Adaptive Noise Filtering (ANF)

• Adaptive Noise Filtering is a family of techniques that update filter coefficients, weights, or noise models from observed data, rather than relying on fixed denoisers.
• Key mechanisms include explicit coefficient adaptation, dynamic relevance adjustment, and online noise model estimation, which improve performance in signal reconstruction and control.
• ANF is applied in domains such as time-domain spectroscopy, active noise control, image restoration, SLAM, and quantum error mitigation, demonstrating robust performance even under extreme noise conditions.

Adaptive Noise Filtering (ANF) denotes a family of adaptive procedures in which noise suppression is achieved by updating a filter, weighting rule, structural sparsity pattern, or noise model from observed data rather than by applying a fixed denoiser. In the cited literature, the term appears in time-domain impedance and admittance spectroscopy, active noise control, image restoration, multimodal representation learning, deep reinforcement learning, LiDAR SLAM, quantum error mitigation, and adaptive state estimation. The shared objective is to separate signal components correlated with the excitation, task, or dynamics from uncorrelated or weakly informative disturbances; the adaptive variable, however, ranges from filter coefficients and gain matrices to region-attention weights, sparse input connectivity, and process-noise covariances (Stupin et al., 2017, Luo et al., 2022, Grooten et al., 2023, Li et al., 5 Jan 2026, Stacey et al., 2019, Koenig et al., 7 May 2025).

1. Terminological scope and recurring mechanisms

The cited literature does not use ANF for a single standardized algorithm. In classical DSP and control, it usually refers to adaptive filters that update coefficients online to reduce residual error, as in FxNLMS, correntropy-based filters, or fractional-order subband algorithms. In inverse problems and sensing, it can mean adaptive identification of a system spectrum from time-domain data. In recent ML systems, it often denotes front-end feature suppression, attention-based reweighting, sparse connectivity adaptation, or outlier filtering before downstream reasoning or extrapolation (Luo et al., 2022, Stupin et al., 2017, Li et al., 5 Jan 2026, Grooten et al., 2023).

Three recurrent mechanisms are visible across these usages. First, ANF may adapt coefficients in an explicit signal model, such as FIR, IIR, semi-IIR, filtered-x, proportionate, or subband structures. Second, it may adapt relevance weights or connectivity so that nuisance dimensions are suppressed before fusion, matching, or policy learning. Third, it may adapt the assumed noise law itself, for example by estimating process-noise covariance online, choosing noise-amplification factors from measured error strength, or regularizing clutter covariance for detection (Gogineni et al., 2017, Koenig et al., 7 May 2025, Stacey et al., 2019, Kammoun et al., 2015).

A common misconception is that ANF always means coefficient adaptation in a classical filter. The cited work shows a broader pattern: GonF in continual VQA is a soft attention-based, denoising front end rather than a binary keep/remove operator; ANF in deep RL is dynamic sparse training restricted to the input layer; and adaptive folding in quantum mitigation filters measurements or entire runs before extrapolation rather than denoising a waveform directly (Li et al., 5 Jan 2026, Grooten et al., 2023, Koenig et al., 7 May 2025).

2. Time-domain adaptive identification in impedance and admittance spectroscopy

A particularly explicit ANF formulation appears in time-domain impedance spectroscopy, where the measured excitation voltage sequence $V_k$ and current response sequence $J_k$ are used to train an adaptive filter instead of taking FFT or DFT images directly. The adaptive predictor is

$$J_k^{a}=\sum_{j=1}^{\ell_d} d_j J_{k-j}^{a}+\sum_{j=0}^{\ell_n} n_j V_{k-j},$$

and the coefficients are obtained by minimizing the squared mismatch between measured and predicted current. For the semi-IIR form used in the paper, the minimization is

$$\sum_{k=\max(\ell_n,\ell_d)}^L\left|J_k-\sum_{j=1}^{\ell_d} d_j J_{k-j}-\sum_{j=0}^{\ell_n} n_j V_{k-j}\right|^2=\min.$$

Using Parseval’s identity, the paper shows that this learning problem is equivalent to fitting the admittance spectrum in the frequency domain, with frequency resolution $\Delta f = 1/T$, and derives the central reconstruction formula

$$Y \approx\left(\sum_{j=0}^{\ell_n} n_j \varphi_j \right)\Bigg/\left( 1-\sum\limits_{j=1}^{\ell_d} d_j \varphi_j\right),$$

so that the complex admittance is encoded directly in the learned adaptive weights (Stupin et al., 2017).

