在轻微模型偏差下，证明非线性表示学习可近似识别
Robustness of Nonlinear Representation Learning
- 基于局部等距性质，利用刚性理论实现表示近似唯一
- 在小扰动下，可近似恢复独立成分分析中的混合矩阵与分量
- 为真实数据的无监督表示学习提供理论支撑，适合研究鲁棒性者
我们研究了在轻微模型误设情况下的无监督表示学习问题，正式提出了非线性表示学习鲁棒性的研究框架。重点关注混合映射在合适距离下接近局部等距的情形，基于现有刚性结果，证明混合结构可被唯一确定至线性变换和小误差范围内。进一步分析独立成分分析（ICA）中观测值 $x=f(s)=As+h(s)$，其中 $A$ 为可逆混合矩阵，$h$ 为小扰动项，证明可近似恢复矩阵 $A$ 和独立成分。两项结果共同表明，在几乎等距混合条件下，非线性 ICA 具有近似可辨识性。这些成果推进了对现实数据无监督表示学习可辨识性的理解，突破了传统严格模型类的限制。
原文摘要 · Abstract (English)
We study the problem of unsupervised representation learning in slightly misspecified settings, and thus formalize the study of robustness of nonlinear representation learning. We focus on the case where the mixing is close to a local isometry in a suitable distance and show based on existing rigidity results that the mixing can be identified up to linear transformations and small errors. In a second step, we investigate Independent Component Analysis (ICA) with observations generated according to $x=f(s)=As+h(s)$ where $A$ is an invertible mixing matrix and $h$ a small perturbation. We show that we can approximately recover the matrix $A$ and the independent components. Together, these two results show approximate identifiability of nonlinear ICA with almost isometric mixing functions. Those results are a step towards identifiability results for unsupervised representation learning for real-world data that do not follow restrictive model classes.
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