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60
A geometric analysis of subspace clustering with outliers
 ANNALS OF STATISTICS
, 2012
"... This paper considers the problem of clustering a collection of unlabeled data points assumed to lie near a union of lower dimensional planes. As is common in computer vision or unsupervised learning applications, we do not know in advance how many subspaces there are nor do we have any information a ..."
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Cited by 66 (3 self)
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This paper considers the problem of clustering a collection of unlabeled data points assumed to lie near a union of lower dimensional planes. As is common in computer vision or unsupervised learning applications, we do not know in advance how many subspaces there are nor do we have any information about their dimensions. We develop a novel geometric analysis of an algorithm named sparse subspace clustering (SSC) [11], which significantly broadens the range of problems where it is provably effective. For instance, we show that SSC can recover multiple subspaces, each of dimension comparable to the ambient dimension. We also prove that SSC can correctly cluster data points even when the subspaces of interest intersect. Further, we develop an extension of SSC that succeeds when the data set is corrupted with possibly overwhelmingly many outliers. Underlying our analysis are clear geometric insights, which may bear on other sparse recovery problems. A numerical study complements our theoretical analysis and demonstrates the effectiveness of these methods.
A boundederror approach to piecewise affine system identification
 IEEE Transactions on Automatic Control
, 2005
"... Abstract — This paper proposes a threestage procedure for ..."
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Cited by 48 (1 self)
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Abstract — This paper proposes a threestage procedure for
A TUTORIAL ON SUBSPACE CLUSTERING
"... The past few years have witnessed an explosion in the availability of data from multiple sources and modalities. For example, millions of cameras have been installed in buildings, streets, airports and cities around the world. This has generated extraordinary advances on how to acquire, compress, st ..."
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Cited by 30 (0 self)
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The past few years have witnessed an explosion in the availability of data from multiple sources and modalities. For example, millions of cameras have been installed in buildings, streets, airports and cities around the world. This has generated extraordinary advances on how to acquire, compress, store, transmit and process massive amounts of complex highdimensional data. Many of these advances have relied on the observation that, even though these data sets are highdimensional, their intrinsic dimension is often much smaller than the dimension of the ambient space. In computer vision, for example, the number of pixels in an image can be rather large, yet most computer vision models use only a few parameters to describe the appearance, geometry and dynamics of a scene. This has motivated the development of a number of techniques for finding a lowdimensional representation
A sparsification approach to set membership identification of a class of affine hybrid systems
 IEEE Transactions on Automatic Control
"... A sparsification approach to set membership identification of a class of affine hybrid systems ..."
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Cited by 27 (12 self)
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A sparsification approach to set membership identification of a class of affine hybrid systems
Bayesian nonparametric inference of switching linear dynamical systems
, 2010
"... Abstract—Many complex dynamical phenomena can be effectively modeled by a system that switches among a set of conditionally linear dynamical modes. We consider two such models: the switching linear dynamical system (SLDS) and the switching vector autoregressive (VAR) process. Our Bayesian nonparamet ..."
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Cited by 25 (4 self)
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Abstract—Many complex dynamical phenomena can be effectively modeled by a system that switches among a set of conditionally linear dynamical modes. We consider two such models: the switching linear dynamical system (SLDS) and the switching vector autoregressive (VAR) process. Our Bayesian nonparametric approach utilizes a hierarchical Dirichlet process prior to learn an unknown number of persistent, smooth dynamical modes. We additionally employ automatic relevance determination to infer a sparse set of dynamic dependencies allowing us to learn SLDS with varying state dimension or switching VAR processes with varying autoregressive order. We develop a sampling algorithm that combines a truncated approximation to the Dirichlet process with efficient joint sampling of the mode and state sequences. The utility and flexibility of our model are demonstrated on synthetic data, sequences of dancing honey bees, the IBOVESPA stock index and a maneuvering target tracking application. Index Terms—Autoregressive processes, Bayesian methods, hidden Markov models, statespace methods, time series analysis,
From videos to verbs: mining videos for activities using a cascade of dynamical systems
 in: IEEE International Conference on Computer Vision and Pattern Recognition
"... Clustering video sequences in order to infer and extract activities from a single video stream is an extremely important problem and has significant potential in video indexing, surveillance, activity discovery and event recognition. Clustering a video sequence into activities requires one to simult ..."
