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Orthogonal Gromov-Wasserstein Discrepancy with Efficient Lower Bound [article]

Hongwei Jin, Zishun Yu, Xinhua Zhang
2022 arXiv   pre-print
To address this issue, we take inspiration from the connection with the quadratic assignment problem, and propose the orthogonal Gromov-Wasserstein (OGW) discrepancy as a surrogate of GW.  ...  It admits an efficient and closed-form lower bound with the complexity of 𝒪(n^3), and directly extends to the fused Gromov-Wasserstein (FGW) distance, incorporating node features into the coupling.  ...  Recently, the Gromov-Wasserstein discrepancy (GW, Peyre et al., 2016) , which extends the Gromov-Wasserstein distance (Memoli, 2011) , has emerged as an effective transportation distance between structured  ... 
arXiv:2205.05838v1 fatcat:ivad3l7hcbexbgyeepuotdzw3y

Sliced Gromov-Wasserstein [article]

Titouan Vayer, Rémi Flamary, Romain Tavenard, Laetitia Chapel, Nicolas Courty
2020 arXiv   pre-print
We illustrate the behavior of this so called Sliced Gromov-Wasserstein (SGW) discrepancy in experiments where we demonstrate its ability to tackle similar problems as GW while being several order of magnitudes  ...  We then define a novel OT discrepancy that can deal with large scale distributions via a slicing approach and we show how it relates to the GW distance while being O(nlog(n)) to compute.  ...  We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Titan X GPU used for this research.  ... 
arXiv:1905.10124v3 fatcat:2kbzjz5o7jbqrpz3lpkco4uq4a

Improving Relational Regularized Autoencoders with Spherical Sliced Fused Gromov Wasserstein [article]

Khai Nguyen and Son Nguyen and Nhat Ho and Tung Pham and Hung Bui
2020 arXiv   pre-print
To improve the discrepancy and consequently the relational regularization, we propose a new relational discrepancy, named spherical sliced fused Gromov Wasserstein (SSFG), that can find an important area  ...  A recent attempt to reduce the inner discrepancy between the prior and aggregated posterior distributions is to incorporate sliced fused Gromov-Wasserstein (SFG) between these distributions.  ...  Spherical sliced fused Gromov Wasserstein We first start with a definition of spherical sliced fused Gromov Wasserstein. Definition 3.  ... 
arXiv:2010.01787v1 fatcat:t3clf5oqhfe45p57vspskmbuli

Theoretical Guarantees for Bridging Metric Measure Embedding and Optimal Transport [article]

Mokhtar Z. Alaya, Maxime Bérar, Gilles Gasso, Alain Rakotomamonjy
2021 arXiv   pre-print
Unlike Gromov-Wasserstein (GW) distance which compares pairwise distances of elements from each distribution, we consider a method allowing to embed the metric measure spaces in a common Euclidean space  ...  This leads to what we call a sub-embedding robust Wasserstein (SERW) distance.  ...  We consider two metric measure spaces (mm-space for short) (Gromov et al., 1999) is a compact metric space and µ is a probability measure with full support, i.e. µ(X) = 1 and supp[µ] = X.  ... 
arXiv:2002.08314v5 fatcat:4j3y2en4mrb33dfmwzarnxgoe4

Heterogeneous Wasserstein Discrepancy for Incomparable Distributions [article]

Mokhtar Z. Alaya, Gilles Gasso, Maxime Berar, Alain Rakotomamonjy
2021 arXiv   pre-print
We show that the embeddings involved in HWD can be efficiently learned.  ...  We provide a theoretical analysis of this new divergence, called heterogeneous Wasserstein discrepancy (HWD), and we show that it preserves several interesting properties including rotation-invariance.  ...  Another approach for scaling up the GW distance is Sliced Gromov-Wasserstein (SGW) discrepancy (Vayer et al., 2019) , which leverages on random projections on 1D and on a closed-form solution of the 1D-Gromov-Wasserstein  ... 
arXiv:2106.02542v2 fatcat:7znxswqvybdnniewpfssi5ay3m

A contribution to Optimal Transport on incomparable spaces [article]

Titouan Vayer
2020 arXiv   pre-print
An important part is notably devoted to the study of the Gromov-Wasserstein distance whose properties allow to define interesting transport problems on incomparable spaces.  ...  We derive a new discrepancy named Sliced Gromov-Wasserstein (SGW ) that relies on these findings for efficient computation.  ...  Computing a lower-bound Originally (2.50) was tackled by computing a lower bound (called the TLB) in [Memoli 2011 ].  ... 
arXiv:2011.04447v1 fatcat:qnmq5pgqqnaphodg7gcn5j2dt4

Reversible Gromov-Monge Sampler for Simulation-Based Inference [article]

YoonHaeng Hur, Wenxuan Guo, Tengyuan Liang
2022 arXiv   pre-print
Motivated by the seminal work on distance and isomorphism between metric measure spaces, we propose a new notion called the Reversible Gromov-Monge (RGM) distance and study how RGM can be used to design  ...  Our RGM sampler can also estimate optimal alignments between two heterogeneous metric measure spaces (𝒳, μ, c_𝒳) and (𝒴, ν, c_𝒴) from empirical data sets, with estimated maps that approximately push  ...  and Gromov-Wasserstein are as follows.  ... 
arXiv:2109.14090v2 fatcat:65qfjqcxnve5ffxhrhmx3tqb4a

Unsupervised Hierarchy Matching with Optimal Transport over Hyperbolic Spaces [article]

