UMAP

Describing Properties of UMAP

• Similarities are computed from distances using a kernel different from the t-SNE kernel

  • i.e. it is not Gaussian
  • Similar to t-SNE, it decays exponentially and has adaptive width

• Similarities are not normalized to sum up to $1$

  • However, the similarity values still end up being normalized to sum up to a constant value
• Similarity values are symmetrized

Defining Assumptions for UMAP

1. The data is uniformly distributed on a Riemannian manifold
2. The Riemannian metric is locally constant (or can be approximated as such)
3. The manifold is locally connected

Comparing UMAP with t-SNE

• UMAP and t-SNE are both useful for visualizations

• UMAP is faster compared to t-SNE

  • UMAP can complete embedding in less than a minute on 70k samples with 784 features
  • t-SNE completes embedding in around 45 minutes on 70k samples with 785 features

• Similar results with random and informative initialization:

  • With random and informative initialization, t-SNE and UMAP both are able to preserve local structures
  • With random initialization, t-SNE and UMAP both struggle to preserve global structures
  • With informative random initialization, t-SNE and UMAP both are able to preserve global structures

References

• Documentation for Using UMAP for Clustering
• Performance Difference between t-SNE and UMAP
• Paper about UMAP Properties
• Paper about Random Initialization in UMAP
• Post about Properties of UMAP
• Illustrations for UMAP and t-SNE