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Topological data analysis requires a tradeoff between two mechanisms: (1) proper subsampling of high dimensional point clouds versus (2) constructing simplicial complexes on this subsampling to extract homological features of these high dimensional point clouds. In this talk, we will discuss optimal strategies of subsampling high dimensional datasets in order to extract these homological features reliably using kernel density estimation. We demonstrate the effectiveness of these methods on neural grid cell data. We also develop alternative constructions of simplicial complexes that significantly reduce computational complexity and extend the application of TDA to more heterogeneous datasets. Lastly, we show how these constructions are compatible with existing algorithms, such as the Village-Net clustering algorithm. We will end the talk discussing future extensions of this work in the space of functional parameters for basic machine learning models.

Zoom Meeting Link: https://ucdavis.zoom.us/j/92691231399 Passcode: tdarocks