The same paper provides a noise-canceling argument in a Toeplitz least-squares formulation. When additive noise is uncorrelated with the excitation and response, perturbation cross-terms vanish in the normal equations, and the learned coefficients are only weakly perturbed, with

$$\frac{|\vec{w}-\vec{w}^*|}{|\vec{w}|}\le \frac{\langle \varepsilon^2\rangle \nu}{A}, \qquad \mathrm{RMSE}\le \frac{\langle \varepsilon^2\rangle \nu}{A\sqrt{\ell_n+\ell_d+1}}.$$

The important distinction from FFT-based processing is that the adaptive filter is not merely transforming noisy data; it is learning the excitation–response relation while rejecting uncorrelated noise (Stupin et al., 2017).

The experimental RLC validation used a linear sweep excitation of 20 mV peak-to-peak, a 10 Hz to 40 kHz range, 500 ms sweep time, 500 kHz sampling, about 250,000 samples, and a semi-IIR model with $\ell_d=49$ and $\ell_n=101$. At 30 dB SNR, both FFT and AF recovered values close to the reference, including $R \approx 256.7\,\Omega$, $J_k$0, and $J_k$1. At 3 dB SNR, AF still yielded $J_k$2, $J_k$3, and $J_k$4, whereas FFT-CNLS became unreliable with $J_k$5, $J_k$6, and $J_k$7. The same study reports that FFT fails for experimental data at around 5 dB SNR, and for artificial digital noise at 0 dB, while AF remains accurate even in these conditions. In living-cell impedance measurements on HeLa cells, FFT-derived spectra were strongly distorted by microscope-related interference, whereas ANF produced cleaner spectra and Nyquist arcs consistent with the Giaever–Keese cell model (Stupin et al., 2017).

3. Active noise control and adaptive cancellation

In active noise control (ANC), ANF typically means learning a control filter that generates anti-noise after the secondary path. The hybrid SFANC-FxNLMS framework is representative: SFANC performs frame-level selection of a pre-trained control filter by a lightweight 1D CNN, and FxNLMS then refines that selected filter at the sampling rate. The problem is formulated through the residual

$J_k$8

with FxNLMS updating coefficients continuously, whereas SFANC treats controller choice as a classification problem over a bank $J_k$9. In the reported implementation, the primary noise is sampled at $$J_k^{a}=\sum_{j=1}^{\ell_d} d_j J_{k-j}^{a}+\sum_{j=0}^{\ell_n} n_j V_{k-j},$$0, each frame contains 16,000 samples, the CNN runs on a co-processor once per frame, and the real-time controller updates the chosen filter at the sampling rate. The network achieves 99.35% test accuracy with 0.21M parameters. During the $$J_k^{a}=\sum_{j=1}^{\ell_d} d_j J_{k-j}^{a}+\sum_{j=0}^{\ell_n} n_j V_{k-j},$$1 transition period on cascaded noise, the hybrid method is 6 dB higher than SFANC and 15 dB higher than FxNLMS in averaged noise reduction; on mixed noise it reaches about 18 dB after 1 s, whereas FxNLMS needs more than 10 s to reach that level, and in the $$J_k^{a}=\sum_{j=1}^{\ell_d} d_j J_{k-j}^{a}+\sum_{j=0}^{\ell_n} n_j V_{k-j},$$2 interval the hybrid is about 10 dB higher than both SFANC and FxNLMS (Luo et al., 2022).

