2,160 research outputs found
Projected Power Iteration for Network Alignment
The network alignment problem asks for the best correspondence between two
given graphs, so that the largest possible number of edges are matched. This
problem appears in many scientific problems (like the study of protein-protein
interactions) and it is very closely related to the quadratic assignment
problem which has graph isomorphism, traveling salesman and minimum bisection
problems as particular cases. The graph matching problem is NP-hard in general.
However, under some restrictive models for the graphs, algorithms can
approximate the alignment efficiently. In that spirit the recent work by Feizi
and collaborators introduce EigenAlign, a fast spectral method with convergence
guarantees for Erd\H{o}s-Reny\'i graphs. In this work we propose the algorithm
Projected Power Alignment, which is a projected power iteration version of
EigenAlign. We numerically show it improves the recovery rates of EigenAlign
and we describe the theory that may be used to provide performance guarantees
for Projected Power Alignment.Comment: 8 page
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
Community detection and stochastic block models: recent developments
The stochastic block model (SBM) is a random graph model with planted
clusters. It is widely employed as a canonical model to study clustering and
community detection, and provides generally a fertile ground to study the
statistical and computational tradeoffs that arise in network and data
sciences.
This note surveys the recent developments that establish the fundamental
limits for community detection in the SBM, both with respect to
information-theoretic and computational thresholds, and for various recovery
requirements such as exact, partial and weak recovery (a.k.a., detection). The
main results discussed are the phase transitions for exact recovery at the
Chernoff-Hellinger threshold, the phase transition for weak recovery at the
Kesten-Stigum threshold, the optimal distortion-SNR tradeoff for partial
recovery, the learning of the SBM parameters and the gap between
information-theoretic and computational thresholds.
The note also covers some of the algorithms developed in the quest of
achieving the limits, in particular two-round algorithms via graph-splitting,
semi-definite programming, linearized belief propagation, classical and
nonbacktracking spectral methods. A few open problems are also discussed
A note on Probably Certifiably Correct algorithms
Many optimization problems of interest are known to be intractable, and while
there are often heuristics that are known to work on typical instances, it is
usually not easy to determine a posteriori whether the optimal solution was
found. In this short note, we discuss algorithms that not only solve the
problem on typical instances, but also provide a posteriori certificates of
optimality, probably certifiably correct (PCC) algorithms. As an illustrative
example, we present a fast PCC algorithm for minimum bisection under the
stochastic block model and briefly discuss other examples
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