Heterogeneous Normalization Module (HNM)

• HNM is a scale-equivariant normalization mechanism designed to decouple internal feature representations from spatially variant noise amplitudes in blind image denoising.
• It combines a Constant Scaling branch for static variance control with a Normalized Self-Modulator for adaptive token-wise correction.
• Empirical ablations demonstrate that HNM outperforms conventional normalization layers, ensuring stable feature distributions under heterogeneous noise conditions.

Searching arXiv for the cited papers to ground the article in current literature. The Heterogeneous Normalization Module (HNM) is a normalization mechanism introduced for robust blind image denoising under out-of-distribution conditions in which test-time noise is spatially non-uniform, even though training is performed on spatially uniform noise. In "Towards Robust Image Denoising with Scale Equivariance" (Zhang et al., 5 Aug 2025), HNM is not presented as a generic stabilization layer, but as a scale-equivariant feature regulation module whose purpose is to decouple internal representations from the local noise amplitude map. Within that framework, HNM is one of the two key architectural components, alongside the Interactive Gating Module (IGM), and its role is to preserve first-order homogeneity while stabilizing and correcting feature distributions under varying noise intensities.

1. Definition and problem setting

In the formulation of blind denoising used in (Zhang et al., 5 Aug 2025), training commonly assumes

$Y = X + \lambda \cdot N,$

with a scalar noise level $\lambda$. The out-of-distribution setting of interest instead involves a spatially varying map $\Lambda$, so corruption becomes position-dependent. The paper argues that standard denoisers tend to entangle content features with noise magnitude: under uniform training noise this can remain tolerable, but under spatially variant test noise the internal feature distributions become inconsistent across locations and the learned mapping breaks down (Zhang et al., 5 Aug 2025).

HNM is introduced precisely to address that mismatch. Its function is to regulate intermediate feature maps so that the network can better transfer from a global training scale to spatially varying local scales at inference. The module is therefore tied to the paper’s broader claim that out-of-distribution denoising robustness can be improved by enforcing first-order homogeneity, or scale equivariance, throughout the network.

This setting distinguishes HNM from normalization layers whose primary role is optimization convenience. The paper explicitly states that BatchNorm/LayerNorm-type normalization is scale-independent and can therefore “inadvertently amplify the entanglement between feature maps and noise levels” (Zhang et al., 5 Aug 2025). A central implication is that the normalization problem here is not only one of variance control, but of maintaining a feature geometry compatible with local noise-amplitude changes.

2. Scale equivariance as the organizing principle

The paper formalizes its target inductive bias through the condition

$\mathcal{F}(kI)=k\mathcal{F}(I),$

for a denoiser $\mathcal F$. In the scalar-noise case, the argument is expressed as

$$\mathcal{F} (X+\lambda\cdot N)=\lambda\cdot \mathcal{F} \left(\frac{X}{\lambda }+N\right)\stackrel{\text{target}{\longrightarrow} } X,$$

and for spatially variant noise as

$$\mathcal{F}(X+\Lambda \odot \tilde{N})=\Lambda \odot \mathcal{F}\left(\frac{X}{\Lambda}+\tilde{N}\right)\stackrel{\text{target}{\longrightarrow} } X.$$

The intended consequence is that internal processing remains structurally consistent after “factoring out” the noise level (Zhang et al., 5 Aug 2025).

HNM is designed in direct response to this requirement. The module is meant to preserve scaling behavior while reducing feature-distribution instability caused by heterogeneous noise. This suggests that HNM should be understood less as a replacement for ordinary normalization in the abstract, and more as a task-specific normalization design for a scale-equivariant denoiser.

The paper’s critique of conventional normalization follows from this same principle. If normalization is scale-independent, then feature amplitudes can be altered in a way that breaks the correspondence between global training-scale variation and local test-scale variation. The LayerNorm ablation is the main explicit experimental support for that claim: replacing HNM with LN substantially degrades OOD denoising performance, which the paper interprets as evidence that scale-independent normalization prevents accurate equivariance (Zhang et al., 5 Aug 2025).

3. Architectural composition: Constant Scaling and Normalized Self-Modulator

HNM consists of two submodules: Constant Scaling (CS) and the Normalized Self-Modulator (NSM). The feature tensor is denoted

$F \in \mathbb{R}^{c \times h \times w},$

and the feature channels are partitioned into two subsets. One subset is processed by CS and the other by NSM; the outputs are then recombined, with the text and figure strongly implying a channel-wise split followed by recombination into a tensor with the same spatial dimensions and channel count as the input (Zhang et al., 5 Aug 2025).

