Topological Data Analysis
Recent developments in machine learning and Artificial Intelligence has catapulted our ability to handle large datasets and extract useful information from them. Topological Data Analysis, or TDA in short, aims to discern the "shape" of data, without the need for training and tuning parameters.
We focus on two aspects in TDA that arise from algebraic and geometric topology.
Persistent Homology:
This is a technique arising from computational topology and aims to evaluate the homology groups, defined in a suitable sense, to a point cloud (usually a collection of points in some Euclidean space.
The method has been applied to a large variety of datasets, such as cancer types, brain imaging, and inter-galactic filament clusters.
Our recent work (with H. Kannan, E. Saucan and A. Samal) applies this method to investigate unweighted complex networks via discrete Morse theory. We show in particular that persistent homology can distinguish between various model and real-world networks.
For more details, we refer you to the our paper "Persistent homology of complex networks via discrete Morse theory" published in the journal Scientific Reports. The arXiv preprint can be found here.
Discrete Ricci curvature:
Classical geometry posits various types of curvature that can be associated with a Riemannian manifold. One of them is the famous Ricci curvature tensor, that appears for example in Einstein's field equations. The Ricci curvature tensor measures dispersal of geodesics emanating from a common source point.
Recently, different discrete versions of Ricci curvature have appeared in the literature. Two of the most well-known are due to Forman, using Bochner-Weitzenboeck methods and due to Ollivier, using optimal transportation and the Wasserstein distance.
Our recent work tries to find relationships between persistent homology and the discretized Ricci curvatures. More details can be found in our paper, which was published in the journal Chaos, Solitons and Fractals:
This is a joint work with A. Samal (IMSc Chennai).