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47
The pyramid match kernel: Discriminative classification with sets of image features
 In ICCV
, 2005
"... Discriminative learning is challenging when examples are sets of features, and the sets vary in cardinality and lack any sort of meaningful ordering. Kernelbased classification methods can learn complex decision boundaries, but a kernel over unordered set inputs must somehow solve for correspondenc ..."
Abstract

Cited by 367 (26 self)
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Discriminative learning is challenging when examples are sets of features, and the sets vary in cardinality and lack any sort of meaningful ordering. Kernelbased classification methods can learn complex decision boundaries, but a kernel over unordered set inputs must somehow solve for correspondences – generally a computationally expensive task that becomes impractical for large set sizes. We present a new fast kernel function which maps unordered feature sets to multiresolution histograms and computes a weighted histogram intersection in this space. This “pyramid match ” computation is linear in the number of features, and it implicitly finds correspondences based on the finest resolution histogram cell where a matched pair first appears. Since the kernel does not penalize the presence of extra features, it is robust to clutter. We show the kernel function is positivedefinite, making it valid for use in learning algorithms whose optimal solutions are guaranteed only for Mercer kernels. We demonstrate our algorithm on object recognition tasks and show it to be accurate and dramatically faster than current approaches. 1.
Random features for largescale kernel machines
 In Neural Infomration Processing Systems
, 2007
"... To accelerate the training of kernel machines, we propose to map the input data to a randomized lowdimensional feature space and then apply existing fast linear methods. Our randomized features are designed so that the inner products of the transformed data are approximately equal to those in the f ..."
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Cited by 115 (3 self)
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To accelerate the training of kernel machines, we propose to map the input data to a randomized lowdimensional feature space and then apply existing fast linear methods. Our randomized features are designed so that the inner products of the transformed data are approximately equal to those in the feature space of a user specified shiftinvariant kernel. We explore two sets of random features, provide convergence bounds on their ability to approximate various radial basis kernels, and show that in largescale classification and regression tasks linear machine learning algorithms that use these features outperform stateoftheart largescale kernel machines. 1
The pyramid match kernel: Efficient learning with sets of features
 Journal of Machine Learning Research
, 2007
"... In numerous domains it is useful to represent a single example by the set of the local features or parts that comprise it. However, this representation poses a challenge to many conventional machine learning techniques, since sets may vary in cardinality and elements lack a meaningful ordering. Kern ..."
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Cited by 94 (8 self)
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In numerous domains it is useful to represent a single example by the set of the local features or parts that comprise it. However, this representation poses a challenge to many conventional machine learning techniques, since sets may vary in cardinality and elements lack a meaningful ordering. Kernel methods can learn complex functions, but a kernel over unordered set inputs must somehow solve for correspondences—generally a computationally expensive task that becomes impractical for large set sizes. We present a new fast kernel function called the pyramid match that measures partial match similarity in time linear in the number of features. The pyramid match maps unordered feature sets to multiresolution histograms and computes a weighted histogram intersection in order to find implicit correspondences based on the finest resolution histogram cell where a matched pair first appears. We show the pyramid match yields a Mercer kernel, and we prove bounds on its error relative to the optimal partial matching cost. We demonstrate our algorithm on both classification and regression tasks, including object recognition, 3D human pose inference, and time of publication estimation for documents, and we show that the proposed method is accurate and significantly more efficient than current approaches.
Efficient image matching with distributions of local invariant features
 In IEEE Conference on Computer Vision and Pattern Recognition
, 2005
"... Sets of local features that are invariant to common image transformations are an effective representation to use when comparing images; current methods typically judge feature sets ’ similarity via a voting scheme (which ignores cooccurrence statistics) or by comparing histograms over a set of prot ..."
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Cited by 50 (1 self)
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Sets of local features that are invariant to common image transformations are an effective representation to use when comparing images; current methods typically judge feature sets ’ similarity via a voting scheme (which ignores cooccurrence statistics) or by comparing histograms over a set of prototypes (which must be found by clustering). We present a method for efficiently comparing images based on their discrete distributions (bags) of distinctive local invariant features, without clustering descriptors. Similarity between images is measured with an approximation of the Earth Mover’s Distance (EMD), which quickly computes minimalcost correspondences between two bags of features. Each image’s feature distribution is mapped into a normed space with a lowdistortion embedding of EMD. Examples most similar to a novel query image are retrieved in time sublinear in the number of examples via approximate nearest neighbor search in the embedded space. We evaluate our method with scene, object, and texture recognition tasks. 1.
Diffusion distance for histogram comparison
 In CVPR06
, 2006
"... In this paper we propose diffusion distance, a new dissimilarity measure between histogrambased descriptors. We define the difference between two histograms to be a temperature field. We then study the relationship between histogram similarity and a diffusion process, showing how diffusion handles ..."
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Cited by 46 (2 self)
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In this paper we propose diffusion distance, a new dissimilarity measure between histogrambased descriptors. We define the difference between two histograms to be a temperature field. We then study the relationship between histogram similarity and a diffusion process, showing how diffusion handles deformation as well as quantization effects. As a result, the diffusion distance is derived as the sum of dissimilarities over scales. Being a crossbin histogram distance, the diffusion distance is robust to deformation, lighting change and noise in histogrambased local descriptors. In addition, it enjoys linear computational complexity which significantly improves previously proposed crossbin distances with quadratic complexity or higher. We tested the proposed approach on both shape recognition and interest point matching tasks using several multidimensional histogrambased descriptors including shape context, SIFT, and spin images. In all experiments, the diffusion distance performs excellently in both accuracy and efficiency in comparison with other stateoftheart distance measures. In particular, it performs as accurately as the Earth Mover’s Distance with much greater efficiency. 1.
