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SECT statistic

The objective of this blog post is to summarize my understanding of the theory and applications behind Smooth Euler Characteristic Transform (SECT) statistic which was first introduced by Crawford et al1 in 2020. The SECT statistic summarizes the shape information associated with an object as a collection of smooth curves that lie in a Hilbert s...

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Mapper Algorithm

The objective of this blog is to document my understanding of the Mapper algorithm1 introduced by Singh et al. along with a review of available statistical literature on the Mapper. The textbook by Prof. Tamal Dey2 is an extremely helpful source for understanding the topological motivation behind Mapper. In particular, chapters 7 and 9 of the te...

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Brain Connectivity Analysis

The purpose of the blog is to summarize the fundamentals of the branch of Connectomics while following the textbook by Fornito et al1. Connectomics deals with quantifying, visualizing, and understanding the organization of brain networks at different scales of space and time. Since nodes and edges can be viewed as the building blocks of a networ...

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Discrete Morse Theory - II

The current post is a continuation of an earlier post Discrete Morse Theory - I. While part one focused mostly on topological concepts, the second part focuses on combinatorial interpretations and constructions associated with a simplicial complex. The primary reference is Chapter 10 of the textbook “Organized Collapse: An Introduction to Discre...

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Abstract and Geometric Simplicial Complexes

Abstract and Geometric Simplicial complexes An abstract simplicial complex $\mathcal{X}$ is a collection of finite sets that are closed under taking subsets. Note that, the empty set $\emptyset$ is assumed to lie in $\mathcal{X}$. Although each element of the abstract simplicial complex must be a finite set, the cardinality of the complex in its...

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Discrete Morse Theory - I

The blog introduces terms and results from Discrete Morse Theory that will eventually be helpful in understanding the paper The Morse Theory of Cech and Delaunay Complexes by U. Bauer and H. Edelsbrunner1. The textbook I adhered to learn the basics is the textbook “Organized Collapse: An Introduction to Discrete Morse Theory” by Dmitry N. Kozlov...

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Generalizing Simplicial Complexes

The fundamental building blocks of TDA are simplices that can be combined to give a simplicial complex approximation of the underlying space $X$. One of the most commonly observed ideas in mathematics is to generalize mathematical objects of interest. In other words, we wish to “peel off” a subset of the object’s properties to understand the str...

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Simplicial Complexes

Introduction and Motivation The contents of this post are predominantly based on the review paper by Otter et al. 1 which discusses the computational aspects of Persistent Homology (PH). To begin with, the first four sections of the paper give a very good introduction to TDA without getting lost in unnecessary mathematical details. I believe tha...

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