  We propose a family of curvature-based regularization terms for deep
generative model learning. Explicit coordinate-invariant formulas for both
intrinsic and extrinsic curvature measures are derived for the case of
arbitrary data manifolds embedded in higher-dimensional Euclidean space.
Because computing the curvature is a highly computation-intensive process
involving the evaluation of second-order derivatives, efficient formulas are
derived for approximately evaluating intrinsic and extrinsic curvatures.
Comparative studies are conducted that compare the relative efficacy of
intrinsic versus extrinsic curvature-based regularization measures, as well as
performance comparisons against existing autoencoder training methods.
Experiments involving noisy motion capture data confirm that curvature-based
methods outperform existing autoencoder regularization methods, with intrinsic
curvature measures slightly more effective than extrinsic curvature measures.
