TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Least-squares NTK flow decays mode by mode and fits distinct sphere data

Statement

Assume AC and use the source-normalized network with scalar output, a finite data set $x_{1}, \dots, x_{N}$ with $N \geq 1$ , and limiting neural tangent Gram matrix $K_{\infty}$ from the sequential infinite-width theorem. For a target vector $y \in R^{N}$ , consider the finite-data least-squares loss $L (f) = \frac{1}{2 N} \sum_{i = 1}^{N} (f_{i} - y_{i})^{2}$ and its limiting kernel-gradient-flow equation

$\dot{f} (t) = - \frac{1}{N} K_{\infty} (f (t) - y), f (0) = f_{0} .$

Let $K_{\infty} = Q diag (λ_{1}, \dots, λ_{N}) Q^{T}$ with $Q$ orthogonal and $λ_{i} \geq 0$ . Then for every $t \geq 0$ ,

$(Q^{T} (f (t) - y))_{i} = e^{- t λ_{i} / N} (Q^{T} (f_{0} - y))_{i} (1 \leq i \leq N) .$

Zero-eigenvalue residual components remain constant. If the inputs are pairwise distinct points on $S^{n_{0} - 1}$ , with $n_{0} \geq 1$ , $L \geq 2$ , $β > 0$ , and a globally Lipschitz nonpolynomial activation, then $K_{\infty}$ is positive definite and $f (t) \to y$ as $t \to \infty$ .

Statement references

Jacot, Gabriel, and Hongler, §5, least-squares NTK dynamics and eigenmode decay

The limiting finite-data kernel is the deterministic recursive NTK

The limiting NTK is strictly positive definite on distinct unit-sphere inputs

Facts & Assumptions

Given: AC; the source-normalized scalar-output network; a finite data set of size $N \geq 1$ and its labels; the limiting kernel $K_{\infty}$ ; and an initial prediction vector $f_{0}$ .

[F1]

The finite-width tangent kernel is a Jacobian Gram matrix, hence every finite-data Gram matrix is symmetric positive semidefinite (The neural tangent kernel of a differentiable parameterized function).

[F2]

Under the AC and source-normalized network hypotheses, the sequential theorem supplies the deterministic limiting Gram matrix as the probability limit of finite-width Gram matrices (Sequential infinite-width NTK limit and finite-time lazy training dynamics).

[F3]

A self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis (Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis).

[F4]

The exponential has derivative equal to itself (The exponential function is smooth and $(\exp)^{'} = \exp$ ).

[F5]

For distinct unit-sphere inputs under the listed assumptions, the recursive limiting NTK Gram matrix is positive definite (The limiting neural tangent kernel is strictly positive definite on every unit sphere).

Proof

technique · diagonalize the symmetric Gram matrix and solve each scalar residual equation

1.1F1F2givenalgebra

For any finite-width parameters $θ$ and coefficients $c_{i}$ , [F1] gives $c^{T} K_{θ} c = {∥ \sum_{i = 1}^{N} c_{i} \nabla_{θ} f_{θ} (x_{i}) ∥}_{2}^{2} \geq 0.$ Thus every $K_{θ}$ is symmetric positive semidefinite. Taking the finite-dimensional probability limit in [F2] preserves symmetry and each quadratic inequality, so $K_{\infty}$ is symmetric positive semidefinite.

2.1F3F4step 1.1algebra

Apply [F3] to $K_{\infty}$ . In the orthogonal coordinates $r (t) = Q^{T} (f (t) - y)$ , the flow equation becomes ${\dot{r}}_{i} (t) = - (λ_{i} / N) r_{i} (t)$ . Differentiating $e^{t λ_{i} / N} r_{i} (t)$ using [F4] gives derivative zero, so it equals its initial value. Therefore $r_{i} (t) = e^{- t λ_{i} / N} r_{i} (0)$ for every $i$ , including $λ_{i} = 0$ .

3.1F5step 2.1algebra∎

Under the sphere hypotheses, [F5] makes every eigenvalue strictly positive. Hence $e^{- t λ_{i} / N} \to 0$ for every $i$ , so each coordinate of $Q^{T} (f (t) - y)$ tends to zero. Since $Q$ is orthogonal, $f (t) - y \to 0$ in Euclidean norm, proving convergence to the target vector.

Depends on

Used by

Dependency tree · two levels

39 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)