Robust filtered-x methods generalize this idea to impulsive, non-Gaussian environments. FXHEKM introduces a hyperbolic tangent exponential kernel M-estimate objective with normalized update

$$J_k^{a}=\sum_{j=1}^{\ell_d} d_j J_{k-j}^{a}+\sum_{j=0}^{\ell_n} n_j V_{k-j},$$3

where the gate

$$J_k^{a}=\sum_{j=1}^{\ell_d} d_j J_{k-j}^{a}+\sum_{j=0}^{\ell_n} n_j V_{k-j},$$4

suppresses updates under large outliers. The paper reports that FXHEKM reaches about $$J_k^{a}=\sum_{j=1}^{\ell_d} d_j J_{k-j}^{a}+\sum_{j=0}^{\ell_n} n_j V_{k-j},$$5 dB mean ANR under Gaussian noise, outperforms FXGR by more than 3 dB up to around $$J_k^{a}=\sum_{j=1}^{\ell_d} d_j J_{k-j}^{a}+\sum_{j=0}^{\ell_n} n_j V_{k-j},$$6 iterations under impulsive conditions, and remains convergent even at $$J_k^{a}=\sum_{j=1}^{\ell_d} d_j J_{k-j}^{a}+\sum_{j=0}^{\ell_n} n_j V_{k-j},$$7, where impulsiveness is severe (Hermont et al., 18 Aug 2025).

A related line replaces quadratic or first-order descent by p-norm and fractional-order subband adaptation. FoNSPN requires the admissible fractional range

$$J_k^{a}=\sum_{j=1}^{\ell_d} d_j J_{k-j}^{a}+\sum_{j=0}^{\ell_n} n_j V_{k-j},$$8

and improves robustness when both inputs and disturbances are impulsive. NKP-FoNSPN and TNKP-FoNSPN combine this fractional-order update with nearest Kronecker product structure and a transformation-based switching strategy; the filtered-x variants, especially TNKP-FxFoNSPN, are reported to outperform FxLMS, FxFoNLMP, FxGMCC, FxAPLEHS, FxFoNSPN, and NKP-FxFoNSPN in simulated and real ANC, including a real single-channel duct experiment and a simulated multi-channel ANC system (Ye et al., 15 Jan 2026, Ye et al., 18 May 2026).

Classical speech-enhancement and noise-cancellation studies form an important historical backdrop. In one comparison, FEDS and FAP were introduced as low-complexity alternatives with reported SNRI values of 22.2623 dB and 24.9078 dB, versus 13.5905 dB for LMS, 16.8679 dB for NLMS, 20.0307 dB for AP, and 29.7355 dB for RLS (Hadei et al., 2011). In another, GAL was reported as best overall in noise cancellation in terms of correlation coefficient, SNR, and convergence time when compared with RLS and FTRLS, even though RLS often converged fastest (Ferdouse et al., 2011). Sparse-system identification under impulsive noise led to IP-MCC, which combines maximum correntropy with improved proportionate gains; its steady-state EMSE is reported as similar to MCC, while at $$J_k^{a}=\sum_{j=1}^{\ell_d} d_j J_{k-j}^{a}+\sum_{j=0}^{\ell_n} n_j V_{k-j},$$9 and $$\sum_{k=\max(\ell_n,\ell_d)}^L\left|J_k-\sum_{j=1}^{\ell_d} d_j J_{k-j}-\sum_{j=0}^{\ell_n} n_j V_{k-j}\right|^2=\min.$$0 it uses 82.6% fewer additions and 71.43% fewer multiplications than MIP-APSA (Gogineni et al., 2017).

4. Image and signal restoration

In image restoration, ANF often appears as a decision-based nonlinear filter whose behavior changes with local corruption density. One such method models the degraded image as

$$\sum_{k=\max(\ell_n,\ell_d)}^L\left|J_k-\sum_{j=1}^{\ell_d} d_j J_{k-j}-\sum_{j=0}^{\ell_n} n_j V_{k-j}\right|^2=\min.$$1

