UMAP
- Uniform Manifold Approximation and Projection
Best for: Fast nonlinear visualization/embedding Aliases: Uniform Manifold Approximation and Projection
How it works
$$C=\sum_{i\ne j}\Big[w_{ij}\log\tfrac{w_{ij}}{v_{ij}}+(1-w_{ij})\log\tfrac{1-w_{ij}}{1-v_{ij}}\Big]$$Builds a fuzzy $k$-nearest-neighbour graph in the input space with weights $w_{ij}=\exp\!\bigl(-(d(x_i,x_j)-\rho_i)/\sigma_i\bigr)$, where $\rho_i$ is the distance to the nearest neighbour and $\sigma_i$ normalises to a fixed local radius. A second fuzzy graph $v_{ij}$ is defined on the low-dimensional points via a smooth kernel $1+a\|y_i-y_j\|^{2b}$. The layout is optimised by minimising the cross-entropy $C=\sum_{i\ne j}\bigl[w_{ij}\log\frac{w_{ij}}{v_{ij}}+(1-w_{ij})\log\frac{1-w_{ij}}{1-v_{ij}}\bigr]$ with force-directed gradient descent, which preserves both local and broader topological structure and scales far better than t-SNE.
When to use
Fast nonlinear embedding for visualization and as a general-purpose feature reducer at scale.
Watch out
Cluster sizes and distances are not directly meaningful; parameter-sensitive; easy to over-interpret.
Common fields
Single-cell biology · text embeddings · image embeddings