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+# Overview
+"UMAP uses local manifold approximations and patches together their fuzzy
+simplicial set representations. This constructs a topological representation
+of the data.
+
+Given a low dimensional representation of the data, a similar process can be
+used to construct an equivalent topological representation.
+
+UMAP then optimizes the layout of the data representation in the low dimensional
+space, minimizing the cross-entropy between the two topological
+representations."
+
+1. Approximate a manifold on which the data is assumed to lie
+2. Construct a fuzzy simplicial set representation of the approximated manifold
+3. Construct another fuzzy simplicial set representation of a low dimensional
+manifold, `R^d`
+4. Optimize the representation
+
+## Estimating the manifold
+Assuming the data points **x** in *X* are uniformly distributed on a manifold
+*M* with respect to the Riemannian metric *g*, then a ball centered at **x_i**
+that contains exactly the *k*-nearest neighbors of **x_i** should have a fixed
+volume regardless of the choice of **x_i**.
+
+From Lemma 1 of the paper, the geodesic distance of **x_i** to its neighbors
+can be approximated by normalizing distances with respect to the distance to the
+*k*-th nearest neighbor of **x_i**.
+
+This results in a family of discrete metric spaces (one for each **x_i**) that
+can be merged into a consistent global structure by converting the metric spaces
+into fuzzy simplicial sets.
+
+## Constructing fuzzy simplicial sets
+"Each metric space in the family can be translated into a fuzzy simplicial set
+via the fuzzy singular set functor, distilling the topological information
+while still retaining metric information in the fuzzy structure. Ironing out
+the incompatibilities of the resulting family of fuzzy simplicial sets can be
+done by simply taking a (fuzzy) union across the entire family. The result is a
+single fuzzy simplicial set which captures the relevant topological and
+underlying metric structure of the manifold *M*."
+
+Lemma 1 only defines distances between the chosen point **x_i** and its *k*
+nearest neighbors; distances between other points **x_j**, **x_k** where i!=k
+and i != j are not well-defined. Therefore, the distances between such points
+in the space local to **x_i** are defined to be infinite.
+
+The manifold is also assumed to be locally connected.
+
+The above results in the fuzzy topological representation of a dataset:
+
+**Definition** Let *X* = {**x_1**, ..., **x_N**} be a dataset in `R^d`. Let
+{(*X*, d_i) | i = 1, ..., N} be a family of extended-pseudo-metric spaces with
+common carrier set X such that
+
+d_i(**x_j**, **x_k**) =
+- d_*M*(**x_j**, **x_k**) - p if i = j or i = k,
+- infinity otherwise
+
+where p is the distance to the nearest neighbor of **x_i** and d_*M* is the
+geodesic distance on the manifold *M*, either known *a priori*, or as
+approximated per Lemma 1.
+
+The **fuzzy topological representation of *X*** is
+
+- UNION i=1:N FinSing((*X*, d_i)).
+
+Dimensionality reduction can be performed by finding low dimensional
+representations that closely match the topological structure of the data.
+
+## Optimizing a low dimensional representation
+...