Matching Point Sets with Respect to the Earth Mover’s Distance

Cabello, Sergio; Giannopoulos, Panos; Knauer, Christian; Rote, Günter

doi:10.1007/11561071_47

Cited by 16 publications

(23 citation statements)

References 11 publications

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“…It has become a metric of choice in computer vision, where images are represented as intensity distributions over a Euclidean grid of pixels. It has also been applied to shape comparison [8], where shapes are represented by point clouds (discrete distributions) in space. While the EMD is a popular way of comparing distributions over a metric space, it is also an expensive one.…”

mentioning

confidence: 99%

Comparing distributions and shapes using the kernel distance

Joshi

Kommaraji

Phillips

et al. 2011

Proceedings of the Twenty-Seventh Annual Symposium on Computational Geometry

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Starting with a similarity function between objects, it is possible to define a distance metric on pairs of objects, and more generally on probability distributions over them. These distance metrics have a deep basis in functional analysis, measure theory and geometric measure theory, and have a rich structure that includes an isometric embedding into a (possibly infinite dimensional) Hilbert space. They have recently been applied to numerous problems in machine learning and shape analysis.In this paper, we provide the first algorithmic analysis of these distance metrics. Our main contributions are as follows: (i) We present fast approximation algorithms for computing the kernel distance between two point sets P and Q that runs in near-linear time in the size of P ∪ Q (note that an explicit calculation would take quadratic time). (ii) We present polynomial-time algorithms for approximately minimizing the kernel distance under rigid transformation; they run in time O(n+ poly(1/ǫ, log n)). (iii) We provide several general techniques for reducing complex objects to convenient sparse representations (specifically to point sets or sets of points sets) which approximately preserve the kernel distance. In particular, this allows us to reduce problems of computing the kernel distance between various types of objects such as curves, surfaces, and distributions to computing the kernel distance between point sets. These take advantage of the reproducing kernel Hilbert space and a new relation linking binary range spaces to continuous range spaces with bounded fat-shattering dimension.

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mentioning

confidence: 99%

Comparing distributions and shapes using the kernel distance

Joshi

Kommaraji

Phillips

et al. 2011

Proceedings of the Twenty-Seventh Annual Symposium on Computational Geometry

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show abstract

“…Thus, the pre-min-cost flow computation of our algorithm incurs O(s 2 n) cost. We focus on the W 1 -distance instead of the general q-Wasserstein distance since we can use the triangle inequality for a guaranteed (1+ε)-spanner [14].…”

Section: Existing Algorithms and Our Approachmentioning

confidence: 99%

“…The node sparsification of G(A, B) gives G δ whose arcs are further sparsified. Using Theorem 1 in [14], we bring the quadratic number of arcs down to a linear number by constructing a geometric (1 + ε)-spanner on the point set Âδ ∪ Bδ . For a point set P ⊂ R 2 , let its complete distance graph be defined with the points in P as nodes where every pair p, q ∈ P , p = q, is joined by an edge with weight equal to p − q 2 .…”

Section: Well Separated Pair Decomposition(arc Sparsification)mentioning

confidence: 99%

Approximating 1-Wasserstein Distance between Persistence Diagrams by Graph Sparsification

Dey¹,

Zhang²

2021

Preprint

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Persistence diagrams (PD)s play a central role in topological data analysis. This analysis requires computing distances among such diagrams such as the 1-Wasserstein distance. Accurate computation of these PD distances for large data sets that render large diagrams may not scale appropriately with the existing methods. The main source of difficulty ensues from the size of the bipartite graph on which a matching needs to be computed for determining these PD distances. We address this problem by making several algorithmic and computational observations in order to obtain an approximation. First, taking advantage of the proximity of PD points, we condense them thereby decreasing the number of nodes in the graph for computation. The increase in point multiplicities is addressed by reducing the matching problem to a min-cost flow problem on a transshipment network. Second, we use Well Separated Pair Decomposition to sparsify the graph to a size that is linear in the number of points. Both node and arc sparsifications contribute to the approximation factor where we leverage a lower bound given by the Relaxed Word Mover's distance. Third, we eliminate bottlenecks during the sparsification procedure by introducing parallelism. Fourth, we develop an open source software called 1 PDoptFlow based on our algorithm, exploiting parallelism by GPU and multicore. We perform extensive experiments and show that the actual empirical error is very low. We also show that we can achieve high performance at low guaranteed relative errors, improving upon the state of the arts.

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“…Our framework will work for other metrics based on (S, d), such as average of aligned distances [41,43,9], averages of all pairs of distances [30], bipartite matchings [2], or partial matchings [17]. Also our framework structurally will work for different base metrics, such as using (R 2 , L2), the standard Euclidean distance defined over the segment end points si.…”

Section: Extension To Other Metricsmentioning

confidence: 99%

Distributed trajectory similarity search

2017

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Mobile and sensing devices have already become ubiquitous. They have made tracking moving objects an easy task. As a result, mobile applications like Uber and many IoT projects have generated massive amounts of trajectory data that can no longer be processed by a single machine efficiently. Among the typical query operations over trajectories, similarity search is a common yet expensive operator in querying trajectory data. It is useful for applications in different domains such as traffic and transportation optimizations, weather forecast and modeling, and sports analytics. It is also a fundamental operator for many important mining operations such as clustering and classification of trajectories. In this paper, we propose a distributed query framework to process trajectory similarity search over a large set of trajectories. We have implemented the proposed framework in Spark, a popular distributed data processing engine, by carefully considering different design choices. Our query framework supports both the Hausdorff distance the Fréchet distance. Extensive experiments have demonstrated the excellent scalability and query efficiency achieved by our design, compared to other methods and design alternatives.

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Matching Point Sets with Respect to the Earth Mover’s Distance

Cited by 16 publications

References 11 publications

Comparing distributions and shapes using the kernel distance

Comparing distributions and shapes using the kernel distance

Approximating 1-Wasserstein Distance between Persistence Diagrams by Graph Sparsification

Distributed trajectory similarity search

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