Figure 2:

Embeddings of a sphere in ${ℝ}^3$ into ${ℝ}^2$. The top and bottom row contain the same plots colored by the height and the azimuthal angle of the sphere (0 − 2π), respectively. LDLE automatically colors the boundary so that the points on the boundary which are adjacent on the sphere have the same color. The arrows are manually drawn to help the reader identify the two pieces of the boundary which are to be stitched together to recover the original sphere. LTSA, UMAP and Laplacian eigenmaps squeezed the sphere into different viewpoints of ${ℝ}^2$ (side or top view of the sphere). t-SNE also tore apart the sphere but the embedding lacks interpretability as it is “unaware” of the boundary.