Bounding the Interleaving Distance for Mapper Graphs with a Loss Function

Explore the concept of bounding the interleaving distance for Mapper graphs using a loss function in this comprehensive lecture by Elizabeth Munch. Delve into the challenges of comparing and clustering graph data with functions to R^d, which are prevalent in various data applications such as Reeb graphs, geometric graphs, and knot embeddings. Examine the interleaving distance on R^d-mapper graph discretizations and understand how functor representations of data can be compared using natural transformation pairs. Investigate the NP-hard nature of computing interleaving distance and discover a novel approach inspired by Robinson's work, which introduces quality measures for map families that don't meet natural transformation criteria. Learn about the concept of assignments and how metric space structure is applied to functor images to define a loss function measuring the deviation from interleaving diagram commutativity. Gain insights into the polynomial computation of this loss function and explore its potential applications in approximating and bounding interleavings across various contexts in applied algebraic topology.