Results 1  10
of
73
The geometry of optimal transportation
 Acta Math
, 1996
"... A classical problem of transporting mass due to Monge and Kantorovich is solved. Given measures µ and ν on R d, we find the measurepreserving map y(x) between them with minimal cost — where cost is measured against h(x − y) withhstrictly convex, or a strictly concave function of x − y. This map i ..."
Abstract

Cited by 245 (33 self)
 Add to MetaCart
(Show Context)
A classical problem of transporting mass due to Monge and Kantorovich is solved. Given measures µ and ν on R d, we find the measurepreserving map y(x) between them with minimal cost — where cost is measured against h(x − y) withhstrictly convex, or a strictly concave function of x − y. This map is unique: it is characterized by the formula y(x) =x−(∇h) −1 (∇ψ(x)) and geometrical restrictions on ψ. Connections with mathematical economics, numerical computations, and the MongeAmpère equation are sketched. ∗ Both authors gratefully acknowledge the support provided by postdoctoral fellowships: WG at
Exponential integrability and transportation cost related to logarithmic Sobolev inequalities
 J. FUNCT. ANAL
, 1999
"... We study some problems on exponential integrability, concentration of measure, and transportation cost related to logarithmic Sobolev inequalities. On the real line, we then give a characterization of those probability measures which satisfy these inequalities. ..."
Abstract

Cited by 175 (9 self)
 Add to MetaCart
(Show Context)
We study some problems on exponential integrability, concentration of measure, and transportation cost related to logarithmic Sobolev inequalities. On the real line, we then give a characterization of those probability measures which satisfy these inequalities.
Differential equations methods for the MongeKantorovich mass transfer problem
 Mem. Amer. Math. Soc
, 1999
"... We demonstrate that a solution to the classical Monge–Kantorovich problem of optimally rearranging the measure µ + = f + dx onto µ − = f − dy can be constructed by studying the pLaplacian equation −div(Dup  p−2 Dup) = f + − f − in the limit as p → ∞. The idea is to show up → u, where u satisfie ..."
Abstract

Cited by 166 (8 self)
 Add to MetaCart
(Show Context)
We demonstrate that a solution to the classical Monge–Kantorovich problem of optimally rearranging the measure µ + = f + dx onto µ − = f − dy can be constructed by studying the pLaplacian equation −div(Dup  p−2 Dup) = f + − f − in the limit as p → ∞. The idea is to show up → u, where u satisfies Du  ≤ 1, −div(aDu) = f + − f − for some density a ≥ 0, and then to build a flow by solving an ODE involving a, Du, f + and f −. Contents 1.
Shape Matching: Similarity Measures and Algorithms
, 2001
"... Shape matching is an important ingredient in shape retrieval, recognition and classification, alignment and registration, and approximation and simplification. This paper treats various aspects that are needed to solve shape matching problems: choosing the precise problem, selecting the properties o ..."
Abstract

Cited by 114 (1 self)
 Add to MetaCart
(Show Context)
Shape matching is an important ingredient in shape retrieval, recognition and classification, alignment and registration, and approximation and simplification. This paper treats various aspects that are needed to solve shape matching problems: choosing the precise problem, selecting the properties of the similarity measure that are needed for the problem, choosing the specific similarity measure, and constructing the algorithm to compute the similarity. The focus is on methods that lie close to the field of computational geometry.
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 ..."
Abstract

Cited by 93 (5 self)
 Add to MetaCart
(Show Context)
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
Perspectives of Monge Properties in Optimization
, 1995
"... An m × n matrix C is called Monge matrix if c ij + c rs c is + c rj for all 1 i ! r m, 1 j ! s n. In this paper we present a survey on Monge matrices and related Monge properties and their role in combinatorial optimization. Specifically, we deal with the following three main topics: (i) f ..."
Abstract

Cited by 69 (3 self)
 Add to MetaCart
An m &times; n matrix C is called Monge matrix if c ij + c rs c is + c rj for all 1 i ! r m, 1 j ! s n. In this paper we present a survey on Monge matrices and related Monge properties and their role in combinatorial optimization. Specifically, we deal with the following three main topics: (i) fundamental combinatorial properties of Monge structures, (ii) applications of Monge properties to optimization problems and (iii) recognition of Monge properties.
Probability Metrics and Recursive Algorithms
"... In this paper it is shown by several examples that probability metrics are a useful tool to study the asymptotic behaviour of (stochastic) recursive algorithms. The basic idea of this approach is to find a `suitable ' probability metric which yields contraction properties of the transformation ..."
Abstract

