1,389 research outputs found
Finding the Graph of Epidemic Cascades
We consider the problem of finding the graph on which an epidemic cascade
spreads, given only the times when each node gets infected. While this is a
problem of importance in several contexts -- offline and online social
networks, e-commerce, epidemiology, vulnerabilities in infrastructure networks
-- there has been very little work, analytical or empirical, on finding the
graph. Clearly, it is impossible to do so from just one cascade; our interest
is in learning the graph from a small number of cascades.
For the classic and popular "independent cascade" SIR epidemics, we
analytically establish the number of cascades required by both the global
maximum-likelihood (ML) estimator, and a natural greedy algorithm. Both results
are based on a key observation: the global graph learning problem decouples
into local problems -- one for each node. For a node of degree , we show
that its neighborhood can be reliably found once it has been infected times (for ML on general graphs) or times (for greedy on
trees). We also provide a corresponding information-theoretic lower bound of
; thus our bounds are essentially tight. Furthermore, if we
are given side-information in the form of a super-graph of the actual graph (as
is often the case), then the number of cascade samples required -- in all cases
-- becomes independent of the network size .
Finally, we show that for a very general SIR epidemic cascade model, the
Markov graph of infection times is obtained via the moralization of the network
graph.Comment: To appear in Proc. ACM SIGMETRICS/Performance 201
Sequential Compressed Sensing
Compressed sensing allows perfect recovery of sparse signals (or signals
sparse in some basis) using only a small number of random measurements.
Existing results in compressed sensing literature have focused on
characterizing the achievable performance by bounding the number of samples
required for a given level of signal sparsity. However, using these bounds to
minimize the number of samples requires a-priori knowledge of the sparsity of
the unknown signal, or the decay structure for near-sparse signals.
Furthermore, there are some popular recovery methods for which no such bounds
are known.
In this paper, we investigate an alternative scenario where observations are
available in sequence. For any recovery method, this means that there is now a
sequence of candidate reconstructions. We propose a method to estimate the
reconstruction error directly from the samples themselves, for every candidate
in this sequence. This estimate is universal in the sense that it is based only
on the measurement ensemble, and not on the recovery method or any assumed
level of sparsity of the unknown signal. With these estimates, one can now stop
observations as soon as there is reasonable certainty of either exact or
sufficiently accurate reconstruction. They also provide a way to obtain
"run-time" guarantees for recovery methods that otherwise lack a-priori
performance bounds.
We investigate both continuous (e.g. Gaussian) and discrete (e.g. Bernoulli)
random measurement ensembles, both for exactly sparse and general near-sparse
signals, and with both noisy and noiseless measurements.Comment: to appear in IEEE transactions on Special Topics in Signal Processin
Low-rank Matrix Completion using Alternating Minimization
Alternating minimization represents a widely applicable and empirically
successful approach for finding low-rank matrices that best fit the given data.
For example, for the problem of low-rank matrix completion, this method is
believed to be one of the most accurate and efficient, and formed a major
component of the winning entry in the Netflix Challenge.
In the alternating minimization approach, the low-rank target matrix is
written in a bi-linear form, i.e. ; the algorithm then alternates
between finding the best and the best . Typically, each alternating step
in isolation is convex and tractable. However the overall problem becomes
non-convex and there has been almost no theoretical understanding of when this
approach yields a good result.
In this paper we present first theoretical analysis of the performance of
alternating minimization for matrix completion, and the related problem of
matrix sensing. For both these problems, celebrated recent results have shown
that they become well-posed and tractable once certain (now standard)
conditions are imposed on the problem. We show that alternating minimization
also succeeds under similar conditions. Moreover, compared to existing results,
our paper shows that alternating minimization guarantees faster (in particular,
geometric) convergence to the true matrix, while allowing a simpler analysis
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