For the Distributional Principal Autoencoder (DPA), we prove an exact identity linking the geometry of the learned encoding to the score of the data distribution, and show that any latent coordinates beyond the data manifold dimension become completely uninformative. This means that the DPA learns nonlinear manifolds shaped locally by the data density, with a clear, testable dimensionality criterion — conditional independence, giving it a natural nonlinear-PCA interpretation.
