Choosing an algorithm

Seven of them, and the honest summary is that three cover most work. Here is how to pick without running all seven.

Start here

Your data	Start with
Cell types you expect to separate	UMAP
A differentiation trajectory, or anything continuous	PHATE
You care about the arrangement of clusters, not just the clusters	PaCMAP
You want the tightest possible local structure and nothing else	TSNE
Relative density in the plot should mean something	DensMAP
You want the spectral embedding itself, not a layout of it	DiffusionMaps

What each one is actually doing

UMAP builds a fuzzy simplicial set from the kNN graph and optimises a cross-entropy against it. The spectral initialisation is where most of its global structure comes from, and it is the default here for that reason. Fast, well understood, and the sensible first thing to run.

densMAP is UMAP plus a term correlating the embedding's local radii with the original space's. Plain UMAP is free to stretch a sparse region and squash a dense one, so relative density in a UMAP plot means nothing at all. If you are about to say "this cluster is tighter, so those cells are more homogeneous", you want densMAP. The term only switches on for the last 30% of the run, so it corrects a settled embedding rather than fighting the layout.

t-SNE optimises a KL divergence between neighbourhood distributions. It is the best of these at local structure and the worst at global: distances between t-SNE clusters carry very little information, and cluster sizes carry none. The learning rate here defaults to max(N / 12, 200) rather than a fixed 200, so it does not need retuning as the dataset grows. Two-dimensional only.

den-SNE is the same density correction applied to t-SNE. The default weight is much smaller than densMAP's because the gradients are on a different scale, not because the effect is meant to be weaker.

PHATE powers a diffusion operator to time t, takes the potential distance between the resulting distributions, and lays those out with MDS. That is a completely different object from a kNN graph layout, and it is why trajectories and branch points survive PHATE that t-SNE tears into blobs. If your biology is continuous, this is the one.

PaCMAP uses three kinds of pair: near pairs pull, further pairs push, and mid-near pairs hold the global arrangement together while their weight decays over the first two phases. That decay is what lets it keep global structure without leaning on a spectral initialisation. PCA init is close to required; random init throws away most of the advantage.

Diffusion maps is the spectral embedding PHATE builds on. Reach for it when you want the diffusion coordinates themselves rather than a 2-D picture of them. The knob worth understanding is alpha: 0 gives the normalised graph Laplacian, 0.5 the Fokker-Planck operator, 1 the Laplace-Beltrami operator, which is the one that removes the influence of sampling density.

Cost

Rough ordering on the same data, dominated by different things:

• TSNE and DensNE pay for the Barnes-Hut tree every epoch.
• UMAP, DensMAP and PaCMAP pay for the neighbour search and then a cheap per-edge update. The parallel Adam optimiser is the default.
• PHATE and DiffusionMaps pay for an eigendecomposition. Set n_landmarks above roughly 50k points or you will wait.

For all of them, past a few hundred thousand points the neighbour search dominates. Switch ann to "nndescent" or "hnsw", or build the graph once with knn_graph and hand it to each estimator.

What none of them do

Project new points. There is no transform. Embedding new data means refitting on the combined set, which moves the existing coordinates too, and any method that appears to avoid that is doing something to the new points that is not the operation the old ones went through.