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Found 11 papers, 5 papers with code
All You Need is Resistance: On the Equivalence of Effective Resistance and Certain Optimal Transport Problems on Graphs
The fields of effective resistance and optimal transport on graphs are filled with rich connections to combinatorics, geometry, machine learning, and beyond.
Comparing Graph Transformers via Positional Encodings
The distinguishing power of graph transformers is closely tied to the choice of positional encoding: features used to augment the base transformer with information about the graph.
Distances for Markov Chains, and Their Differentiation
Recently, in the graph learning and optimization communities, a range of new approaches have been developed for comparing graphs with node attributes, leveraging ideas such as the Optimal Transport (OT) and the Weisfeiler-Lehman (WL) graph isomorphism test.
Understanding Oversquashing in GNNs through the Lens of Effective Resistance
We propose to use total effective resistance as a bound of the total amount of oversquashing in a graph and provide theoretical justification for its use.
The Weisfeiler-Lehman Distance: Reinterpretation and Connection with GNNs
This new interpretation connects the WL distance to the literature on distances for stochastic processes, which also makes the interpretation of the distance more accessible and intuitive.
The Numerical Stability of Hyperbolic Representation Learning
Given the exponential growth of the volume of the ball w. r. t.
Weisfeiler-Lehman meets Gromov-Wasserstein
The WL distance is polynomial time computable and is also compatible with the WL test in the sense that the former is positive if and only if the WL test can distinguish the two involved graphs.
The ultrametric Gromov-Wasserstein distance
In this paper, we investigate compact ultrametric measure spaces which form a subset $\mathcal{U}^w$ of the collection of all metric measure spaces $\mathcal{M}^w$.
Metric Geometry Populations and Evolution
The Gaussian Transform
Our contribution is twofold: (1) theoretically, we establish firstly that GT is stable under perturbations and secondly that in the continuous case, each point possesses an asymptotically ellipsoidal neighborhood with respect to the GT distance; (2) computationally, we accelerate GT both by identifying a strategy for reducing the number of matrix square root computations inherent to the $\ell^2$-Wasserstein distance between Gaussian measures, and by avoiding redundant computations of GT distances between points via enhanced neighborhood mechanisms.
On $p$-metric spaces and the $p$-Gromov-Hausdorff distance
For each given $p\in[1,\infty]$ we investigate certain sub-family $\mathcal{M}_p$ of the collection of all compact metric spaces $\mathcal{M}$ which are characterized by the satisfaction of a strengthened form of the triangle inequality which encompasses, for example, the strong triangle inequality satisfied by ultrametric spaces.
Metric Geometry
The Wasserstein transform
We introduce the Wasserstein transform, a method for enhancing and denoising datasets defined on general metric spaces.
