  We propose a novel framework, called Markov-Lipschitz deep learning (MLDL),
to tackle geometric deterioration caused by collapse, twisting, or crossing in
vector-based neural network transformations for manifold-based representation
learning and manifold data generation. A prior constraint, called locally
isometric smoothness (LIS), is imposed across-layers and encoded into a Markov
random field (MRF)-Gibbs distribution. This leads to the best possible
solutions for local geometry preservation and robustness as measured by locally
geometric distortion and locally bi-Lipschitz continuity. Consequently, the
layer-wise vector transformations are enhanced into well-behaved,
LIS-constrained metric homeomorphisms. Extensive experiments, comparisons, and
ablation study demonstrate significant advantages of MLDL for manifold learning
and manifold data generation. MLDL is general enough to enhance any vector
transformation-based networks. The code is available at
https://github.com/westlake-cairi/Markov-Lipschitz-Deep-Learning.
