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57
Greed is Good: Algorithmic Results for Sparse Approximation
, 2004
"... This article presents new results on using a greedy algorithm, orthogonal matching pursuit (OMP), to solve the sparse approximation problem over redundant dictionaries. It provides a sufficient condition under which both OMP and Donoho’s basis pursuit (BP) paradigm can recover the optimal representa ..."
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Cited by 723 (8 self)
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This article presents new results on using a greedy algorithm, orthogonal matching pursuit (OMP), to solve the sparse approximation problem over redundant dictionaries. It provides a sufficient condition under which both OMP and Donoho’s basis pursuit (BP) paradigm can recover the optimal representation of an exactly sparse signal. It leverages this theory to show that both OMP and BP succeed for every sparse input signal from a wide class of dictionaries. These quasiincoherent dictionaries offer a natural generalization of incoherent dictionaries, and the cumulative coherence function is introduced to quantify the level of incoherence. This analysis unifies all the recent results on BP and extends them to OMP. Furthermore, the paper develops a sufficient condition under which OMP can identify atoms from an optimal approximation of a nonsparse signal. From there, it argues that OMP is an approximation algorithm for the sparse problem over a quasiincoherent dictionary. That is, for every input signal, OMP calculates a sparse approximant whose error is only a small factor worse than the minimal error that can be attained with the same number of terms.
Nearoptimal hashing algorithms for approximate nearest neighbor in high dimensions
, 2008
"... In this article, we give an overview of efficient algorithms for the approximate and exact nearest neighbor problem. The goal is to preprocess a dataset of objects (e.g., images) so that later, given a new query object, one can quickly return the dataset object that is most similar to the query. The ..."
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Cited by 411 (6 self)
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In this article, we give an overview of efficient algorithms for the approximate and exact nearest neighbor problem. The goal is to preprocess a dataset of objects (e.g., images) so that later, given a new query object, one can quickly return the dataset object that is most similar to the query. The problem is of significant interest in a wide variety of areas.
An investigation of practical approximate nearest neighbor algorithms
, 2004
"... This paper concerns approximate nearest neighbor searching algorithms, which have become increasingly important, especially in high dimensional perception areas such as computer vision, with dozens of publications in recent years. Much of this enthusiasm is due to a successful new approximate neares ..."
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Cited by 109 (4 self)
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This paper concerns approximate nearest neighbor searching algorithms, which have become increasingly important, especially in high dimensional perception areas such as computer vision, with dozens of publications in recent years. Much of this enthusiasm is due to a successful new approximate nearest neighbor approach called Locality Sensitive Hashing (LSH). In this paper we ask the question: can earlier spatial data structure approaches to exact nearest neighbor, such as metric trees, be altered to provide approximate answers to proximity queries and if so, how? We introduce a new kind of metric tree that allows overlap: certain datapoints may appear in both the children of a parent. We also introduce new approximate kNN search algorithms on this structure. We show why these structures should be able to exploit the same randomprojectionbased approximations that LSH enjoys, but with a simpler algorithm and perhaps with greater efficiency. We then provide a detailed empirical evaluation on five large, high dimensional datasets which show up to 31fold accelerations over LSH. This result holds true throughout the spectrum of approximation levels.
BoostMap: A Method for Efficient Approximate Similarity Rankings
, 2003
"... This paper introduces BoostMap, a method that can significantly reduce retrieval time in image and video database systems that employ computationally expensive distance measures, metric or nonmetric. Database and query objects are embedded into a Euclidean space, in which similarities can be rapidl ..."
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Cited by 106 (13 self)
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This paper introduces BoostMap, a method that can significantly reduce retrieval time in image and video database systems that employ computationally expensive distance measures, metric or nonmetric. Database and query objects are embedded into a Euclidean space, in which similarities can be rapidly measured using a weighted Manhattan distance. Embedding construction is formulated as a machine learning task, where AdaBoost is used to combine many simple, 1D embeddings into a multidimensional embedding that preserves a significant amount of the proximity structure in the original space. Performance is evaluated in a hand pose estimation system, and a dynamic gesture recognition system, where the proposed method is used to retrieve approximate nearest neighbors under expensive image and video similarity measures. In both systems, BoostMap significantly increases efficiency, with minimal losses in accuracy. Moreover, the experiments indicate that BoostMap compares favorably with existing embedding methods that have been employed in computer vision and database applications, i.e., FastMap and Bourgain embeddings.
