发表机构
CEA List, Université Paris-Saclay; LMS, École Polytechnique, Institut Polytechnique de Paris（CEA List、巴黎-萨克雷大学; LMS、巴黎综合理工学院、巴黎理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究探究自编码器编码器的作用，提出线性编码器自编码器（Lenc-AE）架构，经多数据集验证其重构质量与全非线性自编码器相当，且在简约性、可解释性上更优，明确非线性解码器是流形学习的关键。
AI 中文摘要
自编码器被广泛应用于非线性降维和流形学习。尽管多数常见实现同时依赖非线性编码器与解码器，我们仍探究编码器的特定作用，以及将其约束为线性时在不降低精度的前提下所能达到的程度。我们在四种自编码器架构上开展对比研究：标准全非线性自编码器（AE）、线性编码器自编码器（Lenc-AE）、线性解码器自编码器（Ldec-AE）以及全线性自编码器（LAE），在合成流形、计算力学数据集及含MNIST在内的真实图像数据集上进行评估。我们证明，若解码器保持非线性，施加线性编码器可保留自编码器的大部分表征能力。特别地，Lenc-AE始终优于Ldec-AE和LAE，且达到与全非线性AE相当的重构质量，同时在潜在表征的简约性与可解释性方面具备优势。这些结果表明，非线性解码器是流形学习的关键组件，而非编码器。我们针对该发现提出几何解释，明确了线性编码器足够适用的精确条件，以及暴露其局限性的特定流形构型。
英文摘要
Autoencoders are widely used for nonlinear dimensionality reduction and manifold learning. While most common implementations rely on both nonlinear encoders and decoders, we investigate the specific role of the encoder and the extent to which it can be constrained to be linear without reducing accuracy. We conduct a comparative study on four autoencoder architectures: standard fully nonlinear autoencoders (AE), linear-encoder autoencoders (Lenc-AE), linear-decoder autoencoders (Ldec-AE), and fully linear autoencoders (LAE), evaluated on synthetic manifolds, computational mechanics data sets, and real-world image data sets including MNIST. We demonstrate that imposing a linear encoder preserves most of the representational capacity of the autoencoder, provided the decoder remains nonlinear. In particular, Lenc-AE consistently outperforms both Ldec-AE and LAE, and achieves reconstruction quality comparable to fully nonlinear AE, while offering advantages in terms of parsimony and interpretability of the latent representation. These results suggest that the nonlinear decoder is the critical component for manifold learning, rather than the encoder. A geometric interpretation of this finding is developed, which identifies the precise conditions under which a linear encoder is sufficient, and the specific manifold configurations that expose its limitations.
Comments19 pages, 10 figures, 3 tables. Preprint also available on HAL: https://hal.science/hal-05646575