Cited by 53 (9 self)
 Add to MetaCart
In this paper it is shown by several examples that probability metrics are a useful tool to study the asymptotic behaviour of (stochastic) recursive algorithms. The basic idea of this approach is to find a `suitable ' probability metric which yields contraction properties of the transformations describing the limits of the algorithm. In order to demonstrate the wide range of applicability of this contraction method we investigate examples from various fields, some of them have been analyzed already in the literature.
A linear programming formulation and approximation algorithms for the metric labeling problem
 SIAM J. Discrete Math
"... We consider approximation algorithms for the metric labeling problem. This problem was introduced in a paper by Kleinberg and Tardos [J. ACM, 49 (2002), pp. 616–630] and captures many classification problems that arise in computer vision and related fields. They gave an O(log k log log k) approximat ..."
Abstract

Cited by 43 (1 self)
 Add to MetaCart
(Show Context)
We consider approximation algorithms for the metric labeling problem. This problem was introduced in a paper by Kleinberg and Tardos [J. ACM, 49 (2002), pp. 616–630] and captures many classification problems that arise in computer vision and related fields. They gave an O(log k log log k) approximation for the general case, where k is the number of labels, and a 2approximation for the uniform metric case. (In fact, the bound for general metrics can be improved to O(log k) by the work of Fakcheroenphol, Rao, and Talwar [Proceedings
Video Event Recognition Using Kernel Methods with Multilevel Temporal Alignment
 IEEE Trans. Pattern Analysis and Machine Intelligence
, 1985
"... Abstract—In this work, we systematically study the problem of event recognition in unconstrained news video sequences. We adopt the discriminative kernelbased method for which video clip similarity plays an important role. First, we represent a video clip as a bag of orderless descriptors extracted ..."
Abstract

Cited by 40 (6 self)
 Add to MetaCart
(Show Context)
Abstract—In this work, we systematically study the problem of event recognition in unconstrained news video sequences. We adopt the discriminative kernelbased method for which video clip similarity plays an important role. First, we represent a video clip as a bag of orderless descriptors extracted from all of the constituent frames and apply the earth mover’s distance (EMD) to integrate similarities among frames from two clips. Observing that a video clip is usually comprised of multiple subclips corresponding to event evolution over time, we further build a multilevel temporal pyramid. At each pyramid level, we integrate the information from different subclips with Integervalueconstrained EMD to explicitly align the subclips. By fusing the information from the different pyramid levels, we develop Temporally Aligned Pyramid Matching (TAPM) for measuring video similarity. We conduct comprehensive experiments on the TRECVID 2005 corpus, which contains more than 6,800 clips. Our experiments demonstrate that 1) the TAPM multilevel method clearly outperforms singlelevel EMD (SLEMD) and 2) SLEMD outperforms keyframe and multiframebased detection methods by a large margin. In addition, we conduct indepth investigation of various aspects of the proposed techniques such as weight selection in SLEMD, sensitivity to temporal clustering, the effect of temporal alignment, and possible approaches for speedup. Extensive analysis of the results also reveals intuitive interpretation of video event recognition through video subclip alignment at different levels. Index Terms—Event recognition, news video, concept ontology, Temporally Aligned Pyramid Matching, video indexing, earth mover’s distance. Ç 1
Shape recognition via Wasserstein distance
 Appl. Math
, 1999
"... The KantorovichRubinsteinWasserstein metric defines the distance between two probability measures µ and ν on R d+1 by computing the cheapest way to transport the mass of µ onto ν, where the cost per unit mass transported is a given function c(x, y)onR 2d+2. Motivated by applications to shape recog ..."
Abstract

Cited by 34 (14 self)
 Add to MetaCart
(Show Context)
The KantorovichRubinsteinWasserstein metric defines the distance between two probability measures µ and ν on R d+1 by computing the cheapest way to transport the mass of µ onto ν, where the cost per unit mass transported is a given function c(x, y)onR 2d+2. Motivated by applications to shape recognition, we analyze this transportation problem with the cost c(x, y)=x−y  2 and measures supported on two curves in the plane, or more generally on the boundaries of two domains Ω,Λ ⊂ R d+1. Unlike the theory for measures which are absolutely continuous with respect to Lebesgue, it turns out not to be the case that µa.e. x ∈ ∂Ω is transported to a single image y ∈ ∂Λ; however, we show the images of x are almost surely collinear and parallel the normal to ∂Ω at x. If either domain is strictly convex, we deduce that the solution to the optimization problem is unique. When both domains are uniformly convex, we prove a regularity result showing the images of x ∈ ∂Ω are always collinear, and both images depend on x in a continuous and (continuously) invertible way. This produces some unusual extremal doubly stochastic measures. c○1998 by the authors. Reproduction of this article, in its entirety, is permitted for noncommercial purposes.