Function Spaces Without Kernels: Learning Compact Hilbert Space Representations

• URL: http://arxiv.org/abs/2509.20605v1
• Date: Wed, 24 Sep 2025 22:55:52 GMT
• Title: Function Spaces Without Kernels: Learning Compact Hilbert Space Representations
• Authors: Su Ann Low, Quentin Rommel, Kevin S. Miller, Adam J. Thorpe, Ufuk Topcu,
• Abstract summary: We show that function encoders provide a principled connection to feature learning and kernel methods.<n>We develop two training algorithms that learn compact bases.<n>This work suggests a path toward neural predictors with kernel-level guarantees.
• Score: 17.20282687120807
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Function encoders are a recent technique that learn neural network basis functions to form compact, adaptive representations of Hilbert spaces of functions. We show that function encoders provide a principled connection to feature learning and kernel methods by defining a kernel through an inner product of the learned feature map. This kernel-theoretic perspective explains their ability to scale independently of dataset size while adapting to the intrinsic structure of data, and it enables kernel-style analysis of neural models. Building on this foundation, we develop two training algorithms that learn compact bases: a progressive training approach that constructively grows bases, and a train-then-prune approach that offers a computationally efficient alternative after training. Both approaches use principles from PCA to reveal the intrinsic dimension of the learned space. In parallel, we derive finite-sample generalization bounds using Rademacher complexity and PAC-Bayes techniques, providing inference time guarantees. We validate our approach on a polynomial benchmark with a known intrinsic dimension, and on nonlinear dynamical systems including a Van der Pol oscillator and a two-body orbital model, demonstrating that the same accuracy can be achieved with substantially fewer basis functions. This work suggests a path toward neural predictors with kernel-level guarantees, enabling adaptable models that are both efficient and principled at scale.

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