# Markov-Lipschitz Deep Learning

arXiv: https://arxiv.org/abs/2006.08256 · HTML: https://mlsanity.com/paper/2006.08256/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Jun 2020.

## Built on by (most cited first, 5 of 5)

- [CoNeRF: Controllable Neural Radiance Fields](https://mlsanity.com/paper/2112.01983/) (2021)
- [Co-learning: Learning from Noisy Labels with Self-supervision](https://mlsanity.com/paper/2108.04063/) (2021)
- [Geometric Autoencoders -- What You See is What You Decode](https://mlsanity.com/paper/2306.17638/) (2023)
- [Multi-Agent Semi-Siamese Training for Long-tail and Shallow Face Learning](https://mlsanity.com/paper/2105.04113/) (2021)
- [Deep Multi-Manifold Transformation Based Multivariate Time Series Fault Detection](https://mlsanity.com/paper/2405.16258/) (2024)

## Abstract (arXiv)

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.