Wick Hermite Features Overview

• Wick Hermite features are an orthogonal polynomial system defined via the Wick product and combinatorial posets, offering a structured basis for kernel linearization.
• They leverage the poset of incomplete matchings and Möbius inversion to derive closed product and inversion formulas that simplify high-degree computations.
• This framework enables scalable kernel methods by supporting low-variance random feature expansions for polynomial and Gaussian kernels.

Wick Hermite features are an orthogonal polynomial feature system defined via the Wick product, with a combinatorial structure rooted in the poset of incomplete matchings. This framework provides a basis for efficiently linearizing polynomial and Gaussian kernels and underpins low-variance random feature expansions for shift-invariant kernels whose Fourier transforms are Gaussian mixtures. The key combinatorics derive from the poset $P_{1,2}(n)$ of partitions of $[n]$ into singletons and pairs, ordered by refinement, with Möbius function and product/inversion formulas central to their construction and application (Anshelevich, 2017).

1. Algebraic and Combinatorial Definition

The Wick–Hermite polynomials $H^W_n(x)$ are defined in the context of a unital $*$–algebra $M$, with a projection $a \in M$ ($a^2 = a$), and a state $\langle \cdot \rangle$ such that $\langle a \rangle = t$. The variable $x$ is taken as $x = X(a)$. The central object is the Wick product $W_{1,2}$, recursively defined on $n$ arguments: $$W_{1,2}(a_1, \dots, a_n, a_{n+1}) = W_{1,2}(a_1, \dots, a_n) X(a_{n+1}) - \sum_{i=1}^n W_{1,2}(a_1, \dots, \widehat{a}_i, \dots, a_n, a_{n+1}) \langle a_i a_{n+1} \rangle$$ with $H^W_n(x) = W_{1,2}(a, \dots, a)$ ($n$ copies).

The Wick–Hermite polynomials satisfy the classical three-term recurrence: $$H^W_0(x) = 1, \quad H^W_1(x) = x, \qquad H^W_{n+1}(x) = x H^W_n(x) - n t H^W_{n-1}(x)$$ Combinatorially, $H^W_n(x)$ admit an expansion over $P_{1,2}(n)$: $$H^W_n(x) = \sum_{\pi \in P_{1,2}(n)} \mu(\hat{0}, \pi) \; x^{\,\#\{\text{singletons of }\pi\}} t^{\,\#\{\text{pairs of }\pi\}}$$ where $$\mu(\hat{0}, \pi) = (-1)^{\#\{\text{pairs in }\pi\}}$$.

The closed-form for $H^W_n(x)$ is: $$H^W_n(x) = \sum_{k=0}^{\lfloor n/2 \rfloor} (-1)^k \frac{n!}{k! (n-2k)! 2^k} t^k x^{n-2k}$$

2. Exponential Generating Function and Basis Properties

The ordinary exponential generating function of the Wick–Hermite system is: $$G_W(s; x) = \sum_{n=0}^\infty H^W_n(x) \frac{s^n}{n!} = \exp\left(x s - \frac{t}{2} s^2\right)$$ This function aligns, up to scaling $x \mapsto x/\sqrt{t}$, with the probabilists' Hermite polynomials for variance 1, whose generating function is $e^{x s - s^2 / 2}$. The "physicists'" version, $\He_n(x)$, has generating function $e^{2 x s - s^2}$.

A notable property is that the feature vector $(H^W_0(x), \dots, H^W_D(x))$ can be computed in $O(D)$ time via recurrence, enabling fast high-dimensional polynomial feature computation.

3. Product and Linearization Formulas via Incomplete Matching Posets

Wick–Hermite polynomials possess a closed product formula governed explicitly by the combinatorics of $P_{1,2}(m+n)$. Given $m,n \geq 0$,

$$H^W_m(x) H^W_n(x) = \sum_{k=0}^{\min(m,n)} C_{m,n,k} H^W_{m+n-2k}(x)$$

where

$C_{m,n,k} = \binom{m}{k} \binom{n}{k} k!$

This result follows directly from enumerating matchings where all newly formed pairs cross between the first $m$ and last $n$ elements.

The inversion—expressing monomials $x^n$ in the Wick–Hermite basis—is given by

$$x^n = \sum_{j=0}^{\lfloor n/2 \rfloor} \frac{n!}{j! (n-2j)! 2^j} t^j H^W_{n-2j}(x)$$

This formula reflects the Möbius inversion structure of the incomplete-matching poset.

4. Möbius Function and Poset Structure

The poset $P_{1,2}(n)$ consists of all partitions of $[n]$ into pairs and singletons, ordered by refinement. The Möbius function $\mu(\hat{0}, \pi)$ for this poset is given by $(-1)^{\#\{\text{pairs in }\pi\}}$. All constants arising in the product and inversion formulas are binomial- and factorial-factors originating from the enumeration of matchings in $P_{1,2}(n)$.

The triangular structure of the matrix $\{H^W_n\}_{n \geq 0}$ in the monomial basis enables efficient closed-form inversion between monomial and Wick–Hermite polynomial representations.

5. Efficient Feature Maps and Applications in Kernel Methods

The Wick–Hermite system provides a computationally efficient feature map for polynomial and Gaussian kernels. The expansion of the Gaussian kernel

$$k(x, y) = \exp\left(\frac{x y}{t} - \frac{x^2 + y^2}{2 t}\right) = \sum_{n=0}^\infty \frac{1}{n!} H^W_n(x) H^W_n(y)$$

enables truncation at degree $D$ to yield an $O(D)$-dimensional feature embedding with exact orthogonality—effectively the Mercer expansion in the Wick–Hermite basis.

For polynomial kernels of the form $(\langle x, y \rangle + c)^d$, the inversion and product formulas allow reduction of the full monomial expansion’s $O(d^2)$ cross-terms to $O(d)$ feature computations. More broadly, random-feature schemes sampling $n$ with probability proportional to $t^n/n!$ and evaluating $H^W_n(x)$ deliver unbiased, low-variance approximations for shift-invariant kernels with Gaussian mixture Fourier transforms.

Feature evaluation relies critically on the recurrence

$H^W_{n+1}(x) = x H^W_n(x) - n t H^W_{n-1}(x)$

and the combinatorial identities for rapid reduction of high-degree polynomial terms, facilitating scalable high-dimensional kernel learning.

6. Summary Table: Key Formulas and Combinatorial Factors

Formula Type	Expression	Combinatorial Factor
Polynomial	$$H^W_n(x) = \sum_{k=0}^{\lfloor n/2 \rfloor} (-1)^k \frac{n!}{k! (n-2k)! 2^k} t^k x^{n-2k}$$	binomial/factorial; $P_{1,2}(n)$
Product	$$H^W_m(x) H^W_n(x) = \sum_{k=0}^{\min(m,n)} \binom{m}{k} \binom{n}{k} k! H^W_{m+n-2k}(x)$$	cross-block pairs $P_{1,2}(m+n)$
Inversion	$$x^n = \sum_{j=0}^{\lfloor n/2 \rfloor} \frac{n!}{j! (n-2j)! 2^j} t^j H^W_{n-2j}(x)$$	same as in polynomial formula
Generating Fn	$G_W(s; x) = \exp(x s - \frac{t}{2} s^2)$	combination of exponential series

All constants and structures are fixed by the Möbius function and enumeration principles of the incomplete matching poset $P_{1,2}$ (Anshelevich, 2017).