7,426 research outputs found
On sharp performance bounds for robust sparse signal recoveries
It is well known in compressive sensing that l_1 minimization can recover the sparsest solution for a large class of underdetermined systems of linear equations, provided the signal is sufficiently sparse. In this paper, we compute sharp performance bounds for several different notions of robustness in sparse signal recovery via l_1 minimization. In particular, we determine necessary and sufficient conditions for the measurement matrix A under which l_1 minimization guarantees the robustness of sparse signal recovery in the "weak", "sectional" and "strong" (e.g., robustness for "almost all" approximately sparse signals, or instead for "all" approximately sparse signals). Based on these characterizations, we are able to compute sharp performance bounds on the tradeoff between signal sparsity and signal recovery robustness in these various senses. Our results are based on a high-dimensional geometrical analysis of the null-space of the measurement matrix A. These results generalize the thresholds results for purely sparse signals and also present generalized insights on l_1 minimization for recovering purely sparse signals from a null-space perspective
Statistical mechanics of error exponents for error-correcting codes
Error exponents characterize the exponential decay, when increasing message
length, of the probability of error of many error-correcting codes. To tackle
the long standing problem of computing them exactly, we introduce a general,
thermodynamic, formalism that we illustrate with maximum-likelihood decoding of
low-density parity-check (LDPC) codes on the binary erasure channel (BEC) and
the binary symmetric channel (BSC). In this formalism, we apply the cavity
method for large deviations to derive expressions for both the average and
typical error exponents, which differ by the procedure used to select the codes
from specified ensembles. When decreasing the noise intensity, we find that two
phase transitions take place, at two different levels: a glass to ferromagnetic
transition in the space of codewords, and a paramagnetic to glass transition in
the space of codes.Comment: 32 pages, 13 figure
Channel Detection in Coded Communication
We consider the problem of block-coded communication, where in each block,
the channel law belongs to one of two disjoint sets. The decoder is aimed to
decode only messages that have undergone a channel from one of the sets, and
thus has to detect the set which contains the prevailing channel. We begin with
the simplified case where each of the sets is a singleton. For any given code,
we derive the optimum detection/decoding rule in the sense of the best
trade-off among the probabilities of decoding error, false alarm, and
misdetection, and also introduce sub-optimal detection/decoding rules which are
simpler to implement. Then, various achievable bounds on the error exponents
are derived, including the exact single-letter characterization of the random
coding exponents for the optimal detector/decoder. We then extend the random
coding analysis to general sets of channels, and show that there exists a
universal detector/decoder which performs asymptotically as well as the optimal
detector/decoder, when tuned to detect a channel from a specific pair of
channels. The case of a pair of binary symmetric channels is discussed in
detail.Comment: Submitted to IEEE Transactions on Information Theor
Non-universality of elastic exponents in random bond-bending networks
We numerically investigate the rigidity percolation transition in
two-dimensional flexible, random rod networks with freely rotating cross-links.
Near the transition, networks are dominated by bending modes and the elastic
modulii vanish with an exponent f=3.0\pm0.2, in contrast with central force
percolation which shares the same geometric exponents. This indicates that
universality for geometric quantities does not imply universality for elastic
ones. The implications of this result for actin-fiber networks is discussed.Comment: 4 pages, 3 figures, minor clarifications and amendments. To appear in
PRE Rap. Com
Parallel sparse interpolation using small primes
To interpolate a supersparse polynomial with integer coefficients, two
alternative approaches are the Prony-based "big prime" technique, which acts
over a single large finite field, or the more recently-proposed "small primes"
technique, which reduces the unknown sparse polynomial to many low-degree dense
polynomials. While the latter technique has not yet reached the same
theoretical efficiency as Prony-based methods, it has an obvious potential for
parallelization. We present a heuristic "small primes" interpolation algorithm
and report on a low-level C implementation using FLINT and MPI.Comment: Accepted to PASCO 201
Random Subsets of Structured Deterministic Frames have MANOVA Spectra
We draw a random subset of rows from a frame with rows (vectors) and
columns (dimensions), where and are proportional to . For a
variety of important deterministic equiangular tight frames (ETFs) and tight
non-ETF frames, we consider the distribution of singular values of the
-subset matrix. We observe that for large they can be precisely
described by a known probability distribution -- Wachter's MANOVA spectral
distribution, a phenomenon that was previously known only for two types of
random frames. In terms of convergence to this limit, the -subset matrix
from all these frames is shown to be empirically indistinguishable from the
classical MANOVA (Jacobi) random matrix ensemble. Thus empirically the MANOVA
ensemble offers a universal description of the spectra of randomly selected
-subframes, even those taken from deterministic frames. The same
universality phenomena is shown to hold for notable random frames as well. This
description enables exact calculations of properties of solutions for systems
of linear equations based on a random choice of frame vectors out of
possible vectors, and has a variety of implications for erasure coding,
compressed sensing, and sparse recovery. When the aspect ratio is small,
the MANOVA spectrum tends to the well known Marcenko-Pastur distribution of the
singular values of a Gaussian matrix, in agreement with previous work on highly
redundant frames. Our results are empirical, but they are exhaustive, precise
and fully reproducible
Rank Minimization over Finite Fields: Fundamental Limits and Coding-Theoretic Interpretations
This paper establishes information-theoretic limits in estimating a finite
field low-rank matrix given random linear measurements of it. These linear
measurements are obtained by taking inner products of the low-rank matrix with
random sensing matrices. Necessary and sufficient conditions on the number of
measurements required are provided. It is shown that these conditions are sharp
and the minimum-rank decoder is asymptotically optimal. The reliability
function of this decoder is also derived by appealing to de Caen's lower bound
on the probability of a union. The sufficient condition also holds when the
sensing matrices are sparse - a scenario that may be amenable to efficient
decoding. More precisely, it is shown that if the n\times n-sensing matrices
contain, on average, \Omega(nlog n) entries, the number of measurements
required is the same as that when the sensing matrices are dense and contain
entries drawn uniformly at random from the field. Analogies are drawn between
the above results and rank-metric codes in the coding theory literature. In
fact, we are also strongly motivated by understanding when minimum rank
distance decoding of random rank-metric codes succeeds. To this end, we derive
distance properties of equiprobable and sparse rank-metric codes. These
distance properties provide a precise geometric interpretation of the fact that
the sparse ensemble requires as few measurements as the dense one. Finally, we
provide a non-exhaustive procedure to search for the unknown low-rank matrix.Comment: Accepted to the IEEE Transactions on Information Theory; Presented at
IEEE International Symposium on Information Theory (ISIT) 201
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