Consider a database of @math objects, e.g., a set of videos, where each object has a binary attribute, e.g., a video's suitability for children. The attributes of the objects are to be determined under a crowdsourcing model: a worker is queried about the labels of a chosen subset of objects and either responds with a correct binary answer or declines to respond. Here we propose a parity response model: the worker is asked to check whether the number of objects having a given attribute in the chosen subset is even or odd. For example, if the subset includes two objects, the worker checks whether the two belong to the same class or not. We propose a method for designing the sequence of subsets of objects to be queried so that the attributes of the objects can be identified with high probability using few ( @math ) answers. The method is based on an analogy to the design of Fountain codes for erasure channels. We define the query difficulty @math as the average size of the query subsets and we define the sample complexity @math as the minimum number of collected answers required to attain a given recovery accuracy. We obtain fundamental tradeoffs between recovery accuracy, query difficulty, and sample complexity. In particular, the necessary and sufficient sample complexity required for recovering all @math attributes with high probability is @math and the sample complexity for recovering a fixed proportion @math of the attributes for @math is @math , where @math .
The main contribution in this paper is to characterize the fundamental tradeoffs between recovery accuracy, query difficulty, and sample complexity for this parity crowdsourcing model under the exact and almost exact recovery conditions. These recovery conditions have also been considered for community detection as reviewed in @cite_13 . However, different from almost exact recovery for community detection, where the recovery is said to be successful if any @math fraction of attributes can be recovered with high probability, we consider the exact missing fraction @math and analyze the necessary and sufficient sample complexity, obtaining an explicit expression as a function of @math . Thus, our results show how the missing fraction @math affects the required sample complexity @math in guaranteeing the recovery of the @math attributes.
@cite_13
1980652450 2262666886 1737288045 2085493178
This paper considers the inverse problem with observed variables Y = BGX ⊕Z, where BG 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(e) 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 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-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 α = np log(n), it is shown that exact recovery is possible if and only if α > 2 (1−2e)+o(1 (1−2e)). In other words, 2 (1−2e) is the information theoretic threshold for exact recovery at lowSNR. 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. Full version available in [1]. We consider the problem of identifying underlying community-like structures in graphs. Toward this end, we study the stochastic block model (SBM) on k-clusters: a random model on n = km vertices, partitioned in k equal sized clusters, with edges sampled independently across clusters with probability q and within clusters with probability p, p > q. The goal is to recover the initial “hidden” partition of [n]. We study semidefinite programming (SDP)-based algorithms in this context. In the regime (p = (m) m ) and (q = (m) m ), we show that a certain natural SDP-based algorithm solves the problem of exact recovery in the k-community SBM, with high probability, whenever ( - > 1 ), as long as (k=o( n) ). This threshold is known to be the information theoretically optimal. We also study the case when (k= ( (n)) ). In this case however, we achieve recovery guarantees that no longer match the optimal condition ( - > 1 ), thus leaving achieving optimality for this range an open question. New phase transition phenomena have recently been discovered for the stochastic block model, for the special case of two non-overlapping symmetric communities. This gives raise in particular to new algorithmic challenges driven by the thresholds. This paper investigates whether a general phenomenon takes place for multiple communities, without imposing symmetry. In the general stochastic block model @math , @math vertices are split into @math communities of relative size @math , and vertices in community @math and @math connect independently with probability @math . This paper investigates the partial and exact recovery of communities in the general SBM (in the constant and logarithmic degree regimes), and uses the generality of the results to tackle overlapping communities. The contributions of the paper are: (i) an explicit characterization of the recovery threshold in the general SBM in terms of a new divergence function @math , which generalizes the Hellinger and Chernoff divergences, and which provides an operational meaning to a divergence function analog to the KL-divergence in the channel coding theorem, (ii) the development of an algorithm that recovers the communities all the way down to the optimal threshold and runs in quasi-linear time, showing that exact recovery has no information-theoretic to computational gap for multiple communities, in contrast to the conjectures made for detection with more than 4 communities; note that the algorithm is optimal both in terms of achieving the threshold and in having quasi-linear complexity, (iii) the development of an efficient algorithm that detects communities in the constant degree regime with an explicit accuracy bound that can be made arbitrarily close to 1 when a prescribed signal-to-noise ratio (defined in term of the spectrum of @math ) tends to infinity. Recovery of the sparsity pattern (or support) of an unknown sparse vector from a limited number of noisy linear measurements is an important problem in compressed sensing. In the high-dimensional setting, it is known that recovery with a vanishing fraction of errors is impossible if the measurement rate and the per-sample signal-to-noise ratio (SNR) are finite constants, independent of the vector length. In this paper, it is shown that recovery with an arbitrarily small but constant fraction of errors is, however, possible, and that in some cases computationally simple estimators are near-optimal. Bounds on the measurement rate needed to attain a desired fraction of errors are given in terms of the SNR and various key parameters of the unknown vector for several different recovery algorithms. The tightness of the bounds, in a scaling sense, as a function of the SNR and the fraction of errors, is established by comparison with existing information-theoretic necessary bounds. Near optimality is shown for a wide variety of practically motivated signal models.