  A fundamental task in data exploration is to extract simplified low
dimensional representations that capture intrinsic geometry in data, especially
for faithfully visualizing data in two or three dimensions. Common approaches
to this task use kernel methods for manifold learning. However, these methods
typically only provide an embedding of fixed input data and cannot extend to
new data points. Autoencoders have also recently become popular for
representation learning. But while they naturally compute feature extractors
that are both extendable to new data and invertible (i.e., reconstructing
original features from latent representation), they have limited capabilities
to follow global intrinsic geometry compared to kernel-based manifold learning.
We present a new method for integrating both approaches by incorporating a
geometric regularization term in the bottleneck of the autoencoder. Our
regularization, based on the diffusion potential distances from the
recently-proposed PHATE visualization method, encourages the learned latent
representation to follow intrinsic data geometry, similar to manifold learning
algorithms, while still enabling faithful extension to new data and
reconstruction of data in the original feature space from latent coordinates.
We compare our approach with leading kernel methods and autoencoder models for
manifold learning to provide qualitative and quantitative evidence of our
advantages in preserving intrinsic structure, out of sample extension, and
reconstruction. Our method is easily implemented for big-data applications,
whereas other methods are limited in this regard.
