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235
Robust face recognition via sparse representation
 IEEE TRANS. PATTERN ANALYSIS AND MACHINE INTELLIGENCE
, 2008
"... We consider the problem of automatically recognizing human faces from frontal views with varying expression and illumination, as well as occlusion and disguise. We cast the recognition problem as one of classifying among multiple linear regression models, and argue that new theory from sparse signa ..."
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Cited by 916 (41 self)
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We consider the problem of automatically recognizing human faces from frontal views with varying expression and illumination, as well as occlusion and disguise. We cast the recognition problem as one of classifying among multiple linear regression models, and argue that new theory from sparse signal representation offers the key to addressing this problem. Based on a sparse representation computed by ℓ 1minimization, we propose a general classification algorithm for (imagebased) object recognition. This new framework provides new insights into two crucial issues in face recognition: feature extraction and robustness to occlusion. For feature extraction, we show that if sparsity in the recognition problem is properly harnessed, the choice of features is no longer critical. What is critical, however, is whether the number of features is sufficiently large and whether the sparse representation is correctly computed. Unconventional features such as downsampled images and random projections perform just as well as conventional features such as Eigenfaces and Laplacianfaces, as long as the dimension of the feature space surpasses certain threshold, predicted by the theory of sparse representation. This framework can handle errors due to occlusion and corruption uniformly, by exploiting the fact that these errors are often sparse w.r.t. to the standard (pixel) basis. The theory of sparse representation helps predict how much occlusion the recognition algorithm can handle and how to choose the training images to maximize robustness to occlusion. We conduct extensive experiments on publicly available databases to verify the efficacy of the proposed algorithm, and corroborate the above claims.
A Simple Proof of the Restricted Isometry Property for Random Matrices
 CONSTR APPROX
, 2008
"... We give a simple technique for verifying the Restricted Isometry Property (as introduced by Candès and Tao) for random matrices that underlies Compressed Sensing. Our approach has two main ingredients: (i) concentration inequalities for random inner products that have recently provided algorithmical ..."
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Cited by 636 (69 self)
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We give a simple technique for verifying the Restricted Isometry Property (as introduced by Candès and Tao) for random matrices that underlies Compressed Sensing. Our approach has two main ingredients: (i) concentration inequalities for random inner products that have recently provided algorithmically simple proofs of the Johnson–Lindenstrauss lemma; and (ii) covering numbers for finitedimensional balls in Euclidean space. This leads to an elementary proof of the Restricted Isometry Property and brings out connections between Compressed Sensing and the Johnson–Lindenstrauss lemma. As a result, we obtain simple and direct proofs of Kashin’s theorems on widths of finite balls in Euclidean space (and their improvements due to Gluskin) and proofs of the existence of optimal Compressed Sensing measurement matrices. In the process, we also prove that these measurements have a certain universality with respect to the sparsityinducing basis.
Finding frequent items in data streams
, 2002
"... Abstract. We present a 1pass algorithm for estimating the most frequent items in a data stream using very limited storage space. Our method relies on a novel data structure called a count sketch, which allows us to estimate the frequencies of all the items in the stream. Our algorithm achieves bett ..."
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Cited by 344 (0 self)
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Abstract. We present a 1pass algorithm for estimating the most frequent items in a data stream using very limited storage space. Our method relies on a novel data structure called a count sketch, which allows us to estimate the frequencies of all the items in the stream. Our algorithm achieves better space bounds than the previous best known algorithms for this problem for many natural distributions on the item frequencies. In addition, our algorithm leads directly to a 2pass algorithm for the problem of estimating the items with the largest (absolute) change in frequency between two data streams. To our knowledge, this problem has not been previously studied in the literature. 1
Random projection in dimensionality reduction: Applications to image and text data
 in Knowledge Discovery and Data Mining
, 2001
"... Random projections have recently emerged as a powerful method for dimensionality reduction. Theoretical results indicate that the method preserves distances quite nicely; however, empirical results are sparse. We present experimental results on using random projection as a dimensionality reduction t ..."
