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The limiting neural tangent kernel is strictly positive definite on every unit sphere

Statement

Assume AC. Fix $n_{0} \geq 1$ , $L \geq 2$ , $β > 0$ , and a globally Lipschitz activation $σ : R \to R$ that is not a polynomial. For $x, y$ on the unit sphere $S^{n_{0} - 1} = {x \in R^{n_{0}} : ∥ x ∥_{2} = 1}$ , define $Σ^{1} (x, y) = x \cdot y / n_{0} + β^{2}$ . Recursively, for $r = 2, \dots, L$ , let $(U, V)$ be a centered Gaussian pair whose covariance entries are $Σ^{r - 1} (x, x), Σ^{r - 1} (y, y), Σ^{r - 1} (x, y)$ , and set

$Σ^{r} (x, y) = E [σ (U) σ (V)] + β^{2}, D^{r} (x, y) = E [s (U) s (V)],$

where $s$ is any bounded Borel representative of the almost-everywhere derivative of $σ$ . Set $Θ^{1} = Σ^{1}$ and $Θ^{r} (x, y) = Θ^{r - 1} (x, y) D^{r} (x, y) + Σ^{r} (x, y)$ . Then for every finite list of pairwise distinct points $x_{1}, \dots, x_{N} \in S^{n_{0} - 1}$ and every nonzero $c \in R^{N}$ ,

$\sum_{i, j = 1}^{N} c_{i} c_{j} Θ^{L} (x_{i}, x_{j}) > 0.$

Thus the scalar limiting neural tangent kernel is strictly positive definite on every finite set of distinct sphere points, for every $n_{0} \geq 1$ and $L \geq 2$ . In dimension $n_{0} = 1$ this includes both antipodal points of $S^{0} = {- 1, 1}$ .

Statement references

Jacot, Gabriel, and Hongler, Proposition 2 and Appendix A.4, limiting NTK positivity

Gneiting, Theorem 1 and sphere conventions; this proof handles $S^{0}$ and every dimension directly

Positive bias gives an all-positive power series that separates every finite sphere subset

A nonconstant Lipschitz activation preserves strict covariance positivity at nondegenerate Gaussian inputs

The limiting scalar NTK is given by the stated covariance and tangent recursions

Facts & Assumptions

Given: AC; $n_{0} \geq 1$ ; $L \geq 2$ ; $β > 0$ ; a globally Lipschitz nonpolynomial activation; and finitely many pairwise distinct unit vectors.

[A1]

AC supplies the Gaussian and Hermite probability constructions (The Axiom of Choice, AC supplies countable selections and prescribed serial paths).

[F1]

Every finite positive-semidefinite covariance matrix defines a centered Gaussian vector, singular cases included, and its covariance uniquely determines its law (Multivariate normal law, including singular covariance, Characteristic function of a multivariate normal law).

[F2]

A Lipschitz real function is absolutely continuous on compact intervals ( $C^{1}$ implies Lipschitz, Lipschitz implies absolutely continuous, and absolutely continuous implies continuous and bounded variation).

[F3]

The Gaussian-L² Hermite basis is complete, and its covariance expansion plus the positive-bias coefficient argument give strict activation covariance on the sphere, including $n_{0} = 1$ (Completeness of normalized probabilists’ Hermite polynomials in Gaussian L², Hermite expansion of a correlated Gaussian feature kernel, Positive bias makes the Hermite feature kernel strictly positive definite on every sphere).

[F4]

For a finite nondegenerate Gaussian covariance matrix, a globally Lipschitz nonconstant activation has strictly positive activation covariance; its bounded Borel derivative representative yields a positive-semidefinite derivative Gram, and the Schur product of two positive-semidefinite matrices is positive semidefinite (Gaussian activation covariance and tangent recursion preserve strict positivity).

