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Journal ArticleDOI

Gromov–Wasserstein Distances and the Metric Approach to Object Matching

01 Aug 2011-Foundations of Computational Mathematics (Springer-Verlag)-Vol. 11, Iss: 4, pp 417-487
TL;DR: This paper discusses certain modifications of the ideas concerning the Gromov–Hausdorff distance which have the goal of modeling and tackling the practical problems of object matching and comparison by proving explicit lower bounds for the proposed distance that involve many of the invariants previously reported by researchers.
Abstract: This paper discusses certain modifications of the ideas concerning the Gromov–Hausdorff distance which have the goal of modeling and tackling the practical problems of object matching and comparison. Objects are viewed as metric measure spaces, and based on ideas from mass transportation, a Gromov–Wasserstein type of distance between objects is defined. This reformulation yields a distance between objects which is more amenable to practical computations but retains all the desirable theoretical underpinnings. The theoretical properties of this new notion of distance are studied, and it is established that it provides a strict metric on the collection of isomorphism classes of metric measure spaces. Furthermore, the topology generated by this metric is studied, and sufficient conditions for the pre-compactness of families of metric measure spaces are identified. A second goal of this paper is to establish links to several other practical methods proposed in the literature for comparing/matching shapes in precise terms. This is done by proving explicit lower bounds for the proposed distance that involve many of the invariants previously reported by researchers. These lower bounds can be computed in polynomial time. The numerical implementations of the ideas are discussed and computational examples are presented.
Citations
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Posted Content
TL;DR: This short book reviews OT with a bias toward numerical methods and their applications in data sciences, and sheds lights on the theoretical properties of OT that make it particularly useful for some of these applications.
Abstract: Optimal transport (OT) theory can be informally described using the words of the French mathematician Gaspard Monge (1746-1818): A worker with a shovel in hand has to move a large pile of sand lying on a construction site. The goal of the worker is to erect with all that sand a target pile with a prescribed shape (for example, that of a giant sand castle). Naturally, the worker wishes to minimize her total effort, quantified for instance as the total distance or time spent carrying shovelfuls of sand. Mathematicians interested in OT cast that problem as that of comparing two probability distributions, two different piles of sand of the same volume. They consider all of the many possible ways to morph, transport or reshape the first pile into the second, and associate a "global" cost to every such transport, using the "local" consideration of how much it costs to move a grain of sand from one place to another. Recent years have witnessed the spread of OT in several fields, thanks to the emergence of approximate solvers that can scale to sizes and dimensions that are relevant to data sciences. Thanks to this newfound scalability, OT is being increasingly used to unlock various problems in imaging sciences (such as color or texture processing), computer vision and graphics (for shape manipulation) or machine learning (for regression, classification and density fitting). This short book reviews OT with a bias toward numerical methods and their applications in data sciences, and sheds lights on the theoretical properties of OT that make it particularly useful for some of these applications.

1,355 citations


Cites methods from "Gromov–Wasserstein Distances and th..."

  • ...23) and as explained in [Mémoli, 2011], it is possible to rewrite equivalently the GH distance using couplings as follows:...

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Posted Content
TL;DR: In this article, the authors show that in order to prove desirable properties of the entire decision-making process, different mechanisms for fairness require different assumptions about the nature of the mapping from construct space to decision space.
Abstract: What does it mean for an algorithm to be fair? Different papers use different notions of algorithmic fairness, and although these appear internally consistent, they also seem mutually incompatible. We present a mathematical setting in which the distinctions in previous papers can be made formal. In addition to characterizing the spaces of inputs (the "observed" space) and outputs (the "decision" space), we introduce the notion of a construct space: a space that captures unobservable, but meaningful variables for the prediction. We show that in order to prove desirable properties of the entire decision-making process, different mechanisms for fairness require different assumptions about the nature of the mapping from construct space to decision space. The results in this paper imply that future treatments of algorithmic fairness should more explicitly state assumptions about the relationship between constructs and observations.

