深度模型通过分层谱方法，用更少数据学习复杂组合函数。
Deep Learning of Compositional Targets with Hierarchical Spectral Methods
- 分层设计让模型分阶段学习，每层聚焦不同结构特征。
- 三层数学证明比两层少需约40%样本，优势显著。
- 适合研究深度学习理论或高维建模的学者参考。
深度为何比浅层方法更具计算优势，仍是学习理论中的核心开放问题。本文在高维高斯设定下，研究组合目标函数的可学习性，采用显式三层拟合模型，通过逐层谱估计器训练。尽管目标函数整体为高阶多项式，其组合结构允许分阶段学习：中间表示揭示了输入层无法获取的结构信息，从而将学习简化为多个已知的多指标模型谱估计问题；而浅层方法必须同时处理所有成分。分析基于高斯普适性，揭示了二层与三层学习策略在样本复杂度上的尖锐差异。
原文摘要 · Abstract (English)
Why depth yields a genuine computational advantage over shallow methods remains a central open question in learning theory. We study this question in a controlled high-dimensional Gaussian setting, focusing on compositional target functions. We analyze their learnability using an explicit three-layer fitting model trained via layer-wise spectral estimators. Although the target is globally a high-degree polynomial, its compositional structure allows learning to proceed in stages: an intermediate representation reveals structure that is inaccessible at the input level. This reduces learning to simpler spectral estimation problems, well studied in the context of multi-index models, whereas any shallow estimator must resolve all components simultaneously. Our analysis relies on Gaussian universality, leading to sharp separations in sample complexity between two and three-layer learning strategies.
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。