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Robust subspace clustering via thresholding. arXiv preprint arXiv:1307.4891
, 2013
"... The problem of clustering noisy and incompletely observed highdimensional data points into a union of lowdimensional subspaces and a set of outliers is considered. The number of subspaces, their dimensions, and their orientations are assumed unknown. We propose a simple lowcomplexity subspace clu ..."
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Cited by 10 (3 self)
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The problem of clustering noisy and incompletely observed highdimensional data points into a union of lowdimensional subspaces and a set of outliers is considered. The number of subspaces, their dimensions, and their orientations are assumed unknown. We propose a simple lowcomplexity subspace clustering algorithm, which applies spectral clustering to an adjacency matrix obtained by thresholding the correlations between data points. In other words, the adjacency matrix is constructed from the nearest neighbors of each data point in spherical distance. A statistical performance analysis shows that the algorithm succeeds even when the subspaces intersect and that it exhibits robustness to additive noise. Specifically, our results reveal an explicit tradeoff between the affinity of the subspaces and the tolerable noise level. We furthermore prove that the algorithm succeeds even when the data points are incompletely observed with the number of missing entries allowed to be (up to a logfactor) linear in the ambient dimension. We also propose a simple scheme that provably detects outliers, and we present numerical results on real and synthetic data. 1
Endogenous convolutional sparse representations for translation invariant image subspace models
 Proc. IEEE International Conference on Image Processing (ICIP
, 2014
"... Subspace models for image data sets, constructed by computing sparse representations of each image with respect to other images in the set, have been found to perform very well in a variety of applications, including clustering and classification problems. One of the limitations of these methods, ..."
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Subspace models for image data sets, constructed by computing sparse representations of each image with respect to other images in the set, have been found to perform very well in a variety of applications, including clustering and classification problems. One of the limitations of these methods, however, is that the subspace representation is unable to directly model the effects of nonlinear transformations such as translation, rotation, and dilation that frequently occur in practice. In this paper it is shown that the properties of convolutional sparse representations can be exploited to make these methods translation invariant, thereby simplifying or eliminating the alignment preprocessing task. The potential of the proposed approach is demonstrated in two diverse applications: image clustering and video background modeling.
SUBSPACE CLUSTERING WITH DENSE REPRESENTATIONS
"... Unions of subspaces have recently been shown to provide a powerful nonlinear signal model for collections of highdimensional data, such as large collections of images or videos. In this paper, we introduce a novel datadriven algorithm for learning unions of subspaces directly from a collection of d ..."
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Unions of subspaces have recently been shown to provide a powerful nonlinear signal model for collections of highdimensional data, such as large collections of images or videos. In this paper, we introduce a novel datadriven algorithm for learning unions of subspaces directly from a collection of data; our approach is based upon forming minimum ℓ2norm (leastsquares) representations of a signal with respect to other signals in the collection. The resulting representations are then used as feature vectors to cluster the data in accordance with each signal’s subspace membership. We demonstrate that the proposed leastsquares approach leads to improved classification performance when compared to stateoftheart subspace clustering methods on both synthetic and realworld experiments. This study provides evidence that using leastsquares methods to form datadriven representations of collections of data provide significant advantages over current methods that rely upon sparse representations. Index Terms — Dense representations, subspace clustering, sparsity, pseudoinverse, orthogonal matching pursuit.
Fall Semester 2013 Dimensionality Reduction for Sparse Subspace Clustering
, 2014
"... Subspace clustering addresses the problem of clustering a set of unlabeled highdimensional data points lying near a union of lowdimensional subspaces according to their subspace membership. The number and dimensions of the subspaces, as well as their orientations, are all unknown. Since the comp ..."
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Subspace clustering addresses the problem of clustering a set of unlabeled highdimensional data points lying near a union of lowdimensional subspaces according to their subspace membership. The number and dimensions of the subspaces, as well as their orientations, are all unknown. Since the computational cost of subspace clustering algorithms crucially depends on the ambient space dimension, it is desirable to reduce the dimensionality of the data before clustering. Even when computational cost is not an issue, dimensionality reduction is advantageous because it leads to reduced storage and transmission capacity requirements, and enhances privacy. It is thus important to understand the impact of dimensionality reduction on the performance of subspace clustering algorithms. In this work, we investigate this question analytically by deriving performance guarantees for sparse subspace clustering (SSC) [1–3] applied to
Subspace Clustering Reloaded: Sparse vs. Dense Representations
"... Abstract—Stateoftheart methods for learning unions of subspaces from a collection of data leverage sparsity to form representations of each vector in the dataset with respect to the remaining vectors in the dataset. The resulting sparse representations can be used to form a subspace affinity matr ..."