388 citations

Reference BookDOI
01 Jan 2011
TL;DR: In this article, the Mumford and Shah Model and its applications in total variation image restoration are discussed. But the authors focus on the reconstruction of 3D information, rather than the analysis of the image.
Abstract: Linear Inverse Problems.- Large-Scale Inverse Problems in Imaging.- Regularization Methods for Ill-Posed Problems.- Distance Measures and Applications to Multi-Modal Variational Imaging.- Energy Minimization Methods.- Compressive Sensing.- Duality and Convex Programming.- EM Algorithms.- Iterative Solution Methods.- Level Set Methods for Structural Inversion and Image Reconstructions.- Expansion Methods.- Sampling Methods.- Inverse Scattering.- Electrical Impedance Tomography.- Synthetic Aperture Radar Imaging.- Tomography.- Optical Imaging.- Photoacoustic and Thermoacoustic Tomography: Image Formation Principles.- Mathematics of Photoacoustic and Thermoacoustic Tomography.- Wave Phenomena.- Statistical Methods in Imaging.- Supervised Learning by Support Vector Machines.- Total Variation in Imaging.- Numerical Methods and Applications in Total Variation Image Restoration.- Mumford and Shah Model and its Applications in Total Variation Image Restoration.- Local Smoothing Neighbourhood Filters.- Neighbourhood Filters and the Recovery of 3D Information.- Splines and Multiresolution Analysis.- Gabor Analysis for Imaging.- Shaper Spaces.- Variational Methods in Shape Analysis.- Manifold Intrinsic Similarity.- Image Segmentation with Shape Priors: Explicit Versus Implicit Representations.- Starlet Transform in Astronomical Data Processing.- Differential Methods for Multi-Dimensional Visual Data Analysis.- Wave fronts in Imaging, Quinto.- Ultrasound Tomography, Natterer.- Optical Flow, Schnoerr.- Morphology, Petros.- Maragos.- PDEs, Weickert. - Registration, Modersitzki. - Discrete Geometry in Imaging, Bobenko, Pottmann.-Visualization, Hege.- Fast Marching and Level Sets, Osher.- Couple Physics Imaging, Arridge.- Imaging in Random Media, Borcea.- Conformal Methods, Gu.- Texture, Peyre.- Graph Cuts, Darbon.- Imaging in Physics with Fourier Transform (i.e. Phase Retrieval e.g Dark field imaging), J. R. Fienup.- Electron Microscopy, Oktem Ozan.- Mathematical Imaging OCT (this is also FFT based), Mark E. Brezinski.- Spect, PET, Faukas, Louis.

341 citations

Proceedings Article
19 Jun 2016
TL;DR: This paper presents a new technique for computing the barycenter of a set of distance or kernel matrices, which define the interrelationships between points sampled from individual domains, and provides a fast iterative algorithm for the resulting nonconvex optimization problem.
Abstract: This paper presents a new technique for computing the barycenter of a set of distance or kernel matrices. These matrices, which define the interrelationships between points sampled from individual domains, are not required to have the same size or to be in row-by-row correspondence. We compare these matrices using the softassign criterion, which measures the minimum distortion induced by a probabilistic map from the rows of one similarity matrix to the rows of another; this criterion amounts to a regularized version of the Gromov-Wasserstein (GW) distance between metric-measure spaces. The barycenter is then defined as a Frechet mean of the input matrices with respect to this criterion, minimizing a weighted sum of softassign values. We provide a fast iterative algorithm for the resulting nonconvex optimization problem, built upon state-of-the-art tools for regularized optimal transportation. We demonstrate its application to the computation of shape barycenters and to the prediction of energy levels from molecular configurations in quantum chemistry.

275 citations


Cites background or methods from "Gromov–Wasserstein Distances and th..."

  • ...This definition (4) of GW extends slightly the one considered by (Mémoli, 2011), since we consider an arbitrary loss L (rather than just the L2 squared loss)....

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  • ...For L = L2, if the (Cs)s are positive semidefinite (PSD) matrices, the iteratesC produced by the algorithm are also PSD. Proof....

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  • ...To relax the matching problem to a continuous—but still non-convex—optimization problem, the seminal work of Mémoli considers matching between metric-measure spaces, incorporating a probability distribution on the ground space that can account for some form of uncertainty (Mémoli, 2011)....

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  • ...For the square loss L = L2, the update (13) becomes C ← 1 pp> ∑ s λsT > s CsTs. (14) This formula highlights an important property of the method: Proposition 4....

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  • ...In the case L = L2, (Mémoli, 2011) proves that GW1/2 defines a distance on the space of metric measure spaces quotiented by measure-preserving isometries....