detects corrupted pixels by the extreme-value rule $$\sum_{k=\max(\ell_n,\ell_d)}^L\left|J_k-\sum_{j=1}^{\ell_d} d_j J_{k-j}-\sum_{j=0}^{\ell_n} n_j V_{k-j}\right|^2=\min.$$2 or $$\sum_{k=\max(\ell_n,\ell_d)}^L\left|J_k-\sum_{j=1}^{\ell_d} d_j J_{k-j}-\sum_{j=0}^{\ell_n} n_j V_{k-j}\right|^2=\min.$$3, and then chooses among no filtering, $$\sum_{k=\max(\ell_n,\ell_d)}^L\left|J_k-\sum_{j=1}^{\ell_d} d_j J_{k-j}-\sum_{j=0}^{\ell_n} n_j V_{k-j}\right|^2=\min.$$4 median, $$\sum_{k=\max(\ell_n,\ell_d)}^L\left|J_k-\sum_{j=1}^{\ell_d} d_j J_{k-j}-\sum_{j=0}^{\ell_n} n_j V_{k-j}\right|^2=\min.$$5 median, or the mean of uncorrupted pixels. The switching rules are explicit: if $$\sum_{k=\max(\ell_n,\ell_d)}^L\left|J_k-\sum_{j=1}^{\ell_d} d_j J_{k-j}-\sum_{j=0}^{\ell_n} n_j V_{k-j}\right|^2=\min.$$6, keep the pixel unchanged; if $$\sum_{k=\max(\ell_n,\ell_d)}^L\left|J_k-\sum_{j=1}^{\ell_d} d_j J_{k-j}-\sum_{j=0}^{\ell_n} n_j V_{k-j}\right|^2=\min.$$7, use $$\sum_{k=\max(\ell_n,\ell_d)}^L\left|J_k-\sum_{j=1}^{\ell_d} d_j J_{k-j}-\sum_{j=0}^{\ell_n} n_j V_{k-j}\right|^2=\min.$$8 median filtering; if $$\sum_{k=\max(\ell_n,\ell_d)}^L\left|J_k-\sum_{j=1}^{\ell_d} d_j J_{k-j}-\sum_{j=0}^{\ell_n} n_j V_{k-j}\right|^2=\min.$$9, use $\Delta f = 1/T$0 median filtering; if $\Delta f = 1/T$1, try $\Delta f = 1/T$2 median and, if the output is still corrupted, replace the center by the mean of uncorrupted pixels. On Lena and Goldhill corrupted by 20% impulse noise plus degradations, the reported PSNR values include 35.15 and 35.05 for the proposed filter, against 31.42 and 29.60 for SMF, and 36.54 and 34.42 for ACWMF; the paper emphasizes that the proposed algorithm is particularly effective when impulse noise is mixed with strip lines, drop lines, white bands, black bands, blotches, and band loss (Satpathy et al., 2022).

A second image-restoration variant uses type-2 fuzzy logic for adaptive thresholding. The improved adaptive type-2 fuzzy filter is explicitly two-stage: stage 1 classifies pixels as good or noisy using exactly two primary Gaussian membership functions per filter window, and stage 2 reconstructs noisy pixels by a weighted average over good pixels only. If too few good pixels are available, the half-window size $\Delta f = 1/T$3 is increased by 1 and the process repeats. The method was evaluated on $\Delta f = 1/T$4 USC-SIPI images at 20%, 50%, and 80% salt-and-pepper noise, with visual results also shown at 97% and 99%. The Friedman test yields $\Delta f = 1/T$5 in the main comparison and $\Delta f = 1/T$6 against AT2FF; the proposed method has average rank 1.20 and is reported as statistically better than FM, CEF, PWS, AMEPR, BDND, CM, and SATV, and also better than AT2FF (Singh et al., 2020).

In seismic denoising, the Adaptive Convolutional Filter (ACF) is a three-layer convolutional filter constrained by a zero-shot prior, a global variance prior, and a downsampling constraint. With $\Delta f = 1/T$7, $\Delta f = 1/T$8, and 2464 learnable parameters, it is optimized directly on the test data for 500 iterations with Adam, using learning rate $\Delta f = 1/T$9. On synthetic data it achieves SNR values of 22.86 and 21.99, versus 20.14 and 19.20 for f-x deconvolution, 18.54 and 18.56 for LDRR, and 15.04 and 15.04 for HSLR. On NPRA field lines, it removes the largest amount of noise while maintaining lower average local similarity than f-x deconvolution and LDRR, which the paper interprets as less signal leakage than those methods (Peng et al., 2024).

5. Representation-level filtering in machine learning and SLAM

Recent ML usage shifts ANF from coefficient adaptation toward representation filtering. In continual VQA, MacVQA’s Global Noise Filtering (GonF) module is inserted between the raw visual feature extractor and the multimodal encoder. It scores each region with a learned linