no code implementations • 4 Mar 2024 • Facundo Mémoli, Brantley Vose, Robert C. Williamson
We introduce a notion of distance between supervised learning problems, which we call the Risk distance.
no code implementations • 1 Feb 2023 • Samantha Chen, Sunhyuk Lim, Facundo Mémoli, Zhengchao Wan, Yusu Wang
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.
no code implementations • 5 Feb 2022 • Samantha Chen, Sunhyuk Lim, Facundo Mémoli, Zhengchao Wan, Yusu Wang
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.
no code implementations • 16 Aug 2021 • Gunnar Carlsson, Facundo Mémoli, Santiago Segarra
We begin by introducing three practical properties associated with the notion of robustness in hierarchical clustering: linear scale preservation, stability, and excisiveness.
1 code implementation • 14 Jan 2021 • Facundo Mémoli, Axel Munk, Zhengchao Wan, Christoph Weitkamp
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
no code implementations • 21 Jun 2020 • Kun Jin, Facundo Mémoli, Zhengchao Wan
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.
no code implementations • 1 Jan 2020 • Facundo Mémoli, Guilherme Vituri F. Pinto
Motivated by the concept of network motifs we construct certain clustering methods (functors) which are parametrized by a given collection of motifs (or representers).
1 code implementation • 28 Dec 2019 • Facundo Mémoli, Ling Zhou
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
1 code implementation • 2 Dec 2019 • Facundo Mémoli, Zane Smith, Zhengchao Wan
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
no code implementations • 17 Oct 2018 • Facundo Mémoli, Zane Smith, Zhengchao Wan
We introduce the Wasserstein transform, a method for enhancing and denoising datasets defined on general metric spaces.
1 code implementation • 13 Aug 2018 • Samir Chowdhury, Facundo Mémoli
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
no code implementations • NeurIPS 2016 • Samir Chowdhury, Facundo Mémoli, Zane T. Smith
We consider an embedding of a metric space into a tree proposed by Gromov.
no code implementations • 21 Jul 2016 • Gunnar Carlsson, Facundo Mémoli, Alejandro Ribeiro, Santiago Segarra
We introduce two practical properties of hierarchical clustering methods for (possibly asymmetric) network data: excisiveness and linear scale preservation.
no code implementations • 21 Jul 2016 • Gunnar Carlsson, Facundo Mémoli, Alejandro Ribeiro, Santiago Segarra
This paper characterizes hierarchical clustering methods that abide by two previously introduced axioms -- thus, denominated admissible methods -- and proposes tractable algorithms for their implementation.
no code implementations • 21 Jul 2016 • Gunnar Carlsson, Facundo Mémoli, Alejandro Ribeiro, Santiago Segarra
This paper considers networks where relationships between nodes are represented by directed dissimilarities.
1 code implementation • 15 Sep 2015 • Diego Hernán Díaz Martínez, Facundo Mémoli, Washington Mio
We introduce the notion of multiscale covariance tensor fields (CTF) associated with Euclidean random variables as a gateway to the shape of their distributions.
no code implementations • 17 Apr 2014 • Gunnar Carlsson, Facundo Mémoli, Alejandro Ribeiro, Santiago Segarra
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.
no code implementations • 31 Jan 2013 • Gunnar Carlsson, Facundo Mémoli, Alejandro Ribeiro, Santiago Segarra
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.