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Cited by 23 (6 self)
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Clustering video sequences in order to infer and extract activities from a single video stream is an extremely important problem and has significant potential in video indexing, surveillance, activity discovery and event recognition. Clustering a video sequence into activities requires one to simultaneously recognize activity boundaries (activity consistent subsequences) and cluster these activity subsequences. In order to do this, we build a generative model for activities (in video) using a cascade of dynamical systems and show that this model is able to capture and represent a diverse class of activities. We then derive algorithms to learn the model parameters from a video stream and also show how a single video sequence may be clustered into different clusters where each cluster represents an activity. We also propose a novel technique to build affine, view, rate invariance of the activity into the distance metric for clustering. Experiments show that the clusters found by the algorithm correspond to semantically meaningful activities. 1.
Subspace identification of piecewise linear systems
 In Proceedings of the 43rd IEEE Conference on Decision and Control
, 2005
"... AbstractSubspace identification can be used to obtain models of piecewise linear statespace systems for which the switching is known. The models should not switch faster than the block size of the Hankel matrices used. The nonconsecutive parts of the input and output data that correspond to one o ..."
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Cited by 12 (0 self)
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AbstractSubspace identification can be used to obtain models of piecewise linear statespace systems for which the switching is known. The models should not switch faster than the block size of the Hankel matrices used. The nonconsecutive parts of the input and output data that correspond to one of the local linear systems can be used to obtain the system matrices of that system up to a linear state transformation. The linear systems obtained in this way cannot be combined directly, because the state transformation is different for each of the local linear systems. The transitions between the local linear systems can be used to transform the models to the same state space basis. We show that the necessary transformations can be obtained from the data, if the data contains a sufficiently large number of transitions for which the states at the transition are linearly independent. An algorithm to determine the transformations is presented, and the sensitivity with respect to noise is investigated using a MonteCarlo simulation.
A continuous optimization framework for hybrid system identification
 Automatica
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Learning cyclelinear hybrid automata for excitable cells
 In Proc. of HSCC’07, the 10th International Conference on Hybrid Systems: Computation and Control, volume 4416 of LNCS
, 2007
"... Abstract. We show how to automatically learn the class of Hybrid Automata called CycleLinear Hybrid Automata (CLHA) in order to model the behavior of excitable cells. Such cells, whose main purpose is to amplify and propagate an electrical signal known as the action potential (AP), serve as the “bi ..."
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Cited by 11 (5 self)
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Abstract. We show how to automatically learn the class of Hybrid Automata called CycleLinear Hybrid Automata (CLHA) in order to model the behavior of excitable cells. Such cells, whose main purpose is to amplify and propagate an electrical signal known as the action potential (AP), serve as the “biologic transistors ” of living organisms. The learning algorithm we propose comprises the following three phases: (1) Geometric analysis of the APs in the training set is used to identify, for each AP, the modes and switching logic of the corresponding Linear Hybrid Automata. (2) For each mode, the modified Prony’s method is used to learn the coefficients of the associated linear flows. (3) The modified Prony’s method is used again to learn the functions that adjust, on a percycle basis, the mode dynamics and switching logic of the Linear Hybrid Automata obtained in the first two phases. Our results show that the learned CLHA is able to successfully capture AP morphology and other important excitablecell properties, such as refractoriness and restitution, up to a prescribed approximation error. Our approach is fully implemented in MATLAB and, to the best of our knowledge, provides the most accurate approximation model for ECs to date. 1
Identification of PWARX Hybrid Models with Unknown and Possibly Different Orders
 In Proceedings of IEEE American Control Conference
, 2004
"... We consider the problem of identifying the orders and the model parameters of PWARX hybrid models from noiseless input/output data. We cast the identification problem in an algebraic geometric framework in which the number of discrete states corresponds to the degree of a multivariate polynomial p a ..."
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Cited by 11 (4 self)
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We consider the problem of identifying the orders and the model parameters of PWARX hybrid models from noiseless input/output data. We cast the identification problem in an algebraic geometric framework in which the number of discrete states corresponds to the degree of a multivariate polynomial p and the orders and the model parameters are encoded on the factors of p. We derive a rank constraint on the input/output data from which one can estimate the coefficients of p. Given p, we show that one can estimate the orders and the parameters of each ARX model from the derivatives of p at a collection of regressors that minimize a certain objective function. Our solution does not require previous knowledge about the orders of the ARX models (only an upper bound is needed), nor does it constraint the orders to be equal. Also the switching mechanism can be arbitrary, hence the switches need not be separated by a minimum dwell time. We illustrate our approach with an algebraic example of a switching circuit and with simulation results in the presence of noisy data.