# Bi-Lipschitz Autoencoder With Injectivity Guarantee

arXiv: https://arxiv.org/abs/2604.06701 · HTML: https://mlsanity.com/paper/2604.06701/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 31 references that are in the map.
- First posted on arXiv 8 Apr 2026.

## Builds on (most cited first, 13 of 13)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Generative Moment Matching Networks](https://mlsanity.com/paper/1502.02761/) (2015)
- [Flow Matching in Latent Space](https://mlsanity.com/paper/2307.08698/) (2023)
- [Topological Autoencoders](https://mlsanity.com/paper/1906.00722/) (2019)
- [Parametric UMAP embeddings for representation and semi-supervised learning](https://mlsanity.com/paper/2009.12981/) (2020)
- [Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connections](https://mlsanity.com/paper/2404.02954/) (2024)
- [Learning Flat Latent Manifolds with VAEs](https://mlsanity.com/paper/2002.04881/) (2020)
- [Geometric Autoencoders -- What You See is What You Decode](https://mlsanity.com/paper/2306.17638/) (2023)
- [Heavy-tailed kernels reveal a finer cluster structure in t-SNE visualisations](https://mlsanity.com/paper/1902.05804/) (2019)
- [Diffusion Nets](https://mlsanity.com/paper/1506.07840/) (2015)
- [Isometric Autoencoders](https://mlsanity.com/paper/2006.09289/) (2020)
- [Tangent Space and Dimension Estimation with the Wasserstein Distance](https://mlsanity.com/paper/2110.06357/) (2021)
- [Multi-Scale Geometric Autoencoder](https://mlsanity.com/paper/2509.24168/) (2025)

## Built on by (most cited first, 1 of 1)

- [Sharp Reconstruction Bounds for Autoencoders Using the Same Forward Map](https://mlsanity.com/paper/2609.20333/) (2026)

## Abstract (arXiv)

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.