Bell Polynomial Recursions in Denoising

• Bell Polynomial Recursions are a combinatorial framework that recursively builds higher-order denoisers by encoding corrections via score functions and derivatives.
• They systematically control approximation errors in optimal transport maps, achieving explicit error rates such as O(σ^(2(K+1))) in statistical signal recovery.
• Their computation leverages partial Bell polynomials to inductively isolate map coefficients, enabling practical, data-driven nonparametric denoising.

Bell polynomial recursions arise as a combinatorial and analytic framework underlying the construction of hierarchical or higher-order denoisers in empirical Bayes and optimal transport approaches to statistical signal recovery. They encode the recursive structure of solutions to differential, moment-matching, and transport equations, enabling explicit formulæ for map corrections in series expansions in terms of higher-order score functions. This hierarchy has become prominent in recent denoising theory, particularly for the sequence of optimal transport denoisers interpolating between the observed noisy law and the unobserved clean distribution. These recursions are now central in describing, analyzing, and estimating distributional denoising maps from pure noisy data.

1. Definition and Emergence of Bell Polynomial Recursions

Bell polynomials $B_{n,p}$ are a family of combinatorial polynomials that enumerate the number of ways a set can be partitioned into subsets of given sizes, or algebraically, encode the expansion coefficients of composite function derivatives. In the denoising context, they first appear as the organizing principle in the series expansion of the optimal transport map (OT-map) from the noisy distribution $Q$ to the clean distribution $P$, with the form

$$T_K(y) = y + \sum_{k=1}^{K} \frac{\eta^k}{k!} h_k(y),$$

where each correction term $h_k$ is a polynomial in higher-order scores evaluated at $y$ and defined recursively by Bell polynomial combinations of previous $h_j$ (Liang, 10 Dec 2025).

The core formal recursion is: $$\sum_{\ell=0}^{k-1} (-1)^\ell\frac{k!}{(k-\ell)!\,\ell!} \sum_{j=1}^{k-\ell} G^{(2\ell+j)}(y) B_{k-\ell,j}(h_1, ..., h_{k-\ell-j+1})(y) + (-1)^k G^{(2k)}(y) = 0,$$ in which the only occurrence of $h_k$ is isolated by setting $\ell=0, j=1$, yielding an explicit recursive formula with all terms expressed as Bell polynomials of lower-order corrections and derivatives of the observed density.

2. Hierarchy of Denoisers and Role of Bell Recursions

In the scalar additive Gaussian noise model $Y = X + \sigma Z$, with $Q$ the observed law and $q$ its density, the Bell polynomial recursion constructs a denoiser hierarchy $T_0, T_1, ..., T_\infty$. The $k$-th correction term $h_k$ depends polynomially on the set of score functions up to order $2k-1$: $$s_m(y) = \frac{q^{(m)}(y)}{q(y)}, \qquad 1 \leq m \leq 2k-1.$$ The leading term $T_1$ resembles the half-shrinkage “optimal transport” denoiser while all higher-order terms systematically address moment and density mismatches of order $O(\sigma^{2(K+1)})$ (Liang, 12 Nov 2025, Liang, 10 Dec 2025).

Bell recursions thus provide the mechanism to express each subsequent correction as an explicit function of $q$ and its derivatives without reference to $P$, a critical property for agnostic, empirical-Bayes denoising.

3. Combinatorial and Analytic Structure

The partial Bell polynomials $B_{n,p}(x_1, ..., x_{n-p+1})$ organize the nonlinearity arising from repeated application of chain and product rules in the derivatives of composite functions (through Faa di Bruno's formula). In the optimal transport denoiser expansion, this structure embodies the effects of the nonlinear push-forward and density change under transformation, with the expansion of the OT-map formally written as: $$T_G(y) = y + \sum_{k=1}^{\infty} \frac{\eta^k}{k!} h_k(y),$$ and each $h_k$ recursively built using Bell polynomials in previous $h_j$ (Liang, 10 Dec 2025). The recursion's solution ensures that, to each truncation order, the map $T_K$ matches the smooth moments and density up to $O(\sigma^{2(K+1)})$, with the error terms made explicit by the structure of the recursions.

4. Construction, Solution, and Computational Aspects

The Bell polynomial recursion enables a systematic inductive computation of the map coefficients. For example, to compute $h_k$:

1. All terms involving $\ell = 0, j=1$ are separated to isolate $h_k$.
2. All remaining terms are expressed as combinations of derivatives $G^{(m)}(y)$ and previously determined $h_{j < k}$, themselves Bell polynomials in the lower-order scores.
3. The solution for $h_k(y)$ is

$$h_k(y) = -\frac{1}{G^{(1)}(y)} \left\{ \sum_{\ell=1}^{k-1} (-1)^\ell\frac{k!}{(k-\ell)!\,\ell!} \sum_{j=1}^{k-\ell} G^{(2\ell+j)}(y) B_{k-\ell,j}(h_1, ..., h_{k-\ell-j+1}) + (-1)^k G^{(2k)}(y) \right\}.$$

This produces explicit symbolic formulas or computational graphs for practical evaluation. All derivatives can be computed from data using score-matching or kernel estimation (Liang, 10 Dec 2025).

5. Statistical Interpretation and Estimation

The utility of the Bell recursion is statistical: each higher-order correction corresponds to improved control on the Wasserstein-$r$ distance between the push-forward law $T_K \sharp Q$ and the unknown clean law $P$, yielding exponentially decaying error in the expansion order: $$W_r(T_K \sharp Q, P) \precsim \eta^{K+1} = O(\sigma^{2(K+1)}).$$ Empirical implementation only requires accurate nonparametric estimation of the score sequence $\{s_m\}$ up to order $2K-1$. Both kernel-based plug-in estimators and direct higher-order score-matching estimators (with optimal rates under smoothness constraints) can supply the necessary ingredients for high-order Bell-recursive denoising in practice (Liang, 10 Dec 2025).

6. Connections to Distributional Shrinkage and Related Hierarchies

The Bell polynomial recursion fundamentally underpins the new hierarchy of distributional shrinkage denoisers, providing the algebraic foundation for the progression beyond classical Tweedie (empirical Bayes) denoisers or first-order optimal transport maps. The hierarchy can be recapitulated as:

Level	Denoiser $T_K$	Score Functions Needed	Asymptotic Error
0	Identity	none	$O(1)$
1	$T_1(y) = y+\eta s_1(y)$	first derivative	$O(\eta^2)$
$K$	Bell-recursive OT map	up to order $2K-1$	$O(\eta^{K+1})$
$\infty$	Full OT map	all orders	Exact transport $T_\infty$

This framework has broader implications in constructing functionally agnostic, theoretically optimal denoisers with sharp control over the target distribution at all levels of approximation (Liang, 12 Nov 2025, Liang, 10 Dec 2025).

7. Implications for Theory and Practice

Bell polynomial recursions represent more than a technical mechanism—they define the pathway from classical pointwise and plug-in denoising methods to fully distributional, nonparametric, and structure-exploiting approaches. Their inductive and hierarchical algebraic assembly indexes the entire optimal transport family of denoisers by powers of the noise, systematically correcting for higher-order biases. This approach enables provable, explicit error rates in recovering the latent distribution and offers concrete, data-driven algorithms for constructing denoisers entirely from noisy samples, with statistical guarantees dictated by the recursion's order.