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Abstract—Stateoftheart methods for learning unions of subspaces from a collection of data leverage sparsity to form representations of each vector in the dataset with respect to the remaining vectors in the dataset. The resulting sparse representations can be used to form a subspace affinity matrix to cluster the data into their respective subspaces. While sparsitydriven methods for subspace clustering provide advantages over traditional nearest neighborbased approaches, sparse representations often produce affinity matrices with weakly connected components that are difficult to cluster, even with stateoftheart clustering methods. In this work, we propose a new algorithm that employs dense (leastsquares) representations to extract the subspace affinity between vectors in the dataset. We demonstrate the advantages of the proposed dense subspace clustering algorithm over stateoftheart sparsitydriven methods on real and synthetic data. I. SPARSE SUBSPACE CLUSTERING
Functional Subspace Clustering with Application to Time Series
"... Functional data, where samples are random functions, are increasingly common and important in a variety of applications, such as health care and traffic analysis. They are naturally high dimensional and lie along complex manifolds. These properties warrant use of the subspace assumption, but most ..."
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Functional data, where samples are random functions, are increasingly common and important in a variety of applications, such as health care and traffic analysis. They are naturally high dimensional and lie along complex manifolds. These properties warrant use of the subspace assumption, but most stateoftheart subspace learning algorithms are limited to linear or other simple settings. To address these challenges, we propose a new framework called Functional Subspace Clustering (FSC). FSC assumes that functional samples lie in deformed linear subspaces and formulates the subspace learning problem as a sparse regression over operators. The resulting problem can be efficiently solved via greedy variable selection, given access to a fast deformation oracle. We provide theoretical guarantees for FSC and show how it can be applied to time series with warped alignments. Experimental results on both synthetic data and real clinical time series show that FSC outperforms both standard time series clustering and stateoftheart subspace clustering. 1.
REVISITING ROBUSTNESS OF THE UNIONOFSUBSPACES MODEL FOR DATAADAPTIVE LEARNING OF NONLINEAR SIGNAL MODELS
"... This paper revisits the problem of dataadaptive learning of geometric signal structures based on the UnionofSubspaces (UoS) model. In contrast to prior work, it motivates and investigates an extension of the classical UoS model, termed the MetricConstrained UnionofSubspaces (MCUoS) model. In ..."
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This paper revisits the problem of dataadaptive learning of geometric signal structures based on the UnionofSubspaces (UoS) model. In contrast to prior work, it motivates and investigates an extension of the classical UoS model, termed the MetricConstrained UnionofSubspaces (MCUoS) model. In this regard, it puts forth two iterative methods for dataadaptive learning of an MCUoS in the presence of complete and missing data. The proposed methods outperform existing approaches to learning a UoS in numerical experiments involving both synthetic and real data, which demonstrates effectiveness of both an MCUoS model and the proposed methods. Index Terms — Nonlinear signal models, union of subspaces 1.
Geometric Conditions for SubspaceSparse Recovery
"... Given a dictionary Π and a signal ξ = Πx generated by a few linearly independent columns of Π, classical sparse recovery theory deals with the problem of uniquely recovering the sparse representation x of ξ. In this work, we consider the more general case where ξ lies in a lowdimensional subspace ..."
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Given a dictionary Π and a signal ξ = Πx generated by a few linearly independent columns of Π, classical sparse recovery theory deals with the problem of uniquely recovering the sparse representation x of ξ. In this work, we consider the more general case where ξ lies in a lowdimensional subspace spanned by a few columns of Π, which are possibly linearly dependent. In this case, x may not unique, and the goal is to recover any subset of the columns of Π that spans the subspace containing ξ. We call such a representation x subspacesparse. We study conditions under which existing pursuit methods recover a subspacesparse representation. Such conditions reveal important geometric insights and have implications for the theory of classical sparse recovery as well as subspace clustering. 1.