mlsanityData to 7 Oct 2026

Deep Learning with Kernels through RKHM and the Perron-Frobenius Operator

arXiv 2305.13588 ↗First posted 23 May 2023stat.ML cs.LGcited by 7

Not on the map. Why?

Abstract (arXiv)

Reproducing kernel Hilbert C^*-module (RKHM) is a generalization of reproducing kernel Hilbert space (RKHS) by means of C^*-algebra, and the Perron-Frobenius operator is a linear operator related to the composition of functions. Combining these two concepts, we present deep RKHM, a deep learning framework for kernel methods. We derive a new Rademacher generalization bound in this setting and provide a theoretical interpretation of benign overfitting by means of Perron-Frobenius operators. By virtue of C^*-algebra, the dependency of the bound on output dimension is milder than existing bounds. We show that C^*-algebra is a suitable tool for deep learning with kernels, enabling us to take advantage of the product structure of operators and to provide a clear connection with convolutional neural networks. Our theoretical analysis provides a new lens through which one can design and analyze deep kernel methods.

Key facts. Cited by 7 papers in the map (data through 7 Oct 2026). Builds on 23 of its 40 references that are in the map. First posted on arXiv 23 May 2023.

Builds on

23 of 40 references.

References · 23

  1. 2018Neural Tangent Kernel: Convergence and Generalization in Neural Networkscited by 2,597
  2. 2017Spectrally-normalized margin bounds for neural networkscited by 681
  3. 2019Benign Overfitting in Linear Regressioncited by 544
  4. 2015Norm-Based Capacity Control in Neural Networkscited by 299
  5. 2017Size-Independent Sample Complexity of Neural Networkscited by 289
  6. 2016A vector-contraction inequality for Rademacher complexitiescited by 141
  7. 2018Does data interpolation contradict statistical optimality?cited by 95
  8. 2021The Promises and Pitfalls of Deep Kernel Learningcited by 81
  9. 2020Deep Neural Tangent Kernel and Laplace Kernel Have the Same RKHScited by 74
  10. 2014Convolutional Kernel Networkscited by 47
  11. 2019Compression based bound for non-compressed network: unified generalization error analysis of large compressible deep neural networkcited by 39
  12. 2022Robust Fine-Tuning of Deep Neural Networks with Hessian-based Generalization Guaranteescited by 34
11 more
  1. 2017A representer theorem for deep kernel learningcited by 26
  2. 2014Scalable Matrix-valued Kernel Learning for High-dimensional Nonlinear Multivariate Regression and Granger Causalitycited by 13
  3. 2018Autoencoding any Data through Kernel Autoencoderscited by 10
  4. 2021Reproducing kernel Hilbert C*-module and kernel mean embeddingscited by 10
  5. 2022C^*-algebra Net: A New Approach Generalizing Neural Network Parameters to C^*-algebracited by 9
  6. 2018Metric on Nonlinear Dynamical Systems with Perron-Frobenius Operatorscited by 7
  7. 2019Krylov Subspace Method for Nonlinear Dynamical Systems with Random Noisecited by 7
  8. 2021Entangled Kernels -- Beyond Separabilitycited by 7
  9. 2022Learning in RKHM: a C^*-Algebraic Twist for Kernel Machinescited by 7
  10. 2023Noncommutative C^*-algebra Net: Learning Neural Networks with Powerful Product Structure in C^*-algebracited by 4
  11. 2020Analysis via Orthonormal Systems in Reproducing Kernel Hilbert C^*-Modules and Applicationscited by 1

Built on by

Cited by year

Citing papers per year: 2024: 4; 2025: 3.

7 papers.

  1. 2024C^*-Algebraic Machine Learning: Moving in a New Directioncited by 1
  2. 2024Spectral Truncation Kernels: Noncommutativity in C^*-algebraic Kernel Machinescited by 1
  3. 2025Operator-Based Generalization Bound for Deep Learning: Insights on Multi-Task Learningcited by 1
  4. 2024Quantum Circuit C^*-algebra Netcited by 0
  5. 2024Predicting AI Agent Behavior through Approximation of the Perron-Frobenius Operatorcited by 0
  6. 2025Position: Quantum Kernel Machines Should Move Beyond Scalar-Valued Kernels to Realize Their Potentialcited by 0
  7. 2025Compact Vision Transformer by Reduction of Kernel Complexitycited by 0