# Persistent Homology for Learning Densities with Bounded Support

Florian T. Pokorny, Carl Henrik Ek, Hedvig Kjellström, Danica Kragic
In Advances in Neural Information Processing Systems 25, 2012, pp. 1826--1834

## Abstract

We present a novel method for learning densities with bounded support which enables us to incorporate hard' topological constraints. In particular, we show how emerging techniques from computational algebraic topology and the notion of persistent homology can be combined with kernel-based methods from machine learning for the purpose of density estimation. The proposed formalism facilitates learning of models with bounded support in a principled way, and -- by incorporating persistent homology techniques in our approach -- we are able to encode algebraic-topological constraints which are not addressed in current state of the art probabilistic models. We study the behaviour of our method on two synthetic examples for various sample sizes and exemplify the benefits of the proposed approach on a real-world dataset by learning a motion model for a race car. We show how to learn a model which respects the underlying topological structure of the racetrack, constraining the trajectories of the car.

## Bibtex

@incollection{pokorny2012a, title = {Persistent Homology for Learning Densities with Bounded Support}, author = {Pokorny, Florian T. and Ek, Carl Henrik and Kjellstr{\"o}m, Hedvig and Kragic, Danica}, booktitle = {Advances in Neural Information Processing Systems 25}, editor = {P. Bartlett and F.C.N. Pereira and C.J.C. Burges and L. Bottou and K.Q. Weinberger}, pages = {1826--1834}, year = {2012}, url = {http://www.csc.kth.se/~fpokorny/static/publications/pokorny2012a.pdf}, } `