Fig. 4.

scRNAseq datasets have localized regions of nonzero scalar curvature. (A) Scalar curvatures were computed for a scRNAseq dataset with $N\approx 10^4$ PBMCs collected from a healthy human donor. The ambient ($n$) and manifold ($d$) dimensions were specified to be eight and three, respectively, and variable neighborhood sizes were chosen by setting ${\sigma }_h$ (SI Appendix, Supporting Methods, section F.6). The scalar curvatures are shown here overlaid onto UMAP coordinates, after smoothing the values over $k=250$ nearest neighbors in the ambient space. (B) Scatter plot of (unsmoothed) scalar curvatures, $S$, and associated SEs, ${\sigma }_S$, for each data point in the PBMC dataset. Points enclosed by the red lines reported a 95% CI ($S\pm 2{\sigma }_S$) including zero. (C) As in A, but with scalar curvatures computed by using a fixed neighborhood size, $r$, for all data points. The value of $r$ was set to be the 25th, 50th, and 75th percentile values (left to right) of the neighborhood sizes used in A (SI Appendix, Fig. S4E). Points for which a neighborhood of size $r$ does not include enough neighbors for regression are not shown. (D–F) As in A–C for a mouse gastrulation dataset with $N\approx 1.2\times 10^5$, $d=3$, and $n=11$. (G–I) As in A–C for a mouse brain dataset with $N\approx 1.3\times 10^6$, $d=5$, and $n=9$, plotted on t-SNE coordinates.