David Alvarez-Melis, Youssef Mroueh, Tommi S. Jaakkola
2020 arXiv   pre-print
Alternatively, this problem can be successfully approached (Alvarez-Melis and Jaakkola, 2018) with a generalized version of optimal transport, the Gromov-Wasserstein (GW) distance (Mémoli, 2011) , which  ...  Truly isometric embeddings would yield 0 discrepancy. we can now solve a generalized (hyperbolic) version of the Orthogonal Procrustes problem as before.  ... 
arXiv:1911.02536v2 fatcat:c5tqf2b4gfdylatnavtum7fx7q

The Shape of Data: Intrinsic Distance for Data Distributions [article]

Anton Tsitsulin, Marina Munkhoeva, Davide Mottin, Panagiotis Karras, Alex Bronstein, Ivan Oseledets, Emmanuel Müller
2020 arXiv   pre-print
We develop a first-of-its-kind intrinsic and multi-scale method for characterizing and comparing data manifolds, using a lower-bound of the spectral variant of the Gromov-Wasserstein inter-manifold distance  ...  Figure 10 shows the points sampled from the GAN with some of the points inside the hole.  ...  advances in differential geometry and numerical linear algebra to create IMD (Multi-Scale Intrinsic Distance), a fast, intrinsic method to lower-bound the spectral Gromov-Wasserstein distance between  ... 
arXiv:1905.11141v2 fatcat:i4vyatnjlfg4xkiouqqbtxftg4

Co-clustering through Optimal Transport [article]

Charlotte Laclau, Ievgen Redko, Basarab Matei, Younès Bennani, Vincent Brault
2017 arXiv   pre-print
The algorithm derived for the proposed method and its kernelized version based on the notion of Gromov-Wasserstein distance are fast, accurate and can determine automatically the number of both row and  ...  The entropic Gromov-Wasserstein discrepancy in this case is defined as follows (Peyré et al., 2016) : GW(K r , K c ,μ r ,μ c ) = min γ∈Πμ r ,μc Γ Kr,Kc (γ) − λE(γ) = min T ∈Πμ r ,μc i,j,k,l L(K ri,j ,  ...  Algorithm 2 Co-clustering through Optimal Transport with Gromov-Wasserstein barycenters (CCOT-GW) Input : A -data matrix, λ -regularization parameter, εr, εcweights for barycenter calculation Output: Cr  ... 
arXiv:1705.06189v3 fatcat:toe6dbmnszfmxfxulddqy7yaqq

Sketching Merge Trees for Scientific Data Visualization [article]

Mingzhe Li, Sourabh Palande, Lin Yan, Bei Wang
2021 arXiv   pre-print
We develop a framework for sketching a set of merge trees that combines the Gromov-Wasserstein probabilistic matching with techniques from matrix sketching.  ...  It is shown to be useful in identifying good representatives and outliers with respect to a chosen basis.  ...  The Gromov-Wasserstein discrepancy [50] is defined as D(C) = min C∈C E(C). (2) In this paper, we consider the quadratic loss function L(a, b) = 1 2 |a − b| 2 .  ... 
arXiv:2101.03196v2 fatcat:ex6fu3gs4vbrrehlwa33kidn2i

Shape Correspondence by Aligning Scale-invariant LBO Eigenfunctions

Amit Bracha, Oshri Halim, Ron Kimmel
2020 Eurographics Workshop on 3D Object Retrieval, EG 3DOR  
Such discrepancies include sign ambiguities and possible rotations and reflections within subspaces spanned by eigenfunctions that correspond to similar eigenvalues.  ...  Estimating this transformation allows us to align the eigenfunctions of one shape with those of the other, that could then be used as intrinsic, consistent, and robust descriptors.  ...  [SPKS16] used the related Gromov-Wasserstein distance [Mém11] with a regularization on the entropy of the correspondence.  ... 
doi:10.2312/3dor.20201159 fatcat:t63va6cjxvfehmarqdpmy5xmhu

Computational Optimal Transport [article]

Gabriel Peyré, Marco Cuturi
2020 arXiv   pre-print
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).  ...  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  ...  In [Cuturi, 2013] , lower and upper bounds to approximate the Wasserstein distance between two histograms were proposed.  ... 
arXiv:1803.00567v4 fatcat:zgannw6i6beqde5bx7pj62uyry

Variational Diffusion Autoencoders with Random Walk Sampling [article]

Henry Li, Ofir Lindenbaum, Xiuyuan Cheng, Alexander Cloninger
2020 arXiv   pre-print
Finally, we demonstrate the effectiveness of our method with various real and synthetic datasets.  ...  We report the mean and standard deviation of the Gromov-Wasserstein distance [27] and median bi-Lipschitz over 5 runs in Table 1 .  ...  The lower bound Formally, let us define U x := B d (x, δ) ∩ M X , where B d (x, δ) is the δ-ball around x with respect to d(·, ·), the diffusion distance on M Z .  ... 
arXiv:1905.12724v4 fatcat:ryd2jxanybg2ppgomuqvx2naim

Uniform Distribution Theory and Applications

Michael Gnewuch, Frances Kuo, Harald Niederreiter, Henryk Woźniakowski
2013 Oberwolfach Reports  
The topics of the workshop were recent progress in the theory of uniform distribution theory (also known as discrepancy theory) and new developments in its applications in analysis, approximation theory  ...  Surprisingly, not even a lower bound for the discrepancy of a random point set is known.  ...  We survey the development of the problem beginning with Roth's seminal lower bound on the L 2 discrepancy from 1954 to the recent constructions of point sets and sequences with optimal order of L 2 discrepancy  ... 
doi:10.4171/owr/2013/49 fatcat:wsm65bw7gfh7ndaepflzced3sm
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