Rick Jardine on Fuzzy Sets & Topological Data Analysis

MIT Category Theory Seminar
2020/11/12

©Spifong

Speaker: Rick Jardine
Title: Fuzzy sets, presheaves, and topological data analysis

Abstract:

There was a small buzz among TDA people in 2017-18 about applications of fuzzy set theory. I thought at the time that we should get the foundations right, so that's what the Compositionality paper "Fuzzy sets and presheaves" is about.


The outcome of the early dreams of application have been a bit mixed. I initially believed that fuzzy simplicial sets would lead to an understanding of TDA in terms of local homotopy theory, but we're still waiting for a good sheaf theoretic application. There was also the introduction of the UMAP algorithm by Healy-McInnes in 2018, which depended aggressively on a gluing construction in the fuzzy set category. This algorithm has been highly successful for applications in cluster analysis, but the construction itself was a black box for almost everybody on account of the fuzzy content. It turns out to be far better to take a presheaf theoretic approach to construct and analyze the variations of the UMAP complex directly, and to actually understand what they are.


This talk is part of a series on recently published papers in the journal Compositionality. The paper can be found here: https://compositionality-journal.org/papers/compositionality-1-3/

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