Nearest Neighbors In HighDimensional Spaces
, 2004
"... In this chapter we consider the following problem: given a set P of points in a highdimensional space, construct a data structure which given any query point q nds the point in P closest to q. This problem, called nearest neighbor search is of significant importance to several areas of computer sci ..."
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Cited by 90 (3 self)
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In this chapter we consider the following problem: given a set P of points in a highdimensional space, construct a data structure which given any query point q nds the point in P closest to q. This problem, called nearest neighbor search is of significant importance to several areas of computer science, including pattern recognition, searching in multimedial data, vector compression [GG91], computational statistics [DW82], and data mining. Many of these applications involve data sets which are very large (e.g., a database containing Web documents could contain over one billion documents). Moreover, the dimensionality of the points is usually large as well (e.g., in the order of a few hundred). Therefore, it is crucial to design algorithms which scale well with the database size as well as with the dimension. The nearestneighbor problem is an example of a large class of proximity problems, which, roughly speaking, are problems whose definitions involve the notion of...
Approximation of functions over redundant dictionaries using coherence
 Proc. of SODA
, 2003
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On approximating the depth and related problems
 SIAM J. COMPUT
, 2008
"... We study the question of finding a deepest point in an arrangement of regions, and provide a fast algorithm for this problem using random sampling, showing it sufficient to solve this problem when the deepest point is shallow. This implies, among other results, a fast algorithm for solving linear pr ..."
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Cited by 71 (14 self)
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We study the question of finding a deepest point in an arrangement of regions, and provide a fast algorithm for this problem using random sampling, showing it sufficient to solve this problem when the deepest point is shallow. This implies, among other results, a fast algorithm for solving linear programming with violations approximately. We also use this technique to approximate the disk covering the largest number of red points, while avoiding all the blue points, given two such sets in the plane. Using similar techniques imply that approximate range counting queries have roughly the same time and space complexity as emptiness range queries.
A.: ”Efficient video similarity measurement with video signature
 In: IEEE International Conference on Image Processing (ICIP 02
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An AppearanceBased Framework for 3D Hand Shape Classification and Camera Viewpoint Estimation
, 2002
"... An appearancebased framework for 3D hand shape classification and simultaneous camera viewpoint estimation is presented. Given an input image of a segmented hand, the most similar matches from a large database of synthetic hand images are retrieved. The ground truth labels of those matches, contain ..."
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Cited by 46 (4 self)
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An appearancebased framework for 3D hand shape classification and simultaneous camera viewpoint estimation is presented. Given an input image of a segmented hand, the most similar matches from a large database of synthetic hand images are retrieved. The ground truth labels of those matches, containing hand shape and camera viewpoint information, are returned by the system as estimates for the input image. Database retrieval is done hierarchically, by first quickly rejecting the vast majority of all database views, and then ranking the remaining candidates in order of similarity to the input. Four different similarity measures are employed, based on edge location, edge orientation, finger location and geometric moments.
Entropy based nearest neighbor search in high dimensions
 In SODA ’06: Proceedings of the seventeenth annual ACMSIAM Symposium on Discrete Algorithms
"... In this paper we study the problem of finding the approximate nearest neighbor of a query point in the high dimensional space, focusing on the Euclidean space. The earlier approaches use localitypreserving hash functions (that tend to map nearby points to the same value) to construct several hash ta ..."
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Cited by 45 (5 self)
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In this paper we study the problem of finding the approximate nearest neighbor of a query point in the high dimensional space, focusing on the Euclidean space. The earlier approaches use localitypreserving hash functions (that tend to map nearby points to the same value) to construct several hash tables to ensure that the query point hashes to the same bucket as its nearest neighbor in at least one table. Our approach is different – we use one (or a few) hash table and hash several randomly chosen points in the neighborhood of the query point showing that at least one of them will hash to the bucket containing its nearest neighbor. We show that the number of randomly chosen points in the neighborhood of the query point q required depends on the entropy of the hash value h(p) of a random point p at the same distance from q at its nearest neighbor, given q and the locality preserving hash function h chosen randomly from the hash family. Precisely, we show that if the entropy I(h(p)q, h) = M and g is a bound on the probability that two faroff points will hash to the same bucket, then we can find the approximate nearest neighbor in O(nρ) time and near linear Õ(n) space where ρ = M / log(1/g). Alternatively we can build a data structure of size Õ(n1/(1−ρ)) to answer queries in Õ(d) time. By applying this analysis to the locality preserving hash functions in [15, 19, 6] and adjusting the parameters we show that the c nearest neighbor can be computed in time Õ(nρ) and near linear space where ρ ≈ 2.06/c as c becomes large. 1