(A) Comparison of Diffusion Maps (blue) and PHATE (orange) embeddings on data (black) from a half circle (left, n = 100 data points) and a full circle (right, n = 100 data points). Both the data and the embeddings have been centered about the mean and rescaled by the max Euclidean norm. For the full circle, both embeddings are identical (up to centering & scaling) to the original circle. However, for the half circle, the Diffusion Maps embedding (blue) suffers from instabilities that generate significantly higher densities near the two end points. The PHATE embedding (orange) does not exhibit these instabilities. (B) The α-decaying kernel \(K_{\alpha ,\sigma }\left( x \right) = \exp \left( { - \left( {\frac{{\left| x \right|}}{\sigma }} \right)^\alpha } \right)\) as a function of x for different values of α and σ = 1 (left) and σ = 4 (right). As α increases, \(K_{\alpha ,\sigma }\left( x \right)\) becomes more constant for \(x \in ( - \sigma ,\sigma )\) and the tails of the kernel become lighter (i.e., decay to zero more quickly) for \(x

otin ( - \sigma ,\sigma ).\) (C) Demonstration of the effect of the scale t on the PHATE visualization for the artificial tree data (n = 1440 60-dimensional data points) colored by branch. The first column shows the VNE H(t) (see Eq. 5) of the diffusion affinities as a function of the time scale t. The other columns give the PHATE visualization with different values of t. The red dots in the first column indicate the values of t chosen for the plots. The red dot surrounded by a black box indicate the chosen value of t for the visualization in Figure 1B of the artificial tree data. Values of t that are too low can give noisy visualizations while very high values of t can result in a loss of information in the visualization. (D) Visualization of scRNA-seq data measured from mouse retinal bipolar neurons (Shekhar et al., Cell, vol. 166, no. 5, pp. 1308-1323, 2016), using different informational distances defined via the parameter γ. n = 27499 single cells.