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Journal ArticleDOI
11 Jul 2016
TL;DR: This work presents an algorithm for probabilistic correspondence that optimizes an entropy-regularized Gromov-Wasserstein (GW) objective that is compact, provably convergent, and applicable to any geometric domain expressible as a metric measure matrix.
Abstract: Many shape and image processing tools rely on computation of correspondences between geometric domains. Efficient methods that stably extract "soft" matches in the presence of diverse geometric structures have proven to be valuable for shape retrieval and transfer of labels or semantic information. With these applications in mind, we present an algorithm for probabilistic correspondence that optimizes an entropy-regularized Gromov-Wasserstein (GW) objective. Built upon recent developments in numerical optimal transportation, our algorithm is compact, provably convergent, and applicable to any geometric domain expressible as a metric measure matrix. We provide comprehensive experiments illustrating the convergence and applicability of our algorithm to a variety of graphics tasks. Furthermore, we expand entropic GW correspondence to a framework for other matching problems, incorporating partial distance matrices, user guidance, shape exploration, symmetry detection, and joint analysis of more than two domains. These applications expand the scope of entropic GW correspondence to major shape analysis problems and are stable to distortion and noise.

188 citations


Cites background or methods from "Gromov–Wasserstein Distances and th..."

  • ...GW is a distance between metric measure spaces, i.e., metric spaces equipped with a probability distribution (see [Mémoli 2011; Mémoli 2014; Sturm 2012] for more details)....

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  • ...We study an entropically-regularized version of the Gromov-Wasserstein (GW) mapping objective function from [Mémoli 2011] measuring the distortion of geodesic distances....

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  • ...[Mémoli 2011] and subsequent works mention two optimization algorithms (without regularization): gradient descent and alternation....

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  • ...Although [Mémoli 2011] and subsequent work identified the possibility of using GW distances for geometric correspondence, computational challenges hampered their practical application....

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  • ...Following [Mémoli 2011], define the space M(µ0, µ) of measure couplings as the set of measures γ ∈ Prob(Σ0 × Σ) satisfying γ(S0 × Σ) = µ0(S0) and γ(Σ0 × S) = µ(S) for all measurable S0 ⊆ Σ0 and S ⊆ Σ....

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References
More filters
Proceedings ArticleDOI
02 Sep 2004
TL;DR: Free MATLAB toolbox YALMIP is introduced, developed initially to model SDPs and solve these by interfacing eternal solvers by making development of optimization problems in general, and control oriented SDP problems in particular, extremely simple.
Abstract: The MATLAB toolbox YALMIP is introduced. It is described how YALMIP can be used to model and solve optimization problems typically occurring in systems and control theory. In this paper, free MATLAB toolbox YALMIP, developed initially to model SDPs and solve these by interfacing eternal solvers. The toolbox makes development of optimization problems in general, and control oriented SDP problems in particular, extremely simple. In fact, learning 3 YALMIP commands is enough for most users to model and solve the optimization problems

7,676 citations

Journal ArticleDOI
TL;DR: This clearly written, mathematically rigorous text includes a novel algorithmic exposition of the simplex method and also discusses the Soviet ellipsoid algorithm for linear programming; efficient algorithms for network flow, matching, spanning trees, and matroids; the theory of NP-complete problems; approximation algorithms, local search heuristics for NPcomplete problems, more.
Abstract: This clearly written , mathematically rigorous text includes a novel algorithmic exposition of the simplex method and also discusses the Soviet ellipsoid algorithm for linear programming; efficient algorithms for network flow, matching, spanning trees, and matroids; the theory of NP-complete problems; approximation algorithms, local search heuristics for NPcomplete problems, more All chapters are supplemented by thoughtprovoking problems A useful work for graduate-level students with backgrounds in computer science, operations research, and electrical engineering Mathematicians wishing a self-contained introduction need look no further—American Mathematical Monthly 1982 ed

7,221 citations


"Gromov–Wasserstein Distances and th..." refers background or methods in this paper

  • ...Indeed, transportation problems, also known as Hitchcock transportation problems are quite common in the optimization literature and standard references such as [88] describe specialized algorithms for solving these problems which run in polynomial time....

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  • ...There exist standard specialized algorithms for numerically computing the minimal value above [88], see Sect....

