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61
A unified framework for highdimensional analysis of Mestimators with decomposable regularizers
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Stable principal component pursuit
 In Proc. of International Symposium on Information Theory
, 2010
"... We consider the problem of recovering a target matrix that is a superposition of lowrank and sparse components, from a small set of linear measurements. This problem arises in compressed sensing of structured highdimensional signals such as videos and hyperspectral images, as well as in the analys ..."
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Cited by 94 (3 self)
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We consider the problem of recovering a target matrix that is a superposition of lowrank and sparse components, from a small set of linear measurements. This problem arises in compressed sensing of structured highdimensional signals such as videos and hyperspectral images, as well as in the analysis of transformation invariant lowrank structure recovery. We analyze the performance of the natural convex heuristic for solving this problem, under the assumption that measurements are chosen uniformly at random. We prove that this heuristic exactly recovers lowrank and sparse terms, provided the number of observations exceeds the number of intrinsic degrees of freedom of the component signals by a polylogarithmic factor. Our analysis introduces several ideas that may be of independent interest for the more general problem of compressed sensing and decomposing superpositions of multiple structured signals. 1
Restricted strong convexity and weighted matrix completion: Optimal bounds with noise
, 2012
"... We consider the matrix completion problem under a form of row/column weighted entrywise sampling, including the case of uniform entrywise sampling as a special case. We analyze the associated random observation operator, and prove that with high probability, it satisfies a form of restricted strong ..."
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Cited by 84 (10 self)
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We consider the matrix completion problem under a form of row/column weighted entrywise sampling, including the case of uniform entrywise sampling as a special case. We analyze the associated random observation operator, and prove that with high probability, it satisfies a form of restricted strong convexity with respect to weighted Frobenius norm. Using this property, we obtain as corollaries a number of error bounds on matrix completion in the weighted Frobenius norm under noisy sampling and for both exact and near lowrank matrices. Our results are based on measures of the “spikiness” and “lowrankness” of matrices that are less restrictive than the incoherence conditions imposed in previous work. Our technique involves an Mestimator that includes controls on both the rank and spikiness of the solution, and we establish nonasymptotic error bounds in weighted Frobenius norm for recovering matrices lying with ℓq“balls ” of bounded spikiness. Using informationtheoretic methods, we show that no algorithm can achieve better estimates (up to a logarithmic factor) over these same sets, showing that our conditions on matrices and associated rates are essentially optimal.
Latent Variable Graphical Model Selection via Convex Optimization
, 2010
"... Suppose we have samples of a subset of a collection of random variables. No additional information is provided about the number of latent variables, nor of the relationship between the latent and observed variables. Is it possible to discover the number of hidden components, and to learn a statistic ..."
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Cited by 76 (4 self)
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Suppose we have samples of a subset of a collection of random variables. No additional information is provided about the number of latent variables, nor of the relationship between the latent and observed variables. Is it possible to discover the number of hidden components, and to learn a statistical model over the entire collection of variables? We address this question in the setting in which the latent and observed variables are jointly Gaussian, with the conditional statistics of the observed variables conditioned on the latent variables being specified by a graphical model. As a first step we give natural conditions under which such latentvariable Gaussian graphical models are identifiable given marginal statistics of only the observed variables. Essentially these conditions require that the conditional graphical model among the observed variables is sparse, while the effect of the latent variables is “spread out ” over most of the observed variables. Next we propose a tractable convex program based on regularized maximumlikelihood for model selection in this latentvariable setting; the regularizer uses both the ℓ1 norm and the nuclear norm. Our modeling framework can be viewed as a combination of dimensionality reduction (to identify latent variables) and graphical modeling (to capture remaining statistical structure not attributable to the latent variables), and it consistently estimates both the number of hidden components and the conditional graphical model structure among the observed variables. These results are applicable in the highdimensional setting in which the number of latent/observed variables grows with the number of samples of the observed variables. The geometric properties of the algebraic varieties of sparse matrices and of lowrank matrices play an important role in our analysis.
Simultaneously Structured Models with Application to Sparse and Lowrank Matrices
, 2014
"... The topic of recovery of a structured model given a small number of linear observations has been wellstudied in recent years. Examples include recovering sparse or groupsparse vectors, lowrank matrices, and the sum of sparse and lowrank matrices, among others. In various applications in signal p ..."
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Cited by 41 (5 self)
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The topic of recovery of a structured model given a small number of linear observations has been wellstudied in recent years. Examples include recovering sparse or groupsparse vectors, lowrank matrices, and the sum of sparse and lowrank matrices, among others. In various applications in signal processing and machine learning, the model of interest is known to be structured in several ways at the same time, for example, a matrix that is simultaneously sparse and lowrank. Often norms that promote each individual structure are known, and allow for recovery using an orderwise optimal number of measurements (e.g., `1 norm for sparsity, nuclear norm for matrix rank). Hence, it is reasonable to minimize a combination of such norms. We show that, surprisingly, if we use multiobjective optimization with these norms, then we can do no better, orderwise, than an algorithm that exploits only one of the present structures. This result suggests that to fully exploit the multiple structures, we need an entirely new convex relaxation, i.e. not one that is a function of the convex relaxations used for each structure. We then specialize our results to the case of sparse and lowrank matrices. We show that a nonconvex formulation of the problem can recover the model from very few measurements, which is on the order of the degrees of freedom of the matrix, whereas the convex problem obtained from a combination of the `1 and nuclear norms requires many more measurements. This proves an orderwise gap between the performance of the convex and nonconvex recovery problems in this case. Our framework applies to arbitrary structureinducing norms as well as to a wide range of measurement ensembles. This allows us to give performance bounds for problems such as sparse phase retrieval and lowrank tensor completion.
Dynamic anomalography: Tracking network anomalies via sparsity and low rank
, 2013
"... In the backbone of largescale networks, origintodestination (OD) traffic flows experience abrupt unusual changes known as traffic volume anomalies, which can result in congestion and limit the extent to which enduser quality of service requirements are met. As a means of maintaining seamless en ..."
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Cited by 24 (10 self)
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In the backbone of largescale networks, origintodestination (OD) traffic flows experience abrupt unusual changes known as traffic volume anomalies, which can result in congestion and limit the extent to which enduser quality of service requirements are met. As a means of maintaining seamless enduser experience in dynamic environments, as well as for ensuring network security, this paper deals with a crucial network monitoring task termed dynamic anomalography. Given link traffic measurements (noisy superpositions of unobserved OD flows) periodically acquired by backbone routers, the goal is to construct an estimated map of anomalies in real time, and thus summarize the network ‘health state ’ along both the flow and time dimensions. Leveraging the low intrinsicdimensionality of OD flows and the sparse nature of anomalies, a novel online estimator is proposed based on an exponentiallyweighted leastsquares criterion regularized with the sparsitypromotingnorm of the anomalies, and the nuclear norm of the nominal traffic matrix. After recasting the nonseparable nuclear norm into a form amenable to online optimization, a realtime algorithm for dynamic anomalography is developed and its convergence established under simplifying technical assumptions. For operational conditions where computational complexity reductions are at a premium, a lightweight stochastic gradient algorithm based on Nesterov’s acceleration technique is developed as well. Comprehensive numerical tests with both synthetic and real network data corroborate the effectiveness of the proposed online algorithms and their tracking capabilities, and demonstrate that they outperform stateoftheart approaches developed to diagnose traffic anomalies.