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Robust subspace clustering via thresholding. arXiv preprint arXiv:1307.4891
, 2013
"... The problem of clustering noisy and incompletely observed high-dimensional data points into a union of low-dimensional subspaces and a set of outliers is considered. The number of subspaces, their dimensions, and their orientations are assumed unknown. We propose a simple low-complexity subspace clu ..."
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The problem of clustering noisy and incompletely observed high-dimensional data points into a union of low-dimensional subspaces and a set of outliers is considered. The number of subspaces, their dimensions, and their orientations are assumed unknown. We propose a simple low-complexity 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 log-factor) 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 comput-ing 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 classifica-tion problems. One of the limitations of these methods, ..."
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Subspace models for image data sets, constructed by comput-ing 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 classifica-tion problems. One of the limitations of these methods, how-ever, is that the subspace representation is unable to directly model the effects of non-linear transformations such as trans-lation, 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 meth-ods translation invariant, thereby simplifying or eliminating the alignment pre-processing task. The potential of the pro-posed 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 data-driven 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 data-driven algorithm for learning unions of subspaces directly from a collection of data; our approach is based upon forming minimum ℓ2-norm (least-squares) 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 least-squares approach leads to improved classification performance when compared to stateof-the-art subspace clustering methods on both synthetic and real-world experiments. This study provides evidence that using least-squares methods to form data-driven representations of collections of data provide significant advantages over current methods that rely upon sparse representations. Index Terms — Dense representations, subspace clustering, sparsity, pseudo-inverse, orthogonal matching pursuit.
Fall Semester 2013 Dimensionality Reduction for Sparse Subspace Clustering
, 2014
"... Subspace clustering addresses the problem of clustering a set of unlabeled high-dimensional data points lying near a union of low-dimensional sub-spaces according to their subspace membership. The number and dimen-sions 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 high-dimensional data points lying near a union of low-dimensional sub-spaces according to their subspace membership. The number and dimen-sions of the subspaces, as well as their orientations, are all unknown. Since the computational cost of subspace clustering algorithms cru-cially depends on the ambient space dimension, it is desirable to reduce the dimensionality of the data before clustering. Even when computa-tional cost is not an issue, dimensionality reduction is advantageous be-cause 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 algo-rithms. In this work, we investigate this question analytically by deriving per-formance guarantees for sparse subspace clustering (SSC) [1–3] applied to
Subspace Clustering Reloaded: Sparse vs. Dense Representations
"... Abstract—State-of-the-art 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—State-of-the-art 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 sparsity-driven methods for subspace clustering provide advantages over traditional nearest neighbor-based approaches, sparse representations often produce affinity matrices with weakly connected components that are difficult to cluster, even with state-of-the-art clustering methods. In this work, we propose a new algorithm that employs dense (least-squares) representations to extract the subspace affinity between vectors in the dataset. We demonstrate the advantages of the proposed dense subspace clustering algorithm over state-of-the-art sparsity-driven methods on real and synthetic data. I. SPARSE SUBSPACE CLUSTERING
Functional Subspace Clustering with Application to Time Series
"... Functional data, where samples are random func-tions, are increasingly common and important in a variety of applications, such as health care and traffic analysis. They are naturally high dimen-sional and lie along complex manifolds. These properties warrant use of the subspace assump-tion, but most ..."
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Functional data, where samples are random func-tions, are increasingly common and important in a variety of applications, such as health care and traffic analysis. They are naturally high dimen-sional and lie along complex manifolds. These properties warrant use of the subspace assump-tion, but most state-of-the-art subspace learning algorithms are limited to linear or other simple settings. To address these challenges, we pro-pose a new framework called Functional Sub-space Clustering (FSC). FSC assumes that func-tional samples lie in deformed linear subspaces and formulates the subspace learning problem as a sparse regression over operators. The result-ing problem can be efficiently solved via greedy variable selection, given access to a fast defor-mation oracle. We provide theoretical guaran-tees for FSC and show how it can be applied to time series with warped alignments. Experimen-tal results on both synthetic data and real clini-cal time series show that FSC outperforms both standard time series clustering and state-of-the-art subspace clustering. 1.
REVISITING ROBUSTNESS OF THE UNION-OF-SUBSPACES MODEL FOR DATA-ADAPTIVE LEARNING OF NONLINEAR SIGNAL MODELS
"... This paper revisits the problem of data-adaptive learning of geomet-ric signal structures based on the Union-of-Subspaces (UoS) model. In contrast to prior work, it motivates and investigates an extension of the classical UoS model, termed the Metric-Constrained Union-of-Subspaces (MC-UoS) model. In ..."
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This paper revisits the problem of data-adaptive learning of geomet-ric signal structures based on the Union-of-Subspaces (UoS) model. In contrast to prior work, it motivates and investigates an extension of the classical UoS model, termed the Metric-Constrained Union-of-Subspaces (MC-UoS) model. In this regard, it puts forth two iterative methods for data-adaptive learning of an MC-UoS in the presence of complete and missing data. The proposed methods out-perform existing approaches to learning a UoS in numerical exper-iments involving both synthetic and real data, which demonstrates effectiveness of both an MC-UoS model and the proposed methods. Index Terms — Nonlinear signal models, union of subspaces 1.
Geometric Conditions for Subspace-Sparse Recovery
"... Given a dictionary Π and a signal ξ = Πx gen-erated by a few linearly independent columns of Π, classical sparse recovery theory deals with the problem of uniquely recovering the sparse rep-resentation x of ξ. In this work, we consider the more general case where ξ lies in a low-dimensional subspace ..."
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Given a dictionary Π and a signal ξ = Πx gen-erated by a few linearly independent columns of Π, classical sparse recovery theory deals with the problem of uniquely recovering the sparse rep-resentation x of ξ. In this work, we consider the more general case where ξ lies in a low-dimensional 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 subspace-sparse. We study conditions under which existing pursuit methods recover a subspace-sparse representation. Such conditions reveal important geometric insights and have implications for the theory of classical sparse recovery as well as subspace clustering. 1.
