神经网络在正则化下能自动实现降维，适合处理高维回归问题。
Neural Networks Perform Sufficient Dimension Reduction
- 通过第一层权重学习数据的中心均值子空间，实现降维。
- 理论证明了神经网络估计器在统计上是一致的，结果可靠。
- 比现有方法更优，适合高维数据建模与降维任务。
本文研究神经网络与充分降维（SDR）之间的联系，证明在适当的秩正则化条件下，神经网络在回归任务中天然地执行充分降维。具体而言，第一层权重张成中心均值子空间。我们建立了基于神经网络的中心均值子空间估计器的统计一致性，凸显神经网络解决SDR挑战的适用性。数值实验进一步验证了理论结果，并突出神经网络相较于现有方法在促进充分降维方面的潜在能力。此外，我们还探讨了扩展至揭示中心子空间的可能性，拓宽了研究范围。
原文摘要 · Abstract (English)
This paper investigates the connection between neural networks and sufficient dimension reduction (SDR), demonstrating that neural networks inherently perform SDR in regression tasks under appropriate rank regularizations. Specifically, the weights in the first layer span the central mean subspace. We establish the statistical consistency of the neural network-based estimator for the central mean subspace, underscoring the suitability of neural networks in addressing SDR-related challenges. Numerical experiments further validate our theoretical findings, and highlight the underlying capability of neural networks to facilitate SDR compared to the existing methods. Additionally, we discuss an extension to unravel the central subspace, broadening the scope of our investigation.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。