Markov-Lipschitz Deep Learning

TL;DR

This paper introduces Markov-Lipschitz deep learning (MLDL), a framework that preserves local geometry in neural networks for manifold learning and data generation by imposing a locally isometric smoothness constraint encoded via a Markov random field.

Contribution

The paper proposes a novel MLDL framework that enforces local geometric preservation in neural networks using a Markov random field-based constraint, improving robustness and manifold learning capabilities.

Findings

MLDL enhances manifold learning and data generation.

It maintains local geometric distortion and bi-Lipschitz continuity.

Experiments show significant advantages over existing methods.

Abstract

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.