The CS branch is the static stabilizer: $\text{CS}(F)=\frac{F}{\sqrt c}\odot \eta,$ where $\eta\in\mathbb{R}^c$ is a learnable per-channel parameter initialized to $\lambda$0. CS does not compute sample-dependent mean or variance. Instead, it applies a constant normalization factor $\lambda$1 and then rescales channels with the learned amplitudes $\lambda$2. The rationale given in the paper is that this reduces variance growth without introducing scale distortion; because $\lambda$3 is input-independent, CS preserves first-order homogeneity (Zhang et al., 5 Aug 2025).

The NSM branch is the adaptive component. The main-text NSM equation in the manuscript is described as malformed, but the appendix gives the operative structure more coherently. For a per-pixel feature vector $\lambda$4, the appendix analyzes

$\lambda$5

$\lambda$6

and

$\lambda$7

The coherent appendix form is therefore

$\lambda$8

with $\lambda$9 produced by a $\Lambda$0 convolution of the original feature $\Lambda$1 (Zhang et al., 5 Aug 2025).

The symbols are specified as follows. $\Lambda$2 and $\Lambda$3 are per-token mean and variance computed across channels for each spatial location. $\Lambda$4 are dynamic affine parameters generated by a $\Lambda$5 convolution of $\Lambda$6. The paper explicitly states that $\Lambda$7 and $\Lambda$8 are first-order homogeneous because they are produced by a linear transform of the input (Zhang et al., 5 Aug 2025).

Operationally, CS and NSM divide labor. CS gives lightweight global variance control. NSM performs “temporary local normalization, followed by an adaptive rescaling,” using affine parameters derived from the token’s original pre-normalized state. The heterogeneity in HNM therefore refers to heterogeneous processing across channel groups: one part of the representation receives static stabilization, and the other receives dynamic pixel-wise correction (Zhang et al., 5 Aug 2025).

4. Mechanism, blind operation, and relation to IGM

The module is blind: it does not use a supplied noise level map and is not a separate explicit noise-map branch. Instead, it infers the needed correction from the feature tensor itself (Zhang et al., 5 Aug 2025). This point is important because HNM is designed for the transfer problem from training on uniform noise to testing on spatially non-uniform noise without requiring explicit local noise supervision.

Mechanistically, the paper attributes two complementary effects to HNM. First, CS provides deterministic amplitude regularization in settings where feature amplitudes may grow with depth, especially when standard normalization is avoided to preserve scale equivariance. Second, NSM addresses localized distortions caused by heterogeneous noise intensities across spatial positions. If one region has stronger noise than another, the corresponding token-wise feature distributions are distorted differently; NSM temporarily normalizes each token to remove local amplitude variability, then restores useful magnitude information through self-generated affine parameters derived from the original feature (Zhang et al., 5 Aug 2025).

The derivation of those affine parameters from the original rather than normalized feature is treated as a crucial design decision. In ablation setting (1h), the paper evaluates a “Scale-independent affine” alternative that generates NSM affine parameters from normalized features instead of original features. The reported Kodak24 Speckle $\Lambda$9 result drops from $\mathcal{F}(kI)=k\mathcal{F}(I),$0 for the full model to $\mathcal{F}(kI)=k\mathcal{F}(I),$1, with similar degradations for Poisson, Mixture, and spatially variant Speckle variants. The paper interprets this as evidence that deriving affine parameters from normalized features weakens scale equivariance because the modulation then depends on a scale-independent representation rather than the original feature magnitude (Zhang et al., 5 Aug 2025).

HNM is closely coupled to the Interactive Gating Module. The paper positions HNM as the normalization and stabilization mechanism, and IGM as the expressive scale-equivariant nonlinearity. Their complementarity is reflected in the activation ablations. With HNM present, replacing IGM with ELU or GELU yields $\mathcal{F}(kI)=k\mathcal{F}(I),$2 and $\mathcal{F}(kI)=k\mathcal{F}(I),$3, only modestly below the full model’s $\mathcal{F}(kI)=k\mathcal{F}(I),$4. Without HNM, the same replacements produce $\mathcal{F}(kI)=k\mathcal{F}(I),$5 and $\mathcal{F}(kI)=k\mathcal{F}(I),$6, respectively. The authors explicitly state that this suggests HNM “shifts features to intervals where the exponential effect is less pronounced,” indicating that HNM shapes feature distributions rather than merely normalizing them passively (Zhang et al., 5 Aug 2025).

5. Empirical evidence and ablation results

The most direct evidence for HNM comes from the ablations on Kodak24 OOD denoising. The full model reports the following results: Speckle $\mathcal{F}(kI)=k\mathcal{F}(I),$7, Poisson $\mathcal{F}(kI)=k\mathcal{F}(I),$8, Mixture $\mathcal{F}(kI)=k\mathcal{F}(I),$9, Variant 1 $\mathcal F$0, Variant 2 $\mathcal F$1, and Variant 3 $\mathcal F$2 (Zhang et al., 5 Aug 2025).

The paper compares this against several HNM-related variants:

Setting	Interpretation	Kodak24 OOD trend
Full model (1a)	HNM with CS and NSM	Best across all listed OOD settings
HNM $\mathcal F$3 Full CS (1b)	Replace heterogeneity with CS everywhere	Consistently lower
w/ CS + w/o NSM (1c)	Remove adaptive NSM branch	Consistently lower
Scale-independent affine (1h)	Generate affine parameters from normalized features	Larger degradation

The reported numbers support three claims made explicitly in the paper. First, the heterogeneous combination matters: using CS everywhere is worse than combining CS and NSM. Second, NSM contributes beyond static scaling. Third, preserving scale-equivariant affine generation in NSM matters, with the “Scale-independent affine” alternative producing the clearest degradation (Zhang et al., 5 Aug 2025).

LayerNorm ablations provide a second line of evidence. In the IGM section, the configuration “HNM $\mathcal F$4 LN, w/ IGM (2d)” gives $\mathcal F$5, $\mathcal F$6, and $\mathcal F$7 on the listed OOD settings, clearly below the full HNM model. The paper does not experimentally compare HNM to BatchNorm, InstanceNorm, GroupNorm, AdaIN, or SPADE; it discusses “conventional normalization schemes (e.g., BatchNorm, LayerNorm)” at a high level and performs explicit replacement mainly with LayerNorm (Zhang et al., 5 Aug 2025).

The paper also presents feature-visualization evidence. It shows that the scale-equivariant network produces highly similar final decoder features under different corruption degrees and level maps, whereas designs with LN or exponential activations produce dissimilar features. The visualization is not isolated to HNM alone, but HNM is identified as one of the main modules responsible for that stability (Zhang et al., 5 Aug 2025).

6. Related concepts, interpretive boundaries, and limitations

Within the denoising paper, HNM is a specific module recurring inside SEVNet rather than a generic term for all adaptive normalization. The exact number of HNM insertions, exact stage-by-stage placement, and exact channel split ratio are not fully specified. What is clear is that HNM acts on intermediate feature maps, not on the raw image, and operates in the main feature path of the denoiser (Zhang et al., 5 Aug 2025).

Several nearby works illuminate broader interpretations of heterogeneous normalization. "Attentive Normalization" (Li et al., 2019) provides a conceptually related but structurally different mechanism: it keeps the standardization backbone fixed while replacing a single affine transform with a mixture of affine experts under input-conditioned routing. In that sense, heterogeneity is located in the affine recalibration rather than in the split CS/NSM processing used by HNM. "Adaptive Batch Normalization for Training Data with Heterogeneous Features" (Alsobhi et al., 2022) offers a much coarser, threshold-based batch-level gate that turns BN on or off depending on whether average feature values fall outside class-dependent ranges. "Adaptive Normalization Mamba with Multi Scale Trend Decomposition and Patch MoE Encoding" (Jeon, 7 Dec 2025) uses adaptive multi-scale detrending and channel-wise recalibrated instance-style standardization for time-series forecasting, again showing that “heterogeneous normalization” can refer to heterogeneous conditioning signals rather than to a mixture of classical normalizers.

These related works clarify a common misconception: HNM in (Zhang et al., 5 Aug 2025) is not a general umbrella term for any adaptive normalization layer. It denotes a particular scale-equivariant module composed of CS and NSM for robust blind denoising under spatially heterogeneous noise. Its heterogeneity is architectural—different channel groups receive different treatments—while its motivation is the handling of heterogeneous noise intensity in the data.

The paper also leaves several caveats. The main-text NSM equation is inconsistent or malformed, whereas the appendix gives the more coherent affine-normalization form and is therefore the better source for the intended operation. The exact channel partition ratio is not stated. HNM can partially suppress the adverse effects of non-equivariant operations, but does not eliminate them, as shown by the ELU and GELU ablations. Finally, HNM is not presented as a complete solution for all real-noise phenomena: the real-world extension of the denoiser still requires an external noise translator because the scale-equivariant denoiser assumes independently distributed noise, whereas real camera noise is spatially correlated (Zhang et al., 5 Aug 2025).

Taken together, these points support a precise characterization. HNM is the scale-equivariant normalization block of SEVNet: it mixes a static branch,

$\mathcal F$8

with a dynamic branch that normalizes each token and restores content-dependent magnitude through affine parameters $\mathcal F$9 generated from the original feature. By splitting channels between these two treatments, the module stabilizes feature statistics while allowing local pixel-wise correction. In the paper’s argument, that design is what allows denoising features learned under uniform noise to remain more stable when test noise becomes spatially non-uniform (Zhang et al., 5 Aug 2025).