  UMAP is a non-parametric graph-based dimensionality reduction algorithm using
applied Riemannian geometry and algebraic topology to find low-dimensional
embeddings of structured data. The UMAP algorithm consists of two steps: (1)
Compute a graphical representation of a dataset (fuzzy simplicial complex), and
(2) Through stochastic gradient descent, optimize a low-dimensional embedding
of the graph. Here, we extend the second step of UMAP to a parametric
optimization over neural network weights, learning a parametric relationship
between data and embedding. We first demonstrate that Parametric UMAP performs
comparably to its non-parametric counterpart while conferring the benefit of a
learned parametric mapping (e.g. fast online embeddings for new data). We then
explore UMAP as a regularization, constraining the latent distribution of
autoencoders, parametrically varying global structure preservation, and
improving classifier accuracy for semi-supervised learning by capturing
structure in unlabeled data. Google Colab walkthrough:
https://colab.research.google.com/drive/1WkXVZ5pnMrm17m0YgmtoNjM_XHdnE5Vp?usp=sharing
