Hierarchy of Agnostic Denoisers

• Hierarchy of agnostic denoisers is a structured framework that organizes denoising strategies without requiring detailed signal or noise distribution knowledge.
• It leverages score-based expansions and optimal transport theory to progressively refine recovery accuracy through higher-order derivatives.
• Practical applications integrate classical statistical techniques with modern deep learning, including SURE training and invertible architectures, for robust performance.

A hierarchy of agnostic denoisers refers to a principled organization of denoising strategies that do not assume knowledge of the underlying signal or noise laws and instead leverage structural or distributional properties of the observed noisy data. These hierarchies arise in both classical statistical signal recovery—often through expansions in score functions and optimal transport theory—as well as in modern learning-based and invertible architectures, deep unsupervised and self-supervised learning regimes, and ensemble aggregation. The hierarchy formalizes a progression from pointwise (per-sample) error minimization toward distribution-level fidelity under minimal modeling assumptions.

1. Formal Foundations and Core Definitions

Agnostic denoisers operate in statistical settings where only minimal information is available: typically the noise level (e.g., variance $\sigma^2$), with neither the signal law $P_X$ nor potentially even the noise distribution specified. The observed data follow $Y = X + \sigma Z$, with $Z$ independent and standardized. The principal goal is to construct a mapping $T(Y)$ so that the pushforward law $T\sharp P_Y$ provides an accurate proxy for $P_X$ according to global metrics (moments, densities, or Wasserstein distance), not merely to minimize $E[\|T(Y) - X\|^2]$ pointwise.

The canonical hierarchy consists of:

Denoiser in Hierarchy	Formula	Moment/density accuracy
Bayes-optimal (Tweedie)	$T^*(y) = y + \sigma^2 \nabla \log q(y)$	Exact on mean only, $O(\sigma^2)$ error on higher moments
First-order (OT)	$T_1(y) = y + \frac{\sigma^2}{2}\nabla \log q(y)$	$O(\sigma^4)$ in moments/density (for smooth test $m$)
Second-order	$$T_2(y) = y + \frac{\sigma^2}{2}\nabla\log q(y) - \frac{\sigma^4}{8}\nabla\left[\frac{1}{2}\|\nabla \log q(y)\|^2 + \Delta \log q(y)\right]$$	$O(\sigma^6)$ in moments/density (smooth $m$)

The hierarchy can be extended further in one dimension with the sequence $T_0, T_1, ..., T_K, ..., T_\infty$, where $T_K$ is constructed using higher-order derivatives (“higher-order scores”) of $q$ and achieves $O(\sigma^{2K + 2})$ accuracy in Wasserstein distance, converging to the order-preserving optimal transport map at $T_\infty$ (Liang, 12 Nov 2025, Liang, 10 Dec 2025).

2. Derivation, Theoretical Guarantees, and Shrinkage Behavior

Tweedie's formula ($T^*$) is optimal for mean-squared error under a correct Gaussian noise model but systematically “over-shrinks” the output distribution, producing overly narrow posteriors and underestimating variance by $\Theta(\sigma^2)$. Denoisers in the hierarchy are derived as power-series expansions (in $\sigma^2/2$) of the optimal transport map from $P_Y$ to $P_X$, where each additional term incorporates increasingly global (distributional) features via higher-order derivatives of $\log q(y)$.

• $T_1$ is the first-order correction term, directly interpretable as the midpoint between no denoising and the Tweedie map, and equivalent to a first-order approximation of the Monge–Ampère equation link between $P_Y$ and $P_X$. Under $C^2$ smoothness and $4$th-moment noise, $T_1$ matches second moments to $O(\sigma^4)$ and achieves $O(\sigma^4)$ IPM/density error.
• $T_2$ adds a second-order term, further minimizing distributional mismatch down to $O(\sigma^6)$ provided $C^3$ smoothness and bounded $6$th moments, but with increased sensitivity to estimation error in third derivatives.

This sequence generalizes in one dimension to arbitrary order $K$ via Bell polynomial recursions on higher-order scores (derivatives of $\log q$), with each truncation $T_K$ achieving $O(\sigma^{2K + 2})$ pushforward error in Wasserstein distance (Liang, 10 Dec 2025). The limiting map $T_\infty(y) = F^{-1}(G(y))$ solves the exact optimal transport from $P_Y$ to $P_X$.

3. Score-Based Estimation and Implementation

Denoisers in this hierarchy rely on accurate estimation of score functions $\nabla^j \log q(y)$ for $j \leq 2K-1$. Two main approaches are:

• Parametric/nonparametric score matching: Minimizing Fisher divergence to estimate $\nabla \log q(y)$ (Liang, 12 Nov 2025).
• Higher-order score estimation via plug-in (kernel smoothing): Using derivative-KDEs to form $\hat{q}^{(m)}(y)$ and hence $\nabla^m \log q(y)$. Bias–variance trade-offs are controlled via bandwidth selection $b\asymp n^{-1/(2m+5)}$.
• Direct higher-order score matching: Minimize an objective $$\frac{1}{n} \sum_{i=1}^n [\frac{1}{2}f(Y_i)^2 + (-1)^{m+1}f^{(m)}(Y_i)]$$ over a function class, recovering $f^*_m(y) = q^{(m)}(y)/q(y)$ with convergence rate depending on the smoothness of $q$ (Liang, 10 Dec 2025).

