Math

Nerves and Towers I: Introduction

Fundamental concepts of Mapper and Multiscale Mapper.

Nathaniel Saul

4 minute read

In unsupervised learning, one of the main questions we want to answer is what type of structure does this set or shape have? The field of algebraic topology supplies a few tools to help answer this question. We can ask topological questions about the data, such as how many clusters or connected components are there? or are there gaps, holes, or voids in the data?. These question can be easy to answer if the shape is well defined, but in practice, we instead have data sampled from that shape. If…

Nerves and Towers II: Nerves

An introduction to nerves, one of the fundamental building blocks of TDA.

Nathaniel Saul

3 minute read

The nerve is a simplicial complex built from a cover. It is a discrete summarization of the cover that captures the interesting topological features. Additionally, if the cover is sufficiently refined, then the nerve is guaranteed to preserve the topological features. A nerve of a cover is constructed in a very straight forward way: An (n-1)-dimensional simplex is added for each nonempty n-way intersection of elements of the cover.

Nerves and Towers III: Towers

An introduction to towers, an extension of mapper to the world of persistence.

Nathaniel Saul

6 minute read

Once we understand the tower, we will show that it plays nice with all of the other definitions we have constructed so far and how it allows us to naturally define a multiscale mapper.

What is Mapper?

Understanding of mapper from the ground up.

Nathaniel Saul

3 minute read

As one of the main tools from the field of Topological Data Analysis, Mapper has been shown to be particularly useful for exploring high dimensional point cloud data. This post will walk through the mapper construction from an intuitive perspective and demonstrate how it can be constructed on a toy example.

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