Primoz Skraba, Analysis of Scalar Fields over Point Cloud Data

 
Given a real-valued function defined over some metric space, is itpossible to recover some structural information about the functionfrom the sole information of its values at a finite set of samplepoints, provided pairwise distances are known? This talk will focus ona topological approach to the problem. Specifically, I will discusspersistent homology and how we can approximate it on an underlyingspace using only the sample points. The required algorithms areintuitive and have reasonable complexity, so these tools are also of apractical interest. The theoretical tools required to provecorrectness are themselves of interest and range from the geometric toalgebraic. A few of the topics I will cover to illustrate the proofs includethe construction of some popular simplicial complexes, stability theoremsfor persistent homology, and some variants of algorithm. Finally, I willalso show experimental results over point cloud data.