## Introduction to Persistent Homology (Cech and Vietoris-Rips complex)

Motivation
Data is commonly represented as an unordered sequence of points in the Euclidean space $\mathbb{R}^n$. The global `shape’ of the data may provide important information about the underlying phenomena of the data.

For data points in $\mathbb{R}^2$, determining the global structure is not difficult, but for data in higher dimensions, a planar projection can be hard to decipher.
From point cloud data to simplicial complexes
To convert a collection of points $\{x_\alpha\}$ in a metric space into a global object, one can use the points as the vertices of a graph whose edges are determined by proximity (vertices within some chosen distance $\epsilon$). Then, one completes the graph to a simplicial complex. Two of the most natural methods for doing so are as follows:

Given a set of points $\{x_\alpha\}$ in Euclidean space $\mathbb{R}^n$, the Cech complex (also known as the nerve), $\mathcal{C}_\epsilon$, is the abstract simplicial complex where a set of $k+1$ vertices spans a $k$-simplex whenever the $k+1$ corresponding closed $\epsilon/2$-ball neighborhoods have nonempty intersection.

Given a set of points $\{x_\alpha\}$ in Euclidean space $\mathbb{R}^n$, the Vietoris-Rips complex, $\mathcal{R}_\epsilon$, is the abstract simplicial complex where a set $S$ of $k+1$ vertices spans a $k$-simplex whenever the distance between any pair of points in $S$ is at most $\epsilon$.

Top left: A fixed set of points. Top right: Closed balls of radius $\epsilon/2$ centered at the points. Bottom left: Cech complex has the homotopy type of the $\epsilon/2$ cover ($S^1\vee S^1\vee S^1$) Bottom right: Vietoris-Rips complex has a different homotopy type ($S^1\vee S^2$). Image from R. Ghrist, 2008, Barcodes: The Persistent Topology of Data.

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### 2 Responses to Introduction to Persistent Homology (Cech and Vietoris-Rips complex)

1. tomcircle says:

You can make the math symbols bigger than the text font for easy reading. eg. ${\lambda }$
“&s=3” added after the last } tells latex to make the symbol lambda bigger at size 3 (which is slightly bigger than the text font, so it stands out).

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2. tomcircle says:

Sorry, latex should be:
${\lambda}$“.
For color, you can append behind “&fg=aa0000 ” (brown)

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