Abstract. 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 ⊕ 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(ε) 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 Erdős-Rényi 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 rate of the graph by α = np/ log(n), it is shown that exact recovery is possible if and only if α > 2/(1 − 2ε) 2 + o(1/(1 − 2ε) 2 ). In other words, 2/(1 − 2ε) 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 α = d/ log(n), where d is the minimum degree of the graph, it is shown that the proposed method achieves the rate α > 4((1 + λ)/(1 − λ) 2 )/(1 − 2ε) 2 + o(1/(1 − 2ε) 2 ), where 1 − λ is the spectral gap of the graph G.
We consider the recovery of sparse signals subject to sparse interference, as introduced in Studer et al., IEEE Trans. IT, 2012. We present novel probabilistic recovery guarantees for this framework, covering varying degrees of knowledge of the signal and interference support, which are relevant for a large number of practical applications. Our results assume that the sparsifying dictionaries are solely characterized by coherence parameters and we require randomness only in the signal and/or interference. The obtained recovery guarantees show that one can recover sparsely corrupted signals with overwhelming probability, even if the sparsity of both the signal and interference scale (near) linearly with the number of measurements.
The identification (ID) capacity region of the two-receiver broadcast channel
(BC) is shown to be the set of rate-pairs for which, for some distribution on
the channel input, each receiver's ID rate does not exceed the mutual
information between the channel input and the channel output that it observes.
Moreover, the capacity region's interior is achieved by codes with
deterministic encoders. The results are obtained under the average-error
criterion, which requires that each receiver reliably identify its message
whenever the message intended for the other receiver is drawn at random. They
hold also for channels whose transmission capacity region is to-date unknown.
Key to the proof is a new ID code construction for the single-user channel.
Extensions to the BC with one-sided feedback and the three-receiver BC are also
discussed: inner bounds on their ID capacity regions are obtained, and those
are shown to be in some cases tight.Comment: 83 pages, a shorter version is published in the IEEE Transactions on
Information Theor
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