive mutual information based upper and lower bounds on
the number of nonadaptive group tests required to identify a given number of
"non defective" items from a large population containing a small number of
"defective" items. We show that a reduction in the number of tests is
achievable compared to the approach of first identifying all the defective
items and then picking the required number of non-defective items from the
complement set. In the asymptotic regime with the population size , to identify non-defective items out of a population
containing defective items, when the tests are reliable, our results show
that measurements are
sufficient, where is a constant independent of and , and
is a bounded function of and . Further, in the nonadaptive group
testing setup, we obtain rigorous upper and lower bounds on the number of tests
under both dilution and additive noise models. Our results are derived using a
general sparse signal model, by virtue of which, they are also applicable to
other important sparse signal based applications such as compressive sensing.Comment: 32 pages, 2 figures, 3 tables, revised version of a paper submitted
to IEEE Trans. Inf. Theor
Compressive Demodulation of Mutually Interfering Signals
Multi-User Detection is fundamental not only to cellular wireless
communication but also to Radio-Frequency Identification (RFID) technology that
supports supply chain management. The challenge of Multi-user Detection (MUD)
is that of demodulating mutually interfering signals, and the two biggest
impediments are the asynchronous character of random access and the lack of
channel state information. Given that at any time instant the number of active
users is typically small, the promise of Compressive Sensing (CS) is the
demodulation of sparse superpositions of signature waveforms from very few
measurements. This paper begins by unifying two front-end architectures
proposed for MUD by showing that both lead to the same discrete signal model.
Algorithms are presented for coherent and noncoherent detection that are based
on iterative matching pursuit. Noncoherent detection is all that is needed in
the application to RFID technology where it is only the identity of the active
users that is required. The coherent detector is also able to recover the
transmitted symbols. It is shown that compressive demodulation requires
samples to recover active users whereas
standard MUD requires samples to process total users with a
maximal delay . Performance guarantees are derived for both coherent and
noncoherent detection that are identical in the way they scale with number of
active users. The power profile of the active users is shown to be less
important than the SNR of the weakest user. Gabor frames and Kerdock codes are
proposed as signature waveforms and numerical examples demonstrate the superior
performance of Kerdock codes - the same probability of error with less than
half the samples.Comment: submitted for journal publicatio
Support Recovery with Sparsely Sampled Free Random Matrices
Consider a Bernoulli-Gaussian complex -vector whose components are , with X_i \sim \Cc\Nc(0,\Pc_x) and binary mutually independent
and iid across . This random -sparse vector is multiplied by a square
random matrix \Um, and a randomly chosen subset, of average size , , of the resulting vector components is then observed in additive
Gaussian noise. We extend the scope of conventional noisy compressive sampling
models where \Um is typically %A16 the identity or a matrix with iid
components, to allow \Um satisfying a certain freeness condition. This class
of matrices encompasses Haar matrices and other unitarily invariant matrices.
We use the replica method and the decoupling principle of Guo and Verd\'u, as
well as a number of information theoretic bounds, to study the input-output
mutual information and the support recovery error rate in the limit of . We also extend the scope of the large deviation approach of Rangan,
Fletcher and Goyal and characterize the performance of a class of estimators
encompassing thresholded linear MMSE and relaxation
Limits on Support Recovery with Probabilistic Models: An Information-Theoretic Framework
The support recovery problem consists of determining a sparse subset of a set
of variables that is relevant in generating a set of observations, and arises
in a diverse range of settings such as compressive sensing, and subset
selection in regression, and group testing. In this paper, we take a unified
approach to support recovery problems, considering general probabilistic models
relating a sparse data vector to an observation vector. We study the
information-theoretic limits of both exact and partial support recovery, taking
a novel approach motivated by thresholding techniques in channel coding. We
provide general achievability and converse bounds characterizing the trade-off
between the error probability and number of measurements, and we specialize
these to the linear, 1-bit, and group testing models. In several cases, our
bounds not only provide matching scaling laws in the necessary and sufficient
number of measurements, but also sharp thresholds with matching constant
factors. Our approach has several advantages over previous approaches: For the
achievability part, we obtain sharp thresholds under broader scalings of the
sparsity level and other parameters (e.g., signal-to-noise ratio) compared to
several previous works, and for the converse part, we not only provide
conditions under which the error probability fails to vanish, but also
conditions under which it tends to one.Comment: Accepted to IEEE Transactions on Information Theory; presented in
part at ISIT 2015 and SODA 201
Limits on Sparse Support Recovery via Linear Sketching with Random Expander Matrices
Linear sketching is a powerful tool for the problem of sparse signal recovery, having numerous applications such as compressive sensing, data stream computing, graph sketching, and routing. Motivated by applications where the \emph{positions} of the non-zero entries in a sparse vector are of primary interest, we consider the problem of \emph{support recovery} from a linear sketch taking the form \Yv = \Xv\beta + \Zv. We focus on a widely-used expander-based construction in the columns of the measurement matrix \Xv \in \RR^{n \times p} are random permutations of a sparse binary vector containing ones and zeros. We provide a sharp characterization of the number of measurements required for an information-theoretically optimal decoder, thus permitting a precise comparison to the i.i.d.~Gaussian construction. Our findings reveal both positive and negative results, showing that the performance nearly matches the Gaussian construction at moderate-to-high noise levels, while being worse by an arbitrarily large factor at low noise levels