UMAP is a prominent manifold-learning method especially popular in the analysis and visualization of single-cell, population-genetic, and other biological data. Here, we consider the performance of the first-order approximation of (the true parametric) UMAP objective on our example brain-imaging data. We term this approximation m-umap, and note that it reduces to a generalized modularity.
# Install abct and download abct_utils.py
base = "https://github.com/mikarubi/abct/raw/refs/heads/main"
!wget --no-clobber {base}/docs-code/examples/abct_utils.py
%pip install --quiet abct nilearn
# Import modules
import abct
import numpy as np
from abct_utils import C, ordc, fig_scatter3, fig_scatter, fig_surf, fig_imshowFile ‘abct_utils.py’ already there; not retrieving.
Note: you may need to restart the kernel to use updated packages.UMAP and m-umap ultimately seek to approximate symmetric 𝜅-nearest-neighbor representations. We first visualize the structure of nearest-neighbor correlation networks in our example data.
# Correlation nearest-neighbor network
Cn = abct.kneighbor(C, "nearest", 50).toarray()
fig_imshow(Cn[np.ix_(ordc, ordc)],
"Correlation nearest-neighbor network",
"viridis").show()Simple use of m-umap can lead to runaway solutions. Here, we check this outcome by embedding m-umap solutions on (k-dimensional) spheres. These embeddings have an additional nice property of reducing m-umap with binary constraints to the standard modularity.
We developed a simple algorithm to optimize m-umap. First, we optimized the binary m-umap via modularity maximization of the symmetric 𝜅-nearest-neighbor network. Next, we used the resulting module-indicator matrix to initialize the continuous embedding. Finally, we optimized the continuous embedding directly on the sphere. We now use this algorithm to detect m-umap embeddings.
np.random.seed(1)
U3, M, _ = abct.mumap(C, kappa=50, learnrate=0.01, verbose=False)We now show the three-dimensional m-umap embeddings. Each point represents a node, and colors represent module affiliations (aka binary m-umap embeddings). We also use a Mercator (classic-map) projection to project these embeddings onto a plane.
X3, Y3, Z3 = zip(*U3)
fig = fig_scatter3(X3, Y3, Z3, f"m-umap embedding on a sphere")
fig.update_traces(marker=dict(color=M+1), marker_showscale=True)
fig.show()
U2 = abct.muma.projection(U3)
X2, Y2 = zip(*U2)
fig = fig_scatter(X2, Y2, "", "", f"m-umap projection on a plane")
fig.update_traces(marker=dict(color=M+1), marker_showscale=True)
fig.show()Finally, we show the maps of the individual binary m-umap embeddings.
for i in range(M.max()+1):
fig_surf(1.0 * (M == i), f"Module {i+1}", "viridis")