[F5]

The source-normalized infinite-width theorem gives $Θ_{i j}^{1} = Σ_{i j}^{1}$ and $Θ_{i j}^{r} = Θ_{i j}^{r - 1} {\dot{Σ}}_{i j}^{r} + Σ_{i j}^{r}$ entrywise, where ${\dot{Σ}}_{i j}^{r}$ is the Gaussian expectation of the two hidden-layer derivatives (The neural tangent kernel of a differentiable parameterized function, Jacot–Gabriel–Hongler width-normalized fully connected networks, Sequential infinite-width NTK limit and finite-time lazy training dynamics).

[F6]

Under countable choice, an absolutely continuous real function on a compact interval is differentiable almost everywhere (Absolutely continuous functions have integrable derivatives).

Proof

technique · use the biased Hermite supplier for the potentially singular first covariance, then propagate strictness through positive-definite Gaussian covariances and the tangent recursion

1.1A1F1F2F6givenconstruct

Fix pairwise distinct $x_{1}, \dots, x_{N}$ on the sphere. The matrix $Σ^{1}$ is a Gram matrix because $Σ_{i j}^{1} = ⟨ (x_{i} / \sqrt{n_{0}}, β), (x_{j} / \sqrt{n_{0}}, β) ⟩$ ; it may be singular. Its diagonal is $q = n_{0}^{- 1} + β^{2} > 0$ . By [F2] and [F6], on each compact interval $σ$ is absolutely continuous and differentiable almost everywhere; a countable union gives a null exceptional set, and the pointwise upper limit of bounded continuous difference quotients defines a bounded Borel representative $s_{0}$ equal to $σ'$ wherever it exists. The given representative $s$ agrees with $s_{0}$ outside a null set; include this set and the nondifferentiability set in one null set $E$ . The multivariate Gaussian with covariance $Σ^{1}$ exists by [F1], and every coordinate has a positive-variance normal density and hence avoids $E$ almost surely, so the choice of representative does not change any expectation.

2.1F1F3step 1.1algebra

The covariance $Σ^{2}$ is strictly positive definite by [F3], applied to this finite set, with the same $n_{0}, β, σ$ . The derivative matrix $D^{2}$ is positive semidefinite directly: for every $c \in R^{N}$ , $c^{T} D^{2} c = E [(\sum_{i} c_{i} s (U_{i}))^{2}] \geq 0$ for a common Gaussian vector $U$ of covariance $Σ^{1}$ . Also, writing $u_{i} = (x_{i} / \sqrt{n_{0}}, β)$ , finite expansion gives $c^{T} (Σ^{1} \circ D^{2}) c = E {∥ \sum_{i} c_{i} s (U_{i}) u_{i} ∥}_{2}^{2} \geq 0,$ so $Σ^{1} \circ D^{2}$ is positive semidefinite. Hence $Θ^{2} = Σ^{2} + Σ^{1} \circ D^{2}$ is strictly positive definite.

3.1F4step 2.1givenalgebra

Let $r \geq 3$ and suppose $Σ^{r - 1}$ is strictly positive definite. Its Gaussian vector therefore satisfies the nondegenerate-covariance hypothesis of [F4]; the activation is nonconstant because a nonpolynomial function cannot be constant. Thus $Σ^{r}$ is strictly positive definite and $D^{r}$ is positive semidefinite. If $Θ^{r - 1}$ is positive semidefinite, [F4] makes $Θ^{r - 1} \circ D^{r}$ positive semidefinite; adding $Σ^{r}$ makes $Θ^{r}$ strictly positive definite.

4.1F5step 2.1step 3.1∎

Step 2.1 initializes the induction with both $Σ^{2}$ and $Θ^{2}$ strictly positive definite. Repeated step 3.1 proves $Θ^{L}$ strictly positive definite for each $L \geq 2$ . By [F5], this recursively defined kernel is the limiting NTK of the stated network on the chosen finite data set. Since the list was arbitrary, the claimed strict positive definiteness holds for every finite set of distinct sphere points, including the two points of $S^{0}$ .

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Sources

Arthur Jacot, Franck Gabriel, and Clément Hongler, Neural Tangent Kernel: Convergence and Generalization in Neural Networks (standard reference, not scraped)
Tilmann Gneiting, Strictly and non-strictly positive definite functions on spheres (standard reference, not scraped)