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Deep Networks are Reproducing Kernel Chains

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arxiv 2501.03697 v1 pith:FP3SXFMR submitted 2025-01-07 cs.LG math.FAstat.ML

Tjeerd Jan Heeringa , Len Spek , Christoph Brune This is my paper
classification cs.LGmath.FAstat.ML
keywords neuraldeepfunctionnetworksrkbscrkbskernelnetwork
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abstract

Identifying an appropriate function space for deep neural networks remains a key open question. While shallow neural networks are naturally associated with Reproducing Kernel Banach Spaces (RKBS), deep networks present unique challenges. In this work, we extend RKBS to chain RKBS (cRKBS), a new framework that composes kernels rather than functions, preserving the desirable properties of RKBS. We prove that any deep neural network function is a neural cRKBS function, and conversely, any neural cRKBS function defined on a finite dataset corresponds to a deep neural network. This approach provides a sparse solution to the empirical risk minimization problem, requiring no more than $N$ neurons per layer, where $N$ is the number of data points.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Deep Neural Variation Spaces: A Unifying Perspective on Depth and Complexity

    stat.ML 2026-07 accept novelty 7.5 of 10

    Deep neural variation spaces remain small at any depth; univariate ReLU saturates after depth 2 up to a factor of 2, so norm-controlled deep ReLU nets cannot be highly oscillatory along any direction.

  2. Representation Costs in Data Science: Foundations and the Quasi-Banach Spaces of Deep Neural Networks

    math.FA 2026-06 unverdicted novelty 7.0 of 10

    Develops general framework for representation costs of parametric models, proving that depth-L ReLU networks induce p-normable quasi-Banach spaces with p=2/L.

  3. Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators

    math.FA 2025-09 conditional novelty 7.0 of 10

    Vector-valued neural networks, DeepONets, and hypernetworks are shown to live in integral vector-valued reproducing kernel Banach spaces with representer theorems that recover the architectures.

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