Fig. 3: Demonstration of how data in an observable space \({\cal{X}}\) can be concentrated along a manifold (here just a curve).

Panel a shows how the curve is simply an image of a latent space \({\cal{Z}}\) projected through a generator function \(x_i = g_\theta (z_i)\). Panel b demonstrates noisy data along the circular curve illustrated in (a). Measurement of the Euclidean (straight-line) distance between points A and B implies traversal across a region of \({\cal{X}}\) in which no data lie. The correct approach is instead to measure distance with respect to the data’s manifold of support.