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A Novel Sliced Fused Gromov-Wasserstein Distance

Moritz Piening, Robert Beinert

TL;DR

This work proposes a novel slicing technique for Gromov--Wasserstein as well as for FGW that is based on an appropriate lower bound, hierarchical OT, and suitable quadrature rules for the underlying 1D OT problems.

Abstract

The Gromov--Wasserstein (GW) distance and its fused extension (FGW) are powerful tools for comparing heterogeneous data. Their computation is, however, challenging since both distances are based on non-convex, quadratic optimal transport (OT) problems. Leveraging 1D OT, a sliced version of GW has been proposed to lower the computational burden. Unfortunately, this sliced version is restricted to Euclidean geometry and loses invariance to isometries, strongly limiting its application in practice. To overcome these issues, we propose a novel slicing technique for GW as well as for FGW that is based on an appropriate lower bound, hierarchical OT, and suitable quadrature rules for the underlying 1D OT problems. Our novel sliced FGW significantly reduces the numerical effort while remaining invariant to isometric transformations and allowing the comparison of arbitrary geometries. We show that our new distance actually defines a pseudo-metric for structured spaces that bounds FGW from below and study its interpolation properties between sliced Wasserstein and GW. Since we avoid the underlying quadratic program, our sliced distance is numerically more robust and reliable than the original GW and FGW distance; especially in the context of shape retrieval and graph isomorphism testing.

A Novel Sliced Fused Gromov-Wasserstein Distance

TL;DR

This work proposes a novel slicing technique for Gromov--Wasserstein as well as for FGW that is based on an appropriate lower bound, hierarchical OT, and suitable quadrature rules for the underlying 1D OT problems.

Abstract

The Gromov--Wasserstein (GW) distance and its fused extension (FGW) are powerful tools for comparing heterogeneous data. Their computation is, however, challenging since both distances are based on non-convex, quadratic optimal transport (OT) problems. Leveraging 1D OT, a sliced version of GW has been proposed to lower the computational burden. Unfortunately, this sliced version is restricted to Euclidean geometry and loses invariance to isometries, strongly limiting its application in practice. To overcome these issues, we propose a novel slicing technique for GW as well as for FGW that is based on an appropriate lower bound, hierarchical OT, and suitable quadrature rules for the underlying 1D OT problems. Our novel sliced FGW significantly reduces the numerical effort while remaining invariant to isometric transformations and allowing the comparison of arbitrary geometries. We show that our new distance actually defines a pseudo-metric for structured spaces that bounds FGW from below and study its interpolation properties between sliced Wasserstein and GW. Since we avoid the underlying quadratic program, our sliced distance is numerically more robust and reliable than the original GW and FGW distance; especially in the context of shape retrieval and graph isomorphism testing.

Paper Structure

This paper contains 30 sections, 4 theorems, 36 equations, 6 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Optimal Transport-Based Distances
  3. Wasserstein Distance
  4. Sliced Wasserstein Distance
  5. Gromov--Wasserstein Distance
  6. First Lower Bound
  7. Second Lower Bound
  8. Third Lower Bound
  9. Fused Gromov--Wasserstein Distance
  10. Accelerating the Calculation of FGW
  11. A Lower Bound Using TLB
  12. A Novel Lower Bound via Slicing
  13. Numerical Experiments
  14. Computational and Runtime Analysis
  15. FTLB
  16. ...and 15 more sections

Key Result

Proposition 3.1

For $\mathbb{X}^{\mathcal{S}} \coloneqq (X \times Z, g, \xi)$, $\mathbb{Y}^{\mathcal{S}} \coloneqq (Y \times Z, h, \upsilon)$, $Z \subset \mathbb{R}^d$ compact, let $\xi^{\mathcal{F}} \coloneqq \pi_{Z, \sharp} \, \xi$, $\upsilon^{\mathcal{F}} \coloneqq \pi_{Z, \sharp} \, \upsilon$ and $\mathbb{X} \c

Figures (6)

  • Figure 1: Visualization of our novel sliced fused Grovov--Wasserstein distance called SFTLB: Starting from two structured spaces like labeled graphs equipped with a graph distance, we sort the corresponding distance matrices row-wise. Subsequently, we determine samples of the quantile functions of the local distance distribution for each node. Afterward, we concatenate the sampled quantiles with the original features and interpret the outcome as an empirical Euclidean measure. Finally, we compute the Wasserstein or sliced Wasserstein distance between these measures.
  • Figure 2: Comparison of TLB and FTLB transport plans. Crosses are colored based on the transported mass. FTLB achieves a more regular transport plan by integrating labels.
  • Figure 3: Comparison of FGW, entropic FGW ($\epsilon=0.01$), FTLP transport plan for $\alpha=0.5$. Target nodes are colored according to the transported mass. FGW and FTLB capture the same structural similarities.
  • Figure 4: Euclidean 2D TLB, AE and STLB barycenters for five samples of different shape classes.
  • Figure 5: Impact of projection number $L$ on deviation of Monte Carlo SFTLB Estimate. Each line corresponds to multiple SFTLB estimates for a fixed simulated measure pair.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • proof : Proof of Prop. \ref{['prop:ftlb_prop']}
  • proof : Proof of Prop. \ref{['prop:ftlb_prop']}
  • proof : Proof of Prop. \ref{['prop:prop_sftlb_properties']}
  • proof : Proof of Prop. \ref{['prop:dimensional_dependence_sliced']}
  • proof : Proof of Prop. \ref{['prop:metr-equi']}
  • proof : Extension of Prop. \ref{['prop:metr-equi']}