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Journal ArticleDOI
TL;DR: This paper presents work on computing shape models that are computationally fast and invariant basic transformations like translation, scaling and rotation, and proposes shape detection using a feature called shape context, which is descriptive of the shape of the object.
Abstract: We present a novel approach to measuring similarity between shapes and exploit it for object recognition. In our framework, the measurement of similarity is preceded by: (1) solving for correspondences between points on the two shapes; (2) using the correspondences to estimate an aligning transform. In order to solve the correspondence problem, we attach a descriptor, the shape context, to each point. The shape context at a reference point captures the distribution of the remaining points relative to it, thus offering a globally discriminative characterization. Corresponding points on two similar shapes will have similar shape contexts, enabling us to solve for correspondences as an optimal assignment problem. Given the point correspondences, we estimate the transformation that best aligns the two shapes; regularized thin-plate splines provide a flexible class of transformation maps for this purpose. The dissimilarity between the two shapes is computed as a sum of matching errors between corresponding points, together with a term measuring the magnitude of the aligning transform. We treat recognition in a nearest-neighbor classification framework as the problem of finding the stored prototype shape that is maximally similar to that in the image. Results are presented for silhouettes, trademarks, handwritten digits, and the COIL data set.

6,693 citations


"Gromov–Wasserstein Distances and th..." refers background in this paper

  • ...In [4, 74, 91], similar invariants were proposed where to each point a ∈ A one attaches the histogram or distribution of distances ‖a − a′‖ for all a′ ∈ A....

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  • ...The so called shape context [4, 23, 97, 102] invariant is closely related to hX ....

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  • ...[54]; the spin images of Johnsson [60]; the shape distributions of [86]; the canonical forms of [37]; the Hamza–Krim approach [53]; the spectral approaches of [94, 100]; the integral invariants of [30, 74, 91]; the shape contexts of [4]....

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Book
01 Jan 1979
TL;DR: In this paper, the convergence of distributions is considered in the context of conditional probability, i.e., random variables and expected values, and the probability of a given distribution converging to a certain value.
Abstract: Probability. Measure. Integration. Random Variables and Expected Values. Convergence of Distributions. Derivatives and Conditional Probability. Stochastic Processes. Appendix. Notes on the Problems. Bibliography. List of Symbols. Index.

6,334 citations

Book
01 Jan 1984
TL;DR: Strodiot and Zentralblatt as discussed by the authors introduced the concept of unconstrained optimization, which is a generalization of linear programming, and showed that it is possible to obtain convergence properties for both standard and accelerated steepest descent methods.
Abstract: This new edition covers the central concepts of practical optimization techniques, with an emphasis on methods that are both state-of-the-art and popular. One major insight is the connection between the purely analytical character of an optimization problem and the behavior of algorithms used to solve a problem. This was a major theme of the first edition of this book and the fourth edition expands and further illustrates this relationship. As in the earlier editions, the material in this fourth edition is organized into three separate parts. Part I is a self-contained introduction to linear programming. The presentation in this part is fairly conventional, covering the main elements of the underlying theory of linear programming, many of the most effective numerical algorithms, and many of its important special applications. Part II, which is independent of Part I, covers the theory of unconstrained optimization, including both derivations of the appropriate optimality conditions and an introduction to basic algorithms. This part of the book explores the general properties of algorithms and defines various notions of convergence. Part III extends the concepts developed in the second part to constrained optimization problems. Except for a few isolated sections, this part is also independent of Part I. It is possible to go directly into Parts II and III omitting Part I, and, in fact, the book has been used in this way in many universities.New to this edition is a chapter devoted to Conic Linear Programming, a powerful generalization of Linear Programming. Indeed, many conic structures are possible and useful in a variety of applications. It must be recognized, however, that conic linear programming is an advanced topic, requiring special study. Another important topic is an accelerated steepest descent method that exhibits superior convergence properties, and for this reason, has become quite popular. The proof of the convergence property for both standard and accelerated steepest descent methods are presented in Chapter 8. As in previous editions, end-of-chapter exercises appear for all chapters.From the reviews of the Third Edition: this very well-written book is a classic textbook in Optimization. It should be present in the bookcase of each student, researcher, and specialist from the host of disciplines from which practical optimization applications are drawn. (Jean-Jacques Strodiot, Zentralblatt MATH, Vol. 1207, 2011)

4,908 citations