Applications of Universality in Topological Data Analysis

Supervisor: Dr Omer Bobrowski

Project description:
Recently discovered, the phenomenon of universality in Topological Data Analysis (TDA) offers a new direction to data analysis. This project will explore its applications and develop new methodologies on top of this phenomenon. There are many possible directions but the initial focus will be on:

1. Dimensionality estimation: One of the key challenges in machine learning and data analysis is figuring out how many important features are in a dataset (i.e., its intrinsic dimension). However, traditional methods, such as principal component analysis (PCA), may not be suitable for high-dimensional or noisy data. The project will develop statistical tools for applying universality for estimating dimension from topological features (including mixed dimensional spaces, relaxing the manifold hypothesis).
2. Topological clustering: The connection between clustering and topology is well established with a substantial amount of previous work. This will build on this work to provide a complete framework for proving consistency in different clustering schemes as well as providing a provable approach to estimating the number of clusters from data. This will be driven by applications in a wide range of areas.
3. Quantifying disorder: While TDA is often concerned with global structure, there are many cases where the distributions of smaller features in cases such as quasicrystals or other types of materials plays an important role. The goal of this application is to leverage universality to quantify the amount of order (as a form of regularity) in a point set. This will connect to existing work in sampling and discrepancy theory.
   
   

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