Title: Parametric UMAP: learning embeddings with deep neural networks for representation and semi-supervised learning. Abstract: We propose Parametric  UMAP, a parametric variation of the UMAP (Uniform Manifold Approximation and Projection) algorithm. 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 replace the second step of UMAP with a deep neural network that learns a parametric relationship between data and embedding. We demonstrate that our method performs similarly to its non-parametric counterpart while conferring the benefit of a learned parametric mapping (e.g. fast online embeddings for new data). We then show that UMAP loss can be extended to arbitrary deep learning applications, for example constraining the latent distribution of autoencoders, and improving classifier accuracy for semi-supervised learning by capturing structure in unlabeled data.