An efficient earth mover’s distance algorithm for robust histogram comparison
 PAMI
, 2007
"... DRAFT We propose EMDL1: a fast and exact algorithm for computing the Earth Mover’s Distance (EMD) between a pair of histograms. The efficiency of the new algorithm enables its application to problems that were previously prohibitive due to high time complexities. The proposed EMDL1 significantly s ..."
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Cited by 44 (4 self)
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DRAFT We propose EMDL1: a fast and exact algorithm for computing the Earth Mover’s Distance (EMD) between a pair of histograms. The efficiency of the new algorithm enables its application to problems that were previously prohibitive due to high time complexities. The proposed EMDL1 significantly simplifies the original linear programming formulation of EMD. Exploiting the L1 metric structure, the number of unknown variables in EMDL1 is reduced to O(N) from O(N 2) of the original EMD for a histogram with N bins. In addition, the number of constraints is reduced by half and the objective function of the linear program is simplified. Formally without any approximation, we prove that the EMDL1 formulation is equivalent to the original EMD with a L1 ground distance. To perform the EMDL1 computation, we propose an efficient treebased algorithm, TreeEMD. TreeEMD exploits the fact that a basic feasible solution of the simplex algorithmbased solver forms a spanning tree when we interpret EMDL1 as a network flow optimization problem. We empirically show that this new algorithm has average time complexity of O(N 2), which significantly improves the best reported supercubic complexity of the original EMD. The accuracy of the proposed methods is evaluated by
Nonembeddability theorems via Fourier analysis
"... Various new nonembeddability results (mainly into L1) are proved via Fourier analysis. In particular, it is shown that the Edit Distance on {0, 1}d has L1 distortion (log d) 12o(1). We also give new lower bounds on the L1 distortion of flat tori, quotients of the discrete hypercube under group ac ..."
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Cited by 43 (9 self)
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Various new nonembeddability results (mainly into L1) are proved via Fourier analysis. In particular, it is shown that the Edit Distance on {0, 1}d has L1 distortion (log d) 12o(1). We also give new lower bounds on the L1 distortion of flat tori, quotients of the discrete hypercube under group actions, and the transportation cost (Earthmover) metric.
Approximate correspondences in high dimensions
 In Advances in Neural Information Processing System (NIPS) 19
"... Pyramid intersection is an efficient method for computing an approximate partial matching between two sets of feature vectors. We introduce a novel pyramid embedding based on a hierarchy of nonuniformly shaped bins that takes advantage of the underlying structure of the feature space and remains ac ..."
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Cited by 38 (8 self)
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Pyramid intersection is an efficient method for computing an approximate partial matching between two sets of feature vectors. We introduce a novel pyramid embedding based on a hierarchy of nonuniformly shaped bins that takes advantage of the underlying structure of the feature space and remains accurate even for sets with highdimensional feature vectors. The matching similarity is computed in linear time and forms a Mercer kernel. Whereas previous matching approximation algorithms suffer from distortion factors that increase linearly with the feature dimension, we demonstrate that our approach can maintain constant accuracy even as the feature dimension increases. When used as a kernel in a discriminative classifier, our approach achieves improved object recognition results over a stateoftheart set kernel. 1
Fast and Robust Earth Mover’s Distances
"... We present a new algorithm for a robust family of Earth Mover’s Distances EMDs with thresholded ground distances. The algorithm transforms the flownetwork of the EMD so that the number of edges is reduced by an order of magnitude. As a result, we compute the EMD by an order of magnitude faster tha ..."
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Cited by 32 (6 self)
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We present a new algorithm for a robust family of Earth Mover’s Distances EMDs with thresholded ground distances. The algorithm transforms the flownetwork of the EMD so that the number of edges is reduced by an order of magnitude. As a result, we compute the EMD by an order of magnitude faster than the original algorithm, which makes it possible to compute the EMD on large histograms and databases. In addition, we show that EMDs with thresholded ground distances have many desirable properties. First, they correspond to the way humans perceive distances. Second, they are robust to outlier noise and quantization effects. Third, they are metrics. Finally, experimental results on image retrieval show that thresholding the ground distance of the EMD improves both accuracy and speed. 1.
Fast similarity search for learned metrics
, 2007
"... We propose a method to efficiently index into a large database of examples according to a learned metric. Given a collection of examples, we learn a Mahalanobis distance using an informationtheoretic metric learning technique that adapts prior knowledge about pairwise distances to incorporate simil ..."
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Cited by 29 (4 self)
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We propose a method to efficiently index into a large database of examples according to a learned metric. Given a collection of examples, we learn a Mahalanobis distance using an informationtheoretic metric learning technique that adapts prior knowledge about pairwise distances to incorporate similarity and dissimilarity constraints. To enable sublinear time similarity search under the learned metric, we show how to encode a learned Mahalanobis parameterization into randomized localitysensitive hash functions. We further formulate an indirect solution that enables metric learning and hashing for sparse input vector spaces whose high dimensionality make it infeasible to learn an explicit weighting over the feature dimensions. We demonstrate the approach applied to systems and image datasets, and show that our learned metrics improve accuracy relative to commonlyused metric baselines, while our hashing construction permits efficient indexing with a learned distance and very large databases. 1