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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.
Recursive robust pca or recursive sparse recovery in large but structured noise
 in IEEE Intl. Symp. on Information Theory (ISIT
, 2013
"... This Dissertation is brought to you for free and open access by the Graduate College at Digital Repository @ Iowa State University. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Digital Repository @ Iowa State University. For more informati ..."
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Cited by 22 (17 self)
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This Dissertation is brought to you for free and open access by the Graduate College at Digital Repository @ Iowa State University. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Digital Repository @ Iowa State University. For more information, please contact
Recovery of lowrank plus compressed sparse matrices with application to unveiling traffic anomalies
 IEEE TRANS. INFO. THEORY
, 2013
"... Given the noiseless superposition of a lowrank matrix plus the product of a known fat compression matrix times a sparse matrix, the goal of this paper is to establish deterministic conditions under which exact recovery of the lowrank and sparse components becomes possible. This fundamental identif ..."
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Cited by 21 (5 self)
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Given the noiseless superposition of a lowrank matrix plus the product of a known fat compression matrix times a sparse matrix, the goal of this paper is to establish deterministic conditions under which exact recovery of the lowrank and sparse components becomes possible. This fundamental identifiability issue arises with traffic anomaly detection in backbone networks, and subsumes compressed sensing as well as the timely lowrank plus sparse matrix recovery tasks encountered in matrix decomposition problems. Leveraging the ability of and nuclear norms to recover sparse and lowrank matrices, a convex program is formulated to estimate the unknowns. Analysis and simulations confirm that the said convex program can recover the unknowns for sufficiently lowrank and sparse enough components, along with a compression matrix possessing an isometry property when restricted to operate on sparse vectors. When the lowrank, sparse, and compression matrices are drawn from certain random ensembles, it is established that exact recovery is possible with high probability. Firstorder algorithms are developed to solve the nonsmooth convex optimization problem with provable iteration complexity guarantees. Insightful tests with synthetic and real network data corroborate the effectiveness of the novel approach in unveiling traffic anomalies across flows and time, and its ability to outperform existing alternatives.
An online algorithm for separating sparse and lowdimensional signal sequences from their sum
 IEEE Trans. Signal Process
"... Abstract—This paper designs and extensively evaluates an online algorithm, called practical recursive projected compressive sensing (PracReProCS), for recovering a time sequence of sparse vectors and a time sequence of dense vectors from their sum, , when the ’s lie in a slowly changing lowdimens ..."
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Cited by 10 (8 self)
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Abstract—This paper designs and extensively evaluates an online algorithm, called practical recursive projected compressive sensing (PracReProCS), for recovering a time sequence of sparse vectors and a time sequence of dense vectors from their sum, , when the ’s lie in a slowly changing lowdimensional subspace of the full space. A key application where this problem occurs is in realtime video layering where the goal is to separate a video sequence into a slowly changing background sequence and a sparse foreground sequence that consists of one or more moving regions/objects onthefly. PracReProCS is a practical modification of its theoretical counterpart which was analyzed in our recent work. Extension to the undersampled case is also developed. Extensive experimental comparisons demonstrating the advantage of the approach for both simulated and real videos, over existing batch and recursive methods, are shown. Index Terms—Online robust PCA, recursive sparse recovery, large but structured noise, compressed sensing. I.
Performance guarantees for undersampled recursive sparse recovery in large but structured noise (long version).” [Online]. Available: http://www.public.iastate. edu/%7Eblois/ReProModCSLong.pdf
"... Abstract—We study the problem of recursively reconstructing a time sequence of sparse vectors St from measurements of the form Mt = ASt +BLt where A and B are known measurement matrices, and Lt lies in a slowly changing low dimensional subspace. We assume that the signal of interest (St) is sparse, ..."
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Cited by 8 (7 self)
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Abstract—We study the problem of recursively reconstructing a time sequence of sparse vectors St from measurements of the form Mt = ASt +BLt where A and B are known measurement matrices, and Lt lies in a slowly changing low dimensional subspace. We assume that the signal of interest (St) is sparse, and has support which is correlated over time. We introduce a solution which we call Recursive Projected Modified Compressed Sensing (ReProMoCS), which exploits the correlated support change of St. We show that, under weaker assumptions than previous work, with high probability, ReProMoCS will exactly recover the support set of St and the reconstruction error of St is upper bounded by a small timeinvariant value. A motivating application where the above problem occurs is in functional MRI imaging of the brain to detect regions that are “activated ” in response to stimuli. In this case both measurement matrices are the same (i.e. A = B). The active region image constitutes the sparse vector St and this region changes slowly over time. The background brain image changes are global but the amount of change is very little and hence it can be well modeled as lying in a slowly changing low dimensional subspace, i.e. this constitutes Lt. I.
