DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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The neural tangent kernel of a differentiable parameterized function

Definition

Let $P, q \geq 1$ , let $Ω \subseteq R^{P}$ be open, let $X$ be a set, and let $f : Ω \times X \to R^{q}$ . Fix $θ \in Ω$ and suppose that for each $x \in X$ the map $ϑ \mapsto f (ϑ, x)$ is totally differentiable at $θ$ , meaning it has a linear first-order approximation with remainder $o (∥ ϑ - θ ∥_{2})$ . Write $D_{θ} f (θ, x) \in R^{q \times P}$ for its derivative matrix in the standard Euclidean coordinates. The matrix-valued neural tangent kernel at $θ$ is

$K_{θ} (x, x^{'}) : = D_{θ} f (θ, x) D_{θ} f (θ, x^{'})^{T} \in R^{q \times q} .$

Equivalently, for $1 \leq a, b \leq q$ , $(K_{θ} (x, x^{'}))_{a b} = \sum_{p = 1}^{P} \partial_{θ_{p}} f_{a} (θ, x) \partial_{θ_{p}} f_{b} (θ, x^{'}) .$ For a scalar-output model this is the scalar kernel $\sum_{p = 1}^{P} \partial_{θ_{p}} f (θ, x) \partial_{θ_{p}} f (θ, x^{'})$ .

Definition references

Jacot, Gabriel, and Hongler, §§3–4.1, parameter Jacobian Gram kernel

Chizat, Oyallon, and Bach, §§1.1 and 2.1, differential and tangent operator

Total derivative as the linear first-order approximation

The standard Euclidean inner product

Depends on

Used by

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Arthur Jacot, Franck Gabriel, and Clément Hongler, Neural Tangent Kernel: Convergence and Generalization in Neural Networks (standard reference, not scraped)
Lénaïc Chizat, Edouard Oyallon, and Francis Bach, On Lazy Training in Differentiable Programming (standard reference, not scraped)