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Sublinearly Structured Deep Neural Networks Achieve Feature Learning Consistency for Compositional Functions

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Over the past decade, deep neural networks (DNNs) have achieved remarkable success on complex machine-learning tasks, yet the theoretical foundations of their performance remain incomplete. From a statistical viewpoint, a natural question is: can DNNs attain feature-learning and prediction consistency comparable to that of classical models?

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arXiv Machine Learning
Sep 25

Pointwise Generalization in Deep Neural Networks

The paper introduces a pointwise generalization theory for fully connected deep neural networks, using a pointwise Riemannian Dimension derived from eigenvalues of learned feature representations across layers. This framework provides hypothesis-dependent, representation-aware generalization bounds that are significantly tighter than traditional size- or norm-based approaches, both theoretically and experimentally. The authors analytically identify structural properties that explain deep networks’ tractability and empirically show that the pointwise Riemannian Dimension captures feature compression, over‑parameterization effects, and optimizer bias.

By Shaojie Li, Yunbei Xu
arXiv Machine Learning
Sep 7

Deep Learning as Neural Low-Degree Filtering: A Spectral Theory of Hierarchical Feature Learning

The paper introduces Neural Low-Degree Filtering (Neural LoFi), a stylized limit of gradient-based training that turns hierarchical feature learning into an explicit iterative spectral procedure. In this framework, each layer independently selects directions with maximal low-degree correlation to the label, providing a tractable surrogate for deep learning and a kernel-space interpretation. Experiments on fully connected and convolutional networks show that Neural LoFi outperforms lazy random-feature baselines, recovers meaningful structured filters, and aligns with early gradient-descent feature discovery on real datasets.

By Yatin Dandi, Matteo Vilucchio, Luca Arnaboldi, Hugo Tabanelli, Florent Krzakala