Unsupervised learning of coordinate transformation (e.g. Cartesian to Spherical)

Question

Is it possible to learn the Cartesian to spherical coordinate conversion in an unsupervised manner?

Consider the case of data points sampled from the surface of a sphere. This data lives on the spherical manifold. Assume then that this fact was not known a priori. Would there be a way to learn the coordinate transform from Cartesian to spherical coordinates (effectively identifying the topology) for downstream tasks?

I have tried using an autoencoder, but the true topology is not clear from the learned latent representation.

More context: Let's say I have a scalar function defined on the unit sphere with data in cartesian coordinates. Assume the closed form expression has a nice parsimonious form in terms of theta and phi spherical coordinates. It would be interesting from the perspective of symbolic regression and interpretability if it was possible to discover the best coordinate transform prior to symbolic regression of the function with the data expressed in terms of learned coordinate transform.

Answer 1

If you train an autoencoder (AE) or Isomap/LLE/ICA manifold learning model on data sampled from the surface of a sphere, the network can learn an embedding that is topologically equivalent (diffeomorphic) to the sphere. However, the latent coordinates you obtain will be unique only up to an arbitrary smooth invertible transformation. There isn’t a canonical coordinate system for a manifold like the sphere since any two smooth invertible diffeomorphic coordinate systems are equally valid. Methods like AE, Isomap, and LLE are designed to preserve local intrinsic geometry, so their objective functions don’t force the model to choose one specific global external coordinate. This is a common identifiability theoretical issue in disentangled feature learning without explicit assumption.

For your goal you would likely need to impose additional structure as inductive biases during training. For example, techniques like using a hyperspherical latent space encourage the network to represent the data in a way that more directly aligns with the standard spherical geometry. You can further refer Davidson et al. (2018) "Hyperspherical Variational Auto-Encoders".

But although the default choice of a Gaussian distribution for both the prior and posterior represents a mathematically convenient distribution often leading to competitive results, we show that this parameterization fails to model data with a latent hyperspherical structure. To address this issue we propose using a von Mises-Fisher (vMF) distribution instead, leading to a hyperspherical latent space. Through a series of experiments we show how such a hyperspherical VAE, or S-VAE, is more suitable for capturing data with a hyperspherical latent structure, while outperforming a normal, N -VAE, in low dimensions on other data types.