UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction

Bringing UMAP Closer to the Speed of Light with GPU Acceleration

The Uniform Manifold Approximation and Projection (UMAP) algorithm has become widely popular for its ease of use, quality of results, and support for exploratory, unsupervised, supervised, and semi-supervised learning. While many algorithms can be ported to a GPU in a simple and direct fashion, such efforts have resulted in inefficient and inaccurate versions of UMAP. We show a number of techniques that can be used to make a faster and more faithful GPU version of UMAP, and obtain speedups of up to 100x in practice. Many of these design choices/lessons are general purpose and may inform the co…

UMAP及其他降维方法的比较研究

A Comparative Study of UMAP and Other Dimensionality Reduction Methods

本文通过综合比较分析，评估了UMAP及其监督扩展方法与其他降维方法在回归和分类任务中的性能，揭示了监督UMAP在回归任务中整合响应信息的局限性。

Understanding How Dimension Reduction Tools Work: An Empirical Approach to Deciphering t-SNE, UMAP, TriMAP, and PaCMAP for Data Visualization

Dimension reduction (DR) techniques such as t-SNE, UMAP, and TriMAP have demonstrated impressive visualization performance on many real world datasets. One tension that has always faced these methods is the trade-off between preservation of global structure and preservation of local structure: these methods can either handle one or the other, but not both. In this work, our main goal is to understand what aspects of DR methods are important for preserving both local and global structure: it is difficult to design a better method without a true understanding of the choices we make in our algori…

A dynamical systems based framework for dimension reduction

We propose a novel framework for learning a low-dimensional representation of data based on nonlinear dynamical systems, which we call dynamical dimension reduction (DDR). In the DDR model, each point is evolved via a nonlinear flow towards a lower-dimensional subspace; the projection onto the subspace gives the low-dimensional embedding. Training the model involves identifying the nonlinear flow and the subspace. Following the equation discovery method, we represent the vector field that defines the flow using a linear combination of dictionary elements, where each element is a pre-specified …

Projection-Free Bandit Convex Optimization

In this paper, we propose the first computationally efficient projection-free algorithm for bandit convex optimization (BCO). We show that our algorithm achieves a sublinear regret of $O(nT^{4/5})$ (where $T$ is the horizon and $n$ is the dimension) for any bounded convex functions with uniformly bounded gradients. We also evaluate the performance of our algorithm against baselines on both synthetic and real data sets for quadratic programming, portfolio selection and matrix completion problems.