UMAP is a nonparametric 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) computing a graphical representation of a data set (fuzzy simplicial complex) and (2) through stochastic gradient descent, optimizing 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 nonparametric 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 semisupervised learning by capturing structure in unlabeled data.¹

UMAP 是一种基于非参数图的降维算法，它使用应用的黎曼几何和代数拓扑来查找结构化数据的低维嵌入。UMAP 算法包括两个步骤：(1) 计算数据集的图形表示（模糊单纯复形）和（2）通过随机梯度下降，优化图的低维嵌入。在这里，我们将 UMAP 的第二步扩展到对神经网络权重的参数优化，学习数据和嵌入之间的参数关系。我们首先证明参数 UMAP 的性能与其非参数对应物相当，同时赋予学习参数映射的好处（例如，新数据的快速在线嵌入）。然后我们探索 UMAP 作为正则化，¹