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Hermite expansion of a correlated Gaussian feature kernel

Statement

Assume AC. Let $γ$ be standard Gaussian measure, let $h_{m} = H_{m} / \sqrt{m!}$ be the normalized probabilists’ Hermite basis, and let $f, g \in L^{2} (γ)$ have coefficients $a_{m} = \int f h_{m} d γ$ and $b_{m} = \int g h_{m} d γ$ . For $ρ \in [- 1, 1]$ , let $(X, Y)$ have the centered bivariate normal law with unit variances and correlation $ρ$ . Then $E [f (X) g (Y)] = \sum_{m = 0}^{\infty} a_{m} b_{m} ρ^{m} .$ The series is absolutely convergent, including at $ρ = \pm 1$ . In particular, the Gaussian dual kernel of $f$ is $\sum_{m \geq 0} a_{m}^{2} ρ^{m}$ , and $\sum_{m \geq 0} a_{m}^{2} = ∥ f ∥_{L^{2} (γ)}^{2}$ .

Statement references

Jacot, Gabriel, and Hongler, Appendix A.4, Lemma 2, Hermite expansion of the dual activation kernel

Monic probabilists’ Hermite recurrence and explicit coefficients

Gaussian Hermite orthogonality

Completeness and Parseval in Gaussian L²

Facts & Assumptions

Given: AC; $f, g \in L^{2} (γ)$ ; their normalized Hermite coefficients; and a centered unit-variance Gaussian pair $(X, Y)$ of correlation $ρ \in [- 1, 1]$ .

[A1]

AC supplies the countable and dependent choice required by the Gaussian and Hermite suppliers (The Axiom of Choice, AC supplies countable selections and prescribed serial paths).

[F1]

The functions $h_{m}$ form a complete orthonormal basis of real $L^{2} (γ)$ , so the Hermite expansions converge in norm and satisfy Parseval (Completeness of normalized probabilists’ Hermite polynomials in Gaussian L²).

[F2]

The monic Hermite polynomials satisfy $H_{0} = 1$ , $H_{1} (x) = x$ , $x H_{m} = H_{m + 1} + m H_{m - 1}$ , and $H_{m} (x) = m! \sum_{j = 0}^{⌊ m / 2 ⌋} (- 1 / 2)^{j} x^{m - 2 j} / (j! (m - 2 j)!)$ (The monic probabilists' Hermite polynomials).

[F3]

The standard normal has density $(2 π)^{- 1 / 2} e^{- x^{2} / 2}$ , and each linear projection of a centered Gaussian vector is a scalar normal with the corresponding covariance quadratic form as its variance, including variance zero (Standard normal and normal laws, Multivariate normal law, including singular covariance).

[F4]

Finite linear combinations commute with expectation for integrable random variables (Linearity, monotonicity, and the modulus bound for expectation).

[F5]

Dominated convergence applies under a single integrable majorant (Dominated convergence).

[F6]

The $L^{2}$ Cauchy–Schwarz inequality bounds product expectations (Cauchy-Schwarz for random variables).

[F7]

The real exponential is defined by its power series, has derivative equal to itself, and obeys $\exp (x + y) = \exp (x) \exp (y)$ (The real exponential function and the number $e$ by a power series, The exponential function is smooth and $(\exp)^{'} = \exp$ , The exponential addition formula $\exp (x + y) = \exp (x) \exp (y)$ ).

[F8]

The one-variable chain rule and product rule apply to the real generating function (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

[F9]

A difference quotient of a differentiable real function on a compact interval is bounded in absolute value by a bound for its derivative on that interval (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ ).

Proof

technique · direct

1.1F2F7F8algebra

For real $s$ and $x$ , set $G_{x} (s) = \exp (s x - s^{2} / 2)$ . By [F7]–[F8], $G_{x}^{'} (s) = (x - s) G_{x} (s)$ . Put $P_{m} (x) = G_{x}^{(m)} (0)$ . Then $P_{0} = 1$ and $P_{1} = x$ ; differentiating the identity $G_{x}^{'} = (x - s) G_{x}$ $m$ times and setting $s = 0$ gives $P_{m + 1} = x P_{m} - m P_{m - 1}$ for $m \geq 1$ . The Hermite recurrence in [F2] therefore gives $P_{m} = H_{m}$ for every $m$ .

2.1A1F3F7step 1.1algebra

For real $s, t$ , [F3] gives $s X + t Y \sim N (0, v)$ with $v = s^{2} + 2 ρ s t + t^{2} \geq 0$ . Completing the square in the standard-normal density gives $E e^{a Z} = e^{a^{2} / 2}$ for real $a$ and $Z \sim N (0, 1)$ , so $E e^{s X + t Y} = e^{v / 2}$ by the scalar normal's affine definition; when $v = 0$ , its law is a point mass and the same formula holds. Thus [F7] gives $E [G_{X} (s) G_{Y} (t)] = e^{- (s^{2} + t^{2}) / 2} E e^{s X + t Y} = e^{ρ s t}$ . This uses the given pair directly and includes $ρ = \pm 1$ .

3.1F3F5F6F7F8F9step 1.1step 2.1algebra

For any fixed finite $m, n$ , differentiating the expectation identity in step 2.1 $m$ times in $s$ and $n$ times in $t$ at $(0, 0)$ is justified as follows. On a bounded parameter rectangle, each required derivative of the integrand is bounded by $K (1 + ∣ X ∣)^{q} (1 + ∣ Y ∣)^{q} e^{R (∣ X ∣ + ∣ Y ∣)}$ for finite $K, q, R$ . This is integrable by [F6], since completing the square gives finite second moments for each marginal factor $(1 + ∣ Z ∣)^{q} e^{R ∣ Z ∣}$ under $Z \sim N (0, 1)$ . On a slightly larger rectangle, [F9] gives the same majorant for each difference quotient, so [F5] passes each of the finitely many derivatives through expectation. By step 1.1 the derivative on the left is $E [H_{m} (X) H_{n} (Y)]$ ; differentiating $e^{ρ s t}$ on the right gives $m! ρ^{m}$ if $m = n$ and $0$ otherwise. Hence $E [H_{m} (X) H_{n} (Y)] = m! δ_{m n} ρ^{m}$ .

4.1F1F4step 3.1algebra

For finite partial sums $f_{M} = \sum_{m = 0}^{M} a_{m} h_{m}$ and $g_{M} = \sum_{m = 0}^{M} b_{m} h_{m}$ , step 3.1, finite expectation linearity and orthonormality give $E [f_{M} (X) g_{M} (Y)] = \sum_{m = 0}^{M} a_{m} b_{m} ρ^{m}$ .

5.1F1F6step 4.1∎

By [F1], $f_{M} \to f$ and $g_{M} \to g$ in $L^{2} (γ)$ . Since both marginals of $(X, Y)$ are $γ$ , [F6] bounds the product-expectation error by $∥ f_{M} - f ∥_{2} ∥ g_{M} ∥_{2} + ∥ f ∥_{2} ∥ g_{M} - g ∥_{2}$ , which tends to zero. Also $\sum_{m} ∣ a_{m} b_{m} ρ^{m} ∣ \leq (\sum_{m} a_{m}^{2})^{1 / 2} (\sum_{m} b_{m}^{2})^{1 / 2}$ by [F1] and Cauchy–Schwarz for sequences. Taking $M \to \infty$ proves the stated identity and endpoint absolute convergence; the case $f = g$ gives the dual-kernel formula.

Depends on

Used by

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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)