Markov-Lipschitz Deep Learning

In this paper, we propose a novel framework, called Markov-Lipschitz deep learning (MLDL), for manifold learning and data generation. A prior constraint, called locally isometric smoothness (LIS), is imposed across-layers and encoded into a Markov random field (MRF)-Gibbs distribution. Consequently, the layer-wise vector transformations are enhanced into LIS-constrained metric homeomorphisms. This leads to the best possible solutions for local geometry preservation and robustness as measured by locally geometric distortion and locally bi-Lipschitz continuity. 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. Code is available at: https://github.com/westlake-cairi/Markov-Lipschitz-Deep-Learning.