Automatic differentiation in deep learning platforms enables practical computation of gradient, Hessian, and Laplacian for first and second-order denoisers, albeit with computational cost dominated by Jacobian/Laplacian evaluation rather than by forward inference.

4. Hierarchies from Model-Agnostic and Distribution-Agnostic Learning

Beyond classical score-based hierarchies, broader classes of agnostic denoiser hierarchies arise in modern learning settings:

• SURE-trained deep denoisers: Stein’s unbiased risk estimator (SURE) allows training neural denoisers without clean data, placing SURE-trained DNNs in a practical hierarchy below fully supervised denoisers and above methods like Noise2Noise, Noise2Void, and Deep Image Prior in terms of agnosticism and statistical efficiency. SURE-training achieves within $0.2$–$0.5$ dB of supervised DNNs and outperforms classical pointwise denoisers like BM3D, while requiring only knowledge of the noise parameters (Soltanayev et al., 2018).
• Hierarchical aggregation and universal combination: “Consensus Neural Network” (CsNet) and similar convex-fusion/booster pipelines offer a hierarchical structure for aggregating the outputs of multiple diverse denoisers (CNNs, classical, or hybrid). Each stage constructs convex combinations, with weights determined by learned or unsupervised MSE estimators, and subsequent “booster” nets further refine the reconstruction. The two-stage (or multi-stage) cascade can itself be repeated, forming an agnostic ensemble hierarchy which consistently improves over the best constituent and single deep baselines (Choi et al., 2017).
• Universal master denoisers via loss estimation: Hierarchical combination trees based on unbiased loss estimation can asymptotically track the best performing member of a candidate family of arbitrary denoisers, under constraints on dependency structure or via output randomization (for instance, randomized smoothing in the binary symmetric channel). This perspective yields scalable, universal agnostic denoising even when base denoisers exhibit complex dependencies (Ordentlich, 2020).

5. Hierarchical and Disentangled Deep Architectures

Invertible architectures and Bayesian hierarchical deep models provide a different instance of agnostic denoiser hierarchies.

• Hierarchical disentangled invertible denoisers: Hierarchical invertible neural networks leveraging normalizing flows decompose noisy observations into multi-scale low- and high-frequency components via wavelet transforms, then disentangle noise from signal in latent space in a coarse-to-fine hierarchy. Each level splits off additional structured noise without assuming noise form, and the final reconstruction inverts the flow after zeroing the “noise” subspace. No prior assumptions on the noise distribution are imposed; agnosticism is achieved via explicit distributional regularization on the latent variables (Du et al., 2023).
• Bayesian hierarchical variational models: Variational Deep Image Denoising interprets the denoising process as a two-tier Bayesian hierarchy. Latent variables $c$ encode both degradation and content; the denoiser is conditioned on $c$, itself inferred in an unsupervised manner from observed $y$. This splits the original blind denoising problem into variationally tractable sub-tasks, hierarchically handling different “clusters” of noisy observations through adaptation of the deep architecture, all without explicit noise-level input (Soh et al., 2021).

6. Applications, Practical Guidance, and Limitations

Selection among denoisers in the agnostic hierarchy is application-driven:

• For per-sample MSE minimization, use $T^*$-type (Tweedie) mappings or highly optimized supervised deep models.
• For accurate recovery of $P_X$ at the distribution level—important in downstream generative tasks, uncertainty quantification, and multimodal recovery—prefer higher-order or OT-based denoisers ($T_1, T_2, T_K$), or hierarchical/invertible architectures that capture non-Gaussian statistics and realistic signal properties.
• SURE and MSE-ensemble methods provide high practical performance when labeled data are scarce and multiple denoiser types are available.
• Randomization and hierarchical aggregation procedures guarantee universality in tracking the best candidate denoiser where dependency or channel knowledge is limited.

Limitations arise at higher orders due to the increased sensitivity to estimation error in high-order derivatives, the curse of dimensionality in nonparametric scores, and computational complexity in hierarchical architectures. In practice, $T_1$ (first-order OT) and shallow hierarchical deep ensembles often offer optimal trade-offs between fidelity, stability, and computational cost (Liang, 12 Nov 2025, Liang, 10 Dec 2025, Choi et al., 2017).

7. Outlook and Connections Across Research Paradigms

The hierarchy of agnostic denoisers formalizes the transition from pointwise, model-based denoising to distribution-level, model-agnostic inference. This connects optimal transport theory, empirical Bayes, deep unsupervised learning, and universality in estimator aggregation. The approach is deeply tied to the operational content of higher-order score functions, with Bell-polynomial recursions in the 1D setting and OT-inspired PDE expansions in higher dimensions. Agnostic hierarchies are statistically optimal under minimal assumptions and are consistent with modern requirements in robust unsupervised learning, distribution matching, and generative modeling.

Recent research demonstrates that all information necessary for distribution-level signal recovery from noisy data is encoded in the sequence of higher-order score functions of the noisy observation law $P_Y$; denoisers agnostic to the true signal law $P_X$ can, through hierarchical expansions, approach arbitrarily high accuracy in both theory and practice (Liang, 10 Dec 2025, Liang, 12 Nov 2025).