Homogeneous Autoencoders: Magnitude-Preserving Representation Learning

Standard nonlinear dimension-reduction methods do not in general guarantee preservation of observation magnitude, limiting their use in magnitude-sensitive tasks such as extreme-value analysis. We introduce the Homogeneous Autoencoder (HAE), whose encoder preserves a prescribed magnitude exactly while learning a map from ambient to latent directions. The encoder is positively homogeneous, so rescaling an observation rescales its latent point by the same factor. The decoder preserves the same magnitude while accommodating curved geometry. We identify conditions for homogeneous embeddings and provide a sufficient latent dimension bound for existence. For regularly varying (heavy-tailed) data, we characterise how encoding and decoding transform the limiting distribution of extremes and show that the radial tail index is unchanged. Common threshold-based models of extremes use approximate independence of magnitude and direction to extrapolate from moderate to larger events. HAE preserves this independence whenever it holds in the data, and we prove that, among magnitude-preserving encoders, homogeneity above the threshold is necessary and sufficient to preserve it for every input distribution. Experiments on a synthetic heavy-tailed curved surface and ERA5 precipitation fields show that HAE preserves latent magnitude exactly at reconstruction accuracy comparable to parameter-matched autoencoders, with the resulting latent representation supporting more accurate generation of held-out exceedances than a standard autoencoder.