Search Results for author:
Found 18 papers, 5 papers with code
Geometry and Stability of Supervised Learning Problems
We introduce a notion of distance between supervised learning problems, which we call the Risk distance.
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.
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.
Robust Hierarchical Clustering for Directed Networks: An Axiomatic Approach
We begin by introducing three practical properties associated with the notion of robustness in hierarchical clustering: linear scale preservation, stability, and excisiveness.
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.
Motivic clustering schemes for directed graphs
Motivated by the concept of network motifs we construct certain clustering methods (functors) which are parametrized by a given collection of motifs (or representers).
Persistent Homotopy Groups of Metric Spaces
We pay particular attention to the case of fundamental groups, for which we obtain a more precise description.
Algebraic Topology Computational Geometry 53C23, 51F99, 55N35
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.
The Gromov-Wasserstein distance between networks and stable network invariants
We define a metric---the Network Gromov-Wasserstein distance---on weighted, directed networks that is sensitive to the presence of outliers.
Discrete Mathematics Metric Geometry
Improved Error Bounds for Tree Representations of Metric Spaces
We consider an embedding of a metric space into a tree proposed by Gromov.
Hierarchical Clustering of Asymmetric Networks
This paper considers networks where relationships between nodes are represented by directed dissimilarities.
Excisive Hierarchical Clustering Methods for Network Data
We introduce two practical properties of hierarchical clustering methods for (possibly asymmetric) network data: excisiveness and linear scale preservation.
Admissible Hierarchical Clustering Methods and Algorithms for Asymmetric Networks
This paper characterizes hierarchical clustering methods that abide by two previously introduced axioms -- thus, denominated admissible methods -- and proposes tractable algorithms for their implementation.
The Shape of Data and Probability Measures
We introduce the notion of multiscale covariance tensor fields (CTF) associated with Euclidean random variables as a gateway to the shape of their distributions.
Hierarchical Quasi-Clustering Methods for Asymmetric Networks
This paper introduces hierarchical quasi-clustering methods, a generalization of hierarchical clustering for asymmetric networks where the output structure preserves the asymmetry of the input data.
Axiomatic Construction of Hierarchical Clustering in Asymmetric Networks
Our construction of hierarchical clustering methods is based on defining admissible methods to be those methods that abide by the axioms of value - nodes in a network with two nodes are clustered together at the maximum of the two dissimilarities between them - and transformation - when dissimilarities are reduced, the network may become more clustered but not less.
