  We propose a deep-learning based method for obtaining standardized data
coordinates from scientific measurements.Data observations are modeled as
samples from an unknown, non-linear deformation of an underlying Riemannian
manifold, which is parametrized by a few normalized latent variables. By
leveraging a repeated measurement sampling strategy, we present a method for
learning an embedding in $\mathbb{R}^d$ that is isometric to the latent
variables of the manifold. These data coordinates, being invariant under smooth
changes of variables, enable matching between different instrumental
observations of the same phenomenon. Our embedding is obtained using a LOcal
Conformal Autoencoder (LOCA), an algorithm that constructs an embedding to
rectify deformations by using a local z-scoring procedure while preserving
relevant geometric information. We demonstrate the isometric embedding
properties of LOCA on various model settings and observe that it exhibits
promising interpolation and extrapolation capabilities. Finally, we apply LOCA
to single-site Wi-Fi localization data, and to $3$-dimensional curved surface
estimation based on a $2$-dimensional projection.
