用生成模型解决非线性降维中低维结构识别难题
On Conditional Stochastic Interpolation for Generative Nonlinear Sufficient Dimension Reduction
- 基于生成模型构建新方法GenSDR，可完全恢复中心σ域信息
- 在样本层面证明了估计器的分布一致性，理论保障强
- 拓展至非欧响应场景，适用范围广，适合复杂数据分析
在非线性充分降维（SDR）中，识别低维充分结构一直是基础且困难的问题。现有方法大多缺乏对低维结构识别的完备性理论保证，无论在总体层面还是样本层面。本文提出一种新方法——生成充分降维（GenSDR），利用现代生成模型解决该问题。我们证明GenSDR可在总体和样本层面完全恢复中心σ-域所含信息。尤其在样本层面，从条件分布角度建立了GenSDR估计器的一致性性质，充分利用深度生成模型的分布学习能力。此外，通过引入集成技术，将GenSDR扩展至非欧几里得响应场景，显著提升其适用性。大量数值实验表明GenSDR具有出色的实证性能，展现出解决多种复杂真实任务的强大潜力。
原文摘要 · Abstract (English)
Identifying low-dimensional sufficient structures in nonlinear sufficient dimension reduction (SDR) has long been a fundamental yet challenging problem. Most existing methods lack theoretical guarantees of exhaustiveness in identifying lower dimensional structures, either at the population level or at the sample level. We tackle this issue by proposing a new method, generative sufficient dimension reduction (GenSDR), which leverages modern generative models. We show that GenSDR is able to fully recover the information contained in the central $σ$-field at both the population and sample levels. In particular, at the sample level, we establish a consistency property for the GenSDR estimator from the perspective of conditional distributions, capitalizing on the distributional learning capabilities of deep generative models. Moreover, by incorporating an ensemble technique, we extend GenSDR to accommodate scenarios with non-Euclidean responses, thereby substantially broadening its applicability. Extensive numerical results demonstrate the outstanding empirical performance of GenSDR and highlight its strong potential for addressing a wide range of complex, real-world tasks.
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