From Coarse to Fine and Back Again: Geometry, Topology, and Deep Learning

A large driver contributing to the success of deep-learning models is their ability to synthesize task-specific features from data, which typically outshine hand-crafted features. Thus, for a long time, the predominant belief was that "given enough data, all features can be learned." However, it turns out that certain tasks require imbuing models with *inductive biases* such as invariance or equivariance properties that **cannot** be readily gleaned from the data!
This is particularly true for data sets that model real-world phenomena, including those that involve relations beyond the dyadic, creating a crucial need for different approaches.

In this talk, I will present novel advances in harnessing multi-scale geometrical-topological characteristics of data, focusing in particular on how the tandem of geometry and topology can improve (un)supervised tasks in representation learning. Underscoring the generality of a hybrid geometrical-topological perspective, I will furthermore showcase applications from a diverse set of data domains, including point clouds, graphs, and higher-order combinatorial complexes.