504 research outputs found
Spectral Thresholds in the Bipartite Stochastic Block Model
We consider a bipartite stochastic block model on vertex sets and
, with planted partitions in each, and ask at what densities efficient
algorithms can recover the partition of the smaller vertex set.
When , multiple thresholds emerge. We first locate a sharp
threshold for detection of the partition, in the sense of the results of
\cite{mossel2012stochastic,mossel2013proof} and \cite{massoulie2014community}
for the stochastic block model. We then show that at a higher edge density, the
singular vectors of the rectangular biadjacency matrix exhibit a localization /
delocalization phase transition, giving recovery above the threshold and no
recovery below. Nevertheless, we propose a simple spectral algorithm, Diagonal
Deletion SVD, which recovers the partition at a nearly optimal edge density.
The bipartite stochastic block model studied here was used by
\cite{feldman2014algorithm} to give a unified algorithm for recovering planted
partitions and assignments in random hypergraphs and random -SAT formulae
respectively. Our results give the best known bounds for the clause density at
which solutions can be found efficiently in these models as well as showing a
barrier to further improvement via this reduction to the bipartite block model.Comment: updated version, will appear in COLT 201
Subsampled Power Iteration: a Unified Algorithm for Block Models and Planted CSP's
We present an algorithm for recovering planted solutions in two well-known
models, the stochastic block model and planted constraint satisfaction
problems, via a common generalization in terms of random bipartite graphs. Our
algorithm matches up to a constant factor the best-known bounds for the number
of edges (or constraints) needed for perfect recovery and its running time is
linear in the number of edges used. The time complexity is significantly better
than both spectral and SDP-based approaches.
The main contribution of the algorithm is in the case of unequal sizes in the
bipartition (corresponding to odd uniformity in the CSP). Here our algorithm
succeeds at a significantly lower density than the spectral approaches,
surpassing a barrier based on the spectral norm of a random matrix.
Other significant features of the algorithm and analysis include (i) the
critical use of power iteration with subsampling, which might be of independent
interest; its analysis requires keeping track of multiple norms of an evolving
solution (ii) it can be implemented statistically, i.e., with very limited
access to the input distribution (iii) the algorithm is extremely simple to
implement and runs in linear time, and thus is practical even for very large
instances
Clustering Partially Observed Graphs via Convex Optimization
This paper considers the problem of clustering a partially observed
unweighted graph---i.e., one where for some node pairs we know there is an edge
between them, for some others we know there is no edge, and for the remaining
we do not know whether or not there is an edge. We want to organize the nodes
into disjoint clusters so that there is relatively dense (observed)
connectivity within clusters, and sparse across clusters.
We take a novel yet natural approach to this problem, by focusing on finding
the clustering that minimizes the number of "disagreements"---i.e., the sum of
the number of (observed) missing edges within clusters, and (observed) present
edges across clusters. Our algorithm uses convex optimization; its basis is a
reduction of disagreement minimization to the problem of recovering an
(unknown) low-rank matrix and an (unknown) sparse matrix from their partially
observed sum. We evaluate the performance of our algorithm on the classical
Planted Partition/Stochastic Block Model. Our main theorem provides sufficient
conditions for the success of our algorithm as a function of the minimum
cluster size, edge density and observation probability; in particular, the
results characterize the tradeoff between the observation probability and the
edge density gap. When there are a constant number of clusters of equal size,
our results are optimal up to logarithmic factors.Comment: This is the final version published in Journal of Machine Learning
Research (JMLR). Partial results appeared in International Conference on
Machine Learning (ICML) 201
Spectral partitioning of time-varying networks with unobserved edges
We discuss a variant of `blind' community detection, in which we aim to
partition an unobserved network from the observation of a (dynamical) graph
signal defined on the network. We consider a scenario where our observed graph
signals are obtained by filtering white noise input, and the underlying network
is different for every observation. In this fashion, the filtered graph signals
can be interpreted as defined on a time-varying network. We model each of the
underlying network realizations as generated by an independent draw from a
latent stochastic blockmodel (SBM). To infer the partition of the latent SBM,
we propose a simple spectral algorithm for which we provide a theoretical
analysis and establish consistency guarantees for the recovery. We illustrate
our results using numerical experiments on synthetic and real data,
highlighting the efficacy of our approach.Comment: 5 pages, 2 figure
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