Fix the definition of the Vietoris-Rips Complex

src/article/Supplement.svx

@@ -47,7 +47,7 @@ In order to construct the initial high-dimensional graph, UMAP relies on constru
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-We begin by considering each point in our data as a sample from a continuous, high-dimensional shape (our topology). We can think of each point as a 0-simplex. By extending out from each point some radius r, and connecting points that overlap, we can construct sets of 1-, 2-, and higher-dimensional simplices. This simplicial complex does a reasonable job of approximating the fundamental topology of the dataset, explained by the [Nerve Theorem](https://en.wikipedia.org/wiki/Nerve_of_a_covering). It turns out that the bulk of the work of representing the topology is actually being done by the 0- and 1-simplices, which constitute what's known as a **Vietoris-Rips Complex**. Most importantly, the **Vietoris-Rips Complex** is much easier to work with computationally, especially for large datasets. By considering just the 0- and 1-simplices, we’ve effectively just constructed a graph, which can be readily projected into a lower-dimensional analogue.
+We begin by considering each point in our data as a sample from a continuous, high-dimensional shape (our topology). We can think of each point as a 0-simplex. By extending out from each point some radius r, and connecting sets of points whose neighborhoods have a non-empty intersection, we can construct sets of 1-, 2-, and higher-dimensional simplices. This simplicial complex does a reasonable job of approximating the fundamental topology of the dataset, explained by the [Nerve Theorem](https://en.wikipedia.org/wiki/Nerve_of_a_covering). It turns out that there is a closely related construction, the **Vietoris-Rips Complex**, which is easier to work with computationally. Specifically, the **Vietoris-Rips Complex** is completely determined by its 0- and 1-simplices: for any set of vertices where every pair has an edge between them, the corresponding higher dimensional simplex is automatically included. Thus, the **Vietoris-Rips Complex** can be represented by a graph, which can be readily projected into a lower-dimensional analogue.
 
 Unfortunately, real-world high-dimensional data presents a problem that UMAP needs to overcome - picking the right sized radius. Too small a radius and we’ll tend towards isolated, local clusters of points. Too large, and everything becomes connected. This problem is exacerbated by the **curse of dimensionality**, where distances between points become increasingly similar in higher dimensions. UMAP solves this problem in a clever way: Rather than using a fixed radius, UMAP uses a variable radius determined for each point based on the distance to its kth nearest neighbor. Within this local radius, connectedness is then made “fuzzy” by making each connection a probability, with further points less likely to be connected. Since we don’t want any points to be completely isolated, a constraint is added that all points must be connected to at least its closest neighboring point. The final output of this process is a weighted graph, with edge weights representing the likelihood that two points are “connected” in our high-dimensional manifold.