# Deep regularization and direct training of the inner layers of Neural Networks with Kernel Flows

arXiv: https://arxiv.org/abs/2002.08335 · HTML: https://mlsanity.com/paper/2002.08335/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Feb 2020.

## Built on by (most cited first, 10 of 10)

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- [Data-driven geophysical forecasting: Simple, low-cost, and accurate baselines with kernel methods](https://mlsanity.com/paper/2103.10935/) (2021)
- [Learning Dynamical Systems from Data: A Simple Cross-Validation Perspective, Part V: Sparse Kernel Flows for 132 Chaotic Dynamical Systems](https://mlsanity.com/paper/2301.10321/) (2023)
- [Learning dynamical systems from data: A simple cross-validation perspective, part III: Irregularly-Sampled Time Series](https://mlsanity.com/paper/2111.13037/) (2021)
- [DeepParticle: learning invariant measure by a deep neural network minimizing Wasserstein distance on data generated from an interacting particle method](https://mlsanity.com/paper/2111.01356/) (2021)
- [Learning "best" kernels from data in Gaussian process regression. With application to aerodynamics](https://mlsanity.com/paper/2206.02563/) (2022)
- [Kernel Sum of Squares for Data Adapted Kernel Learning of Dynamical Systems from Data: A global optimization approach](https://mlsanity.com/paper/2408.06465/) (2024)
- [Deep Learning with Kernel Flow Regularization for Time Series Forecasting](https://mlsanity.com/paper/2109.11649/) (2021)
- [Likelihood-based generalization of Markov parameter estimation and multiple shooting objectives in system identification](https://mlsanity.com/paper/2212.13902/) (2022)
- [Neural means and kernel corrections for operator learning](https://mlsanity.com/paper/2609.00389/) (2026)

## Abstract (arXiv)

We introduce a new regularization method for Artificial Neural Networks (ANNs) based on Kernel Flows (KFs). KFs were introduced as a method for kernel selection in regression/kriging based on the minimization of the loss of accuracy incurred by halving the number of interpolation points in random batches of the dataset. Writing fθ(x) = big(f(n)θncirc f(n-1)_θn-1 circ dots circ f(1)θ1big)(x) for the functional representation of compositional structure of the ANN, the inner layers outputs h(i)(x) = big(f(i)θicirc f(i-1)_θi-1 circ dots circ f(1)θ1big)(x) define a hierarchy of feature maps and kernels k(i)(x,x')=exp(- γi |h(i)(x)-h(i)(x')|22). When combined with a batch of the dataset these kernels produce KF losses e2(i) (the L2 regression error incurred by using a random half of the batch to predict the other half) depending on parameters of inner layers θ1,ldots,θi (and γi). The proposed method simply consists in aggregating a subset of these KF losses with a classical output loss. We test the proposed method on CNNs and WRNs without alteration of structure nor output classifier and report reduced test errors, decreased generalization gaps, and increased robustness to distribution shift without significant increase in computational complexity. We suspect that these results might be explained by the fact that while conventional training only employs a linear functional (a generalized moment) of the empirical distribution defined by the dataset and can be prone to trapping in the Neural Tangent Kernel regime (under over-parameterizations), the proposed loss function (defined as a nonlinear functional of the empirical distribution) effectively trains the underlying kernel defined by the CNN beyond regressing the data with that kernel.