RKHS
Reproducing kernel Hilbert spaces in deep learning: feature maps, the softmax kernel, positive-definite vs nonnegative kernels, and Mercer theory.
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Solving It and Descending It, in JAX/Flax NNX
A runnable companion to the solve-vs-descend post: the Yat kernel and its Gram matrix, the exact kernel ridge solve via Cholesky, the same kernel as a Flax NNX module trained by AdamW with LR sweeps and best-epoch selection, the measured timing wall, minibatching through 511k rows, and the conv trunk the solve can never train. Every number is from the real Kaggle runs.
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One Kernel Family, Fitted Two Ways
A dense kernel-ridge solve and a learned-center Yat expansion use the same kernel family but optimize different hypothesis classes. Their predictions correlate at 0.95 on housing; the compressed model then scales through datasets the dense baseline cannot hold.
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What Can a Weight Be?
A kernel is a spectral price list: it decides which functions are affordable, and regularization sets the budget. Compare Sobolev, Gaussian, spherical, and finite examples, then connect their eigenvalues to kernel ridge shrinkage and effective dimension.
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Where a Weight Lives, in JAX/Flax NNX
A runnable companion: build the representer-theorem weight in JAX. A positive-definite kernel, the Gram matrix, a single linear solve for the coefficients, and the weight comes out as a combination of the data, f = sum alpha_i k(x_i, .). A linear weight cannot separate nested rings; the placed kernel weight does, read purely through the kernel as a similarity-weighted vote of the data.
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Where Does a Weight Live?
A standard neuron's weight and its input never actually meet: one is a point you can see, the other an arrow off in its own space, joined only by a shadow. This is what a reproducing kernel Hilbert space fixes: it gives input and weight one shared address, where the optimal weight is built from the data itself and sits right next to it. Four interactive panels.
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A Kernel's Price List, in JAX
Construct valid periodic kernel spectra, audit truncated RKHS norms for convergence, then solve kernel ridge regression and read regularization as spectral shrinkage, effective dimension, and a measured generalization curve.
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The Yat-Kernel MLP in JAX/Flax NNX
Build a finite bank of Yat-kernel sections in JAX/Flax NNX, verify the kernel, train it on two moons and Fashion-MNIST, inspect exact prototype contributions, and test the initialization control that separates visible centers from noisy ones.
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What a Finite Kernel Buys an MLP
Replace the activation with a finite bank of learned kernel sections. The resulting MLP exposes prototypes, exact layer-local contributions, measurable geometry, and the conditions those claims require, then tests the construction on arithmetic and Fashion-MNIST.
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Cheap Attention: Linear-Time Kernel Approximation
A 128K-token context creates billions of pairwise questions per attention head. But the N×N matrix is not the essence of attention; it is the receipt for an infinite feature map we never wrote down. Approximate that feature map with random features, reassociate the sum, and softmax attention becomes linear-time kernel attention.
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Self-Attention as Kernel Regression in JAX/Flax NNX
A runnable companion to Attention is Explainable Because it is a Kernel: build scaled dot-product attention from scratch in Flax NNX, prove in code that it is exactly a Nadaraya–Watson kernel smoother, watch the separate q/k projections break positive-definiteness numerically, swap the exp-dot-product kernel for Gaussian, Yat, and linear kernels to see which keep the weights a convex partition of unity, read the temperature as a kernel bandwidth, and train a single head end-to-end to route to a marked token.
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What Attention Weights Can Explain
Self-attention has the normalized weighted-average form of a compatibility smoother. That exposes exact routing arithmetic, but it does not make attention weights causal explanations or guarantee a Mercer kernel on tokens.