Bi-Lipschitz Autoencoder With Injectivity Guarantee

2026-04-08

AI Summary

This work addresses the challenges in autoencoder-based dimensionality reduction, where non-injective encoders often lead to convergence difficulties and distorted latent representations. Existing regularization techniques are typically overly restrictive and lack robustness. To overcome these limitations, the authors propose the Bi-Lipschitz Autoencoder (BLAE), which integrates injectivity-promoting regularization based on a separation criterion with a bi-Lipschitz relaxation mechanism. This approach theoretically ensures approximate invertibility of the mapping while effectively preserving the geometric structure of the underlying data manifold. By eliminating ill-conditioned local minima, BLAE significantly enhances robustness to sparse sampling and distributional shifts. Experimental results across multiple datasets demonstrate that BLAE outperforms current methods in accurately recovering the intrinsic manifold geometry.

Abstract

Autoencoders are widely used for dimensionality reduction, based on the assumption that high-dimensional data lies on low-dimensional manifolds. Regularized autoencoders aim to preserve manifold geometry during dimensionality reduction, but existing approaches often suffer from non-injective mappings and overly rigid constraints that limit their effectiveness and robustness. In this work, we identify encoder non-injectivity as a core bottleneck that leads to poor convergence and distorted latent representations. To ensure robustness across data distributions, we formalize the concept of admissible regularization and provide sufficient conditions for its satisfaction. In this work, we propose the Bi-Lipschitz Autoencoder (BLAE), which introduces two key innovations: (1) an injective regularization scheme based on a separation criterion to eliminate pathological local minima, and (2) a bi-Lipschitz relaxation that preserves geometry and exhibits robustness to data distribution drift. Empirical results on diverse datasets show that BLAE consistently outperforms existing methods in preserving manifold structure while remaining resilient to sampling sparsity and distribution shifts. Code is available at https://github.com/qipengz/BLAE.

Problem

• autoencoder
• injectivity
• manifold geometry
• dimensionality reduction
• regularization

Innovation

• Bi-Lipschitz
• injectivity
• manifold geometry
• regularization
• autoencoder