SEPARATING SPARSE AND LOWDIMENSIONAL SIGNAL SEQUENCES FROM TIMEVARYING UNDERSAMPLED PROJECTIONS OF THEIR SUMS
"... The goal of this work is to recover a sequence of sparse vectors, st; and a sequence of dense vectors, ℓt, that lie in a “slowly changing” low dimensional subspace, from timevarying undersampled linear projections of their sum. This type of problem typically occurs when the quantity being imaged ca ..."
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Cited by 4 (3 self)
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The goal of this work is to recover a sequence of sparse vectors, st; and a sequence of dense vectors, ℓt, that lie in a “slowly changing” low dimensional subspace, from timevarying undersampled linear projections of their sum. This type of problem typically occurs when the quantity being imaged can be split into a sum of two layers, one of which is sparse and the other is lowdimensional. A key application where this problem occurs is in undersampled functional magnetic resonance imaging (fMRI) to detect brain activation patterns in response to a stimulus. The brain image at time t can be modeled as being a sum of the active region image, st, (equal to the activation in the active region and zero everywhere else) and the background brain image, ℓt, which can be accurately modeled as lying in a slowly changing low dimensional subspace. We introduce a novel solution approach called matrix completion projected compressive sensing or MatComProCS. Significantly improved performance of MatComProCS over existing work is shown for the undersampled fMRI based brain active region detection problem. Index Terms — matrix completion, compressive sensing, fMRI 1.
PRACTICAL REPROCS FOR SEPARATING SPARSE AND LOWDIMENSIONAL SIGNAL SEQUENCES FROM THEIR SUM – PART 1
"... This paper designs and evaluates a practical algorithm, called PracReProCS, for recovering a time sequence of sparse vectors St and a time sequence of dense vectors Lt from their sum, Mt: = St + Lt, when any subsequence of the Lt’s lies in a slowly changing lowdimensional subspace. A key applica ..."
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Cited by 3 (3 self)
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This paper designs and evaluates a practical algorithm, called PracReProCS, for recovering a time sequence of sparse vectors St and a time sequence of dense vectors Lt from their sum, Mt: = St + Lt, when any subsequence of the Lt’s lies in a slowly changing lowdimensional subspace. A key application where this problem occurs is in video layering where the goal is to separate a video sequence into a slowly changing background sequence and a sparse foreground sequence that consists of one or more moving regions/objects. PracReProCS is the practical analog of its theoretical counterpart that was studied in our recent work. Index Terms — robust PCA, robust matrix completion, sparse recovery, compressed sensing 1.
Time Invariant Error Bounds for ModifiedCS based Sparse Signal Sequence Recovery
"... PAPER AWARD”. In this work, we obtain performance guarantees for modifiedCS and for its improved version, modifiedCSAddLSDel, for recursive reconstruction of sparse signal sequences from noisy measurements. Under mild assumptions, and for a realistic signal change model, we show that the suppor ..."
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Cited by 2 (1 self)
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PAPER AWARD”. In this work, we obtain performance guarantees for modifiedCS and for its improved version, modifiedCSAddLSDel, for recursive reconstruction of sparse signal sequences from noisy measurements. Under mild assumptions, and for a realistic signal change model, we show that the support recovery error of both algorithms is bounded by a timeinvariant and small value at all times. The same is also true for the reconstruction error. Under a slow support change assumption, our results hold under weaker assumptions on the number of measurements than what simple compressive sensing (basis pursuit denoising) needs. Also, the result for modifiedCSaddLSdel holds under weaker assumptions on the signal magnitude increase rate than the result for modifiedCS. Similar results were obtained in an earlier work, however the signal change model assumed there was very simple and not practically valid. I.
Recursive Reconstruction of Sparse Signal Sequences
"... In this chapter, we describe our recent work on the design and analysis of recursive algorithms for causally reconstructing a time sequence of (approximately) sparse signals from a greatly reduced number of linear projection measurements. The signals are sparse in some transform domain referred to a ..."
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In this chapter, we describe our recent work on the design and analysis of recursive algorithms for causally reconstructing a time sequence of (approximately) sparse signals from a greatly reduced number of linear projection measurements. The signals are sparse in some transform domain referred to as the sparsity basis and their sparsity patterns (support set of the sparsity basis coefficients) can change with time. By “recursive”, we mean use only the previous signal’s estimate and the current measurements to get the current signal’s estimate. We also briefly summarize our exact reconstruction results for the noisefree case and our error bounds and error stability results (conditions under which a timeinvariant and small bound on the reconstruction error holds at all times) for the noisy case. Connections with related work are also discussed. A key example application where the above problem occurs is dynamic magnetic resonance imaging (MRI) for realtime medical applications such as interventional radiology and MRIguided surgery, or in functional MRI to track brain activation changes. Crosssectional images of the brain, heart, larynx or other human organ images are piecewise smooth, and thus approximately sparse in the wavelet domain. In a time sequence, their sparsity pattern changes with time, but quite slowly. The same is also often true for the nonzero signal values. This simple fact, which was first observed in our work, is the key reason that our proposed recursive algorithms can achieve provably exact or accurate reconstruction from very few measurements. I.