This is a broad category of work. For now, all I'll put here is our "Mahalanobis Autoencoder" project, which is one approach for doing something like Mahalanobis-distance diffusion maps in a neural network.
This was published in PNAS as "Local conformal autoencoder for standardized data coordinates". My main contribution was an application to localization from WIFI signal strengths (show below). This was done in simulation, but, because of the way the embedding is constructed specifically to keep locally accurate metric information ("a one-meter circle remains a one-meter circle, and not an ellipse"), it should be robust to local variation in the propagation of the signal strength in real uses.
We propose a local conformal autoencoder (LOCA) for standardized data coordinates. LOCA is a deep learning-based method for obtaining standardized data coordinates from scientific measurements. Data observations are modeled as samples from an unknown, nonlinear deformation of an underlying Riemannian manifold, which is parametrized by a few normalized, latent variables. We assume a repeated measurement sampling strategy, common in scientific measurements, and present a method for learning an embedding in Rd that is isometric to the latent variables of the manifold. The coordinates recovered by our method are invariant to diffeomorphisms of the manifold, making it possible to match between different instrumental observations of the same phenomenon. Our embedding is obtained using LOCA, which is an algorithm that learns to rectify deformations by using a local z-scoring procedure, while preserving relevant geometric information. We demonstrate the isometric embedding properties of LOCA in various model settings and observe that it exhibits promising interpolation and extrapolation capabilities, superior to the current state of the art. Finally, we demonstrate LOCAs efficacy in single-site Wi-Fi localization data and for the reconstruction of three-dimensional curved surfaces from two-dimensional projections.