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Register a new userDecoding binary node labels from censored edge measurements: Phase transition and efficient recovery
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Added by
Afonso S. Bandeira
Publication date
2014
fields
Informatics Engineering
and research's language is
English
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We consider the problem of clustering a graph $G$ into two communities by observing a subset of the vertex correlations. Specifically, we consider the inverse problem with observed variables $Y=B_G x oplus Z$, where $B_G$ is the incidence matrix of a graph $G$, $x$ is the vector of unknown vertex variables (with a uniform prior) and $Z$ is a noise vector with Bernoulli$(varepsilon)$ i.i.d. entries. All variables and operations are Boolean. This model is motivated by coding, synchronization, and community detection problems. In particular, it corresponds to a stochastic block model or a correlation clustering problem with two communities and censored edges. Without noise, exact recovery (up to global flip) of $x$ is possible if and only the graph $G$ is connected, with a sharp threshold at the edge probability $log(n)/n$ for ErdH{o}s-Renyi random graphs. The first goal of this paper is to determine how the edge probability $p$ needs to scale to allow exact recovery in the presence of noise. Defining the degree (oversampling) rate of the graph by $alpha =np/log(n)$, it is shown that exact recovery is possible if and only if $alpha >2/(1-2varepsilon)^2+ o(1/(1-2varepsilon)^2)$. In other words, $2/(1-2varepsilon)^2$ is the information theoretic threshold for exact recovery at low-SNR. In addition, an efficient recovery algorithm based on semidefinite programming is proposed and shown to succeed in the threshold regime up to twice the optimal rate. For a deterministic graph $G$, defining the degree rate as $alpha=d/log(n)$, where $d$ is the minimum degree of the graph, it is shown that the proposed method achieves the rate $alpha> 4((1+lambda)/(1-lambda)^2)/(1-2varepsilon)^2+ o(1/(1-2varepsilon)^2)$, where $1-lambda$ is the spectral gap of the graph $G$.
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We consider the problem of decoding a discrete signal of categorical variables from the observation of several histograms of pooled subsets of it. We present an Approximate Message Passing (AMP) algorithm for recovering the signal in the random dense setting where each observed histogram involves a random subset of entries of size proportional to n. We characterize the performance of the algorithm in the asymptotic regime where the number of observations $m$ tends to infinity proportionally to n, by deriving the corresponding State Evolution (SE) equations and studying their dynamics. We initiate the analysis of the multi-dimensional SE dynamics by proving their convergence to a fixed point, along with some further properties of the iterates. The analysis reveals sharp phase transition phenomena where the behavior of AMP changes from exact recovery to weak correlation with the signal as m/n crosses a threshold. We derive formulae for the threshold in some special cases and show that they accurately match experimental behavior.
In the long-studied problem of combinatorial group testing, one is asked to detect a set of $k$ defective items out of a population of size $n$, using $m ll n$ disjunctive measurements. In the non-adaptive setting, the most widely used combinatorial objects are disjunct and list-disjunct matrices, which define incidence matrices of test schemes. Disjunct matrices allow the identification of the exact set of defectives, whereas list disjunct matrices identify a small superset of the defectives. Apart from the combinatorial guarantees, it is often of key interest to equip measurement designs with efficient decoding algorithms. The most efficient decoders should run in sublinear time in $n$, and ideally near-linear in the number of measurements $m$. In this work, we give several constructions with an optimal number of measurements and near-optimal decoding time for the most fundamental group testing tasks, as well as for central tasks in the compressed sensing and heavy hitters literature. For many of those tasks, the previous measurement-optimal constructions needed time either quadratic in the number of measurements or linear in the universe size. Most of our results are obtained via a clean and novel approach which avoids list-recoverable codes or related complex techniques which were present in almost every state-of-the-art work on efficiently decodable constructions of such objects.
This paper is concerned with the problem of recovering a structured signal from a relatively small number of corrupted random measurements. Sharp phase transitions have been numerically observed in practice when different convex programming procedures are used to solve this problem. This paper is devoted to presenting theoretical explanations for these phenomenons by employing some basic tools from Gaussian process theory. Specifically, we identify the precise locations of the phase transitions for both constrained and penalized recovery procedures. Our theoretical results show that these phase transitions are determined by some geometric measures of structure, e.g., the spherical Gaussian width of a tangent cone and the Gaussian (squared) distance to a scaled subdifferential. By utilizing the established phase transition theory, we further investigate the relationship between these two kinds of recovery procedures, which also reveals an optimal strategy (in the sense of Lagrange theory) for choosing the tradeoff parameter in the penalized recovery procedure. Numerical experiments are provided to verify our theoretical results.
Reed-Muller codes are some of the oldest and most widely studied error-correcting codes, of interest for both their algebraic structure as well as their many algorithmic properties. A recent beautiful result of Saptharishi, Shpilka and Volk showed that for binary Reed-Muller codes of length $n$ and distance $d = O(1)$, one can correct $operatorname{polylog}(n)$ random errors in $operatorname{poly}(n)$ time (which is well beyond the worst-case error tolerance of $O(1)$). In this paper, we consider the problem of `syndrome decoding Reed-Muller codes from random errors. More specifically, given the $operatorname{polylog}(n)$-bit long syndrome vector of a codeword corrupted in $operatorname{polylog}(n)$ random coordinates, we would like to compute the locations of the codeword corruptions. This problem turns out to be equivalent to a basic question about computing tensor decomposition of random low-rank tensors over finite fields. Our main result is that syndrome decoding of Reed-Muller codes (and the equivalent tensor decomposition problem) can be solved efficiently, i.e., in $operatorname{polylog}(n)$ time. We give two algorithms for this problem: 1. The first algorithm is a finite field variant of a classical algorithm for tensor decomposition over real numbers due to Jennrich. This also gives an alternate proof for the main result of Saptharishi et al. 2. The second algorithm is obtained by implementing the steps of the Berlekamp-Welch-style decoding algorithm of Saptharishi et al. in sublinear-time. The main new ingredient is an algorithm for solving certain kinds of systems of polynomial equations.
A new family of operators, coined hierarchical measurement operators, is introduced and discussed within the well-known hierarchical sparse recovery framework. Such operator is a composition of block and mixing operations and notably contains the Kronecker product as a special case. Results on their hierarchical restricted isometry property (HiRIP) are derived, generalizing prior work on recovery of hierarchically sparse signals from Kronecker-structured linear measurements. Specifically, these results show that, very surprisingly, sparsity properties of the block and mixing part can be traded against each other. The measurement structure is well-motivated by a massive random access channel design in communication engineering. Numerical evaluation of user detection rates demonstrate the huge benefit of the theoretical framework.
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