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Cited by 239 (0 self)
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Random projections have recently emerged as a powerful method for dimensionality reduction. Theoretical results indicate that the method preserves distances quite nicely; however, empirical results are sparse. We present experimental results on using random projection as a dimensionality reduction tool in a number of cases, where the high dimensionality of the data would otherwise lead to burdensome computations. Our application areas are the processing of both noisy and noiseless images, and information retrieval in text documents. We show that projecting the data onto a random lowerdimensional subspace yields results comparable to conventional dimensionality reduction methods such as principal component analysis: the similarity of data vectors is preserved well under random projection. However, using random projections is computationally signicantly less expensive than using, e.g., principal component analysis. We also show experimentally that using a sparse random matrix gives additional computational savings in random projection.
Subspace clustering for high dimensional data: a review
 ACM SIGKDD Explorations Newsletter
, 2004
"... Subspace clustering for high dimensional data: ..."
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Random projections of smooth manifolds
 Foundations of Computational Mathematics
, 2006
"... We propose a new approach for nonadaptive dimensionality reduction of manifoldmodeled data, demonstrating that a small number of random linear projections can preserve key information about a manifoldmodeled signal. We center our analysis on the effect of a random linear projection operator Φ: R N ..."
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Cited by 144 (25 self)
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We propose a new approach for nonadaptive dimensionality reduction of manifoldmodeled data, demonstrating that a small number of random linear projections can preserve key information about a manifoldmodeled signal. We center our analysis on the effect of a random linear projection operator Φ: R N → R M, M < N, on a smooth wellconditioned Kdimensional submanifold M ⊂ R N. As our main theoretical contribution, we establish a sufficient number M of random projections to guarantee that, with high probability, all pairwise Euclidean and geodesic distances between points on M are wellpreserved under the mapping Φ. Our results bear strong resemblance to the emerging theory of Compressed Sensing (CS), in which sparse signals can be recovered from small numbers of random linear measurements. As in CS, the random measurements we propose can be used to recover the original data in R N. Moreover, like the fundamental bound in CS, our requisite M is linear in the “information level” K and logarithmic in the ambient dimension N; we also identify a logarithmic dependence on the volume and conditioning of the manifold. In addition to recovering faithful approximations to manifoldmodeled signals, however, the random projections we propose can also be used to discern key properties about the manifold. We discuss connections and contrasts with existing techniques in manifold learning, a setting where dimensionality reducing mappings are typically nonlinear and constructed adaptively from a set of sampled training data.
Graph sparsification by effective resistances
 SIAM J. Comput
"... We present a nearlylinear time algorithm that produces highquality sparsifiers of weighted graphs. Given as input a weighted graph G = (V, E, w) and a parameter ǫ> 0, we produce a weighted subgraph H = (V, ˜ E, ˜w) of G such that  ˜ E  = O(n log n/ǫ 2) and for all vectors x ∈ R V (1 − ǫ) ∑ ..."
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Cited by 142 (9 self)
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We present a nearlylinear time algorithm that produces highquality sparsifiers of weighted graphs. Given as input a weighted graph G = (V, E, w) and a parameter ǫ> 0, we produce a weighted subgraph H = (V, ˜ E, ˜w) of G such that  ˜ E  = O(n log n/ǫ 2) and for all vectors x ∈ R V (1 − ǫ) ∑ (x(u) − x(v)) 2 wuv ≤ ∑ (x(u) − x(v)) 2 ˜wuv ≤ (1 + ǫ) ∑ (x(u) − x(v)) 2 wuv. (1) uv∈E uv ∈ ˜ E This improves upon the sparsifiers constructed by Spielman and Teng, which had O(n log c n) edges for some large constant c, and upon those of Benczúr and Karger, which only satisfied (1) for x ∈ {0, 1} V. We conjecture the existence of sparsifiers with O(n) edges, noting that these would generalize the notion of expander graphs, which are constantdegree sparsifiers for the complete graph. A key ingredient in our algorithm is a subroutine of independent interest: a nearlylinear time algorithm that builds a data structure from which we can query the approximate effective resistance between any two vertices in a graph in O(log n) time. uv∈E
Compressed Sensing and Redundant Dictionaries
"... This article extends the concept of compressed sensing to signals that are not sparse in an orthonormal basis but rather in a redundant dictionary. It is shown that a matrix, which is a composition of a random matrix of certain type and a deterministic dictionary, has small restricted isometry con ..."
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Cited by 140 (16 self)
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This article extends the concept of compressed sensing to signals that are not sparse in an orthonormal basis but rather in a redundant dictionary. It is shown that a matrix, which is a composition of a random matrix of certain type and a deterministic dictionary, has small restricted isometry constants. Thus, signals that are sparse with respect to the dictionary can be recovered via Basis Pursuit from a small number of random measurements. Further, thresholding is investigated as recovery algorithm for compressed sensing and conditions are provided that guarantee reconstruction with high probability. The different schemes are compared by numerical experiments.
Random projection for high dimensional data clustering: A cluster ensemble approach
 In: Proceedings of the 20th International Conference on Machine Learning (ICML
"... We investigate how random projection can best be used for clustering high dimensional data. Random projection has been shown to have promising theoretical properties. In practice, however, we find that it results in highly unstable clustering performance. Our solution is to use random projection in ..."
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Cited by 138 (4 self)
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We investigate how random projection can best be used for clustering high dimensional data. Random projection has been shown to have promising theoretical properties. In practice, however, we find that it results in highly unstable clustering performance. Our solution is to use random projection in a cluster ensemble approach. Empirical results show that the proposed approach achieves better and more robust clustering performance compared to not only single runs of random projection/clustering but also clustering with PCA, a traditional data reduction method for high dimensional data. To gain insights into the performance improvement obtained by our ensemble method, we analyze and identify the influence of the quality and the diversity of the individual clustering solutions on the final ensemble performance. 1.
Properties of embedding methods for similarity searching in metric spaces
 PAMI
, 2003
"... Complex data types—such as images, documents, DNA sequences, etc.—are becoming increasingly important in modern database applications. A typical query in many of these applications seeks to find objects that are similar to some target object, where (dis)similarity is defined by some distance functi ..."
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Cited by 110 (5 self)
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Complex data types—such as images, documents, DNA sequences, etc.—are becoming increasingly important in modern database applications. A typical query in many of these applications seeks to find objects that are similar to some target object, where (dis)similarity is defined by some distance function. Often, the cost of evaluating the distance between two objects is very high. Thus, the number of distance evaluations should be kept at a minimum, while (ideally) maintaining the quality of the result. One way to approach this goal is to embed the data objects in a vector space so that the distances of the embedded objects approximates the actual distances. Thus, queries can be performed (for the most part) on the embedded objects. In this paper, we are especially interested in examining the issue of whether or not the embedding methods will ensure that no relevant objects are left out (i.e., there are no false dismissals and, hence, the correct result is reported). Particular attention is paid to the SparseMap, FastMap, and MetricMap embedding methods. SparseMap is a variant of Lipschitz embeddings, while FastMap and MetricMap are inspired by dimension reduction methods for Euclidean spaces (using KLT or the related PCA and SVD). We show that, in general, none of these embedding methods guarantee that queries on the embedded objects have no false dismissals, while also demonstrating the limited cases in which the guarantee does hold. Moreover, we describe a variant of SparseMap that allows queries with no false dismissals. In addition, we show that with FastMap and MetricMap, the distances of the embedded objects can be much greater than the actual distances. This makes it impossible (or at least impractical) to modify FastMap and MetricMap to guarantee no false dismissals.