Foundation Lab

Reproducing Kernel Hilbert Spaces

Unifies SVMs, Gaussian processes, kernel regression, and NTK under one framework

Concept 71 of 100TheoryPhase 10

#71RKHSTheory

key equation

f(x) = \langle f, k(x, \cdot) \rangle_{\mathcal{H}}

Selected Foundation Object

Keep the equation fixed; move through the evidence.

Concept 71 of 100RKHSTheory / Phase 10: Mathematical foundations & information geometry

Current question

Kernels implicitly define an (often infinite-dimensional) feature space

f(x) = \langle f, k(x, \cdot) \rangle_{\mathcal{H}}

PredictionCommit before tracing the equation.

Ask what should change under a concrete input, then trace that expectation through the equation.

EvidenceCompare the equation and source.

Use the key equation and canonical papers as the available witnesses, without implying that a runnable panel exists.

InvariantName what survives notation changes.

The useful learning product is the reusable mechanism you can carry into another model, paper, or engineering tradeoff.

Next moveContinue through the atlas.

Use prerequisites, dependents, and semantic links to repair the next gap without leaving the object behind.

Unifies SVMs, Gaussian processes, kernel regression, and NTK under one framework
Explains why "similarity functions" must be positive definite—they define inner products
NTK shows neural networks are kernel machines in the infinite-width limit

What is still poorly explained in textbooks and papers:

Kernels implicitly define an (often infinite-dimensional) feature space
Positive definiteness = you can build a Hilbert space where the kernel is an inner product
Attention can be viewed as a learned, data-dependent kernel

If you want intuition first, start with the key equation and cited sources. Come back here for the full walkthrough.

Key Equation

f(x) = \langle f, k(x, \cdot) \rangle_{\mathcal{H}}

Reproducing property: evaluation is an inner product:

f(x) = \langle f, k(x, \cdot) \rangle_{\mathcal{H}}

Kernel trick: inner product in feature space without explicit computation:

k(x, x') = \langle \varphi(x), \varphi(x') \rangle

Mercer decomposition (spectral):

k(x, x') = \sum_{m=1}^\infty \lambda_m e_m(x) e_m(x')

Representer theorem: optimal function is a linear combination of kernel evaluations:

f^*(x) = \sum_{i=1}^n \alpha_i k(x_i, x)

Shawe-Taylor & Cristianini2004Cambridge University Press

Jacot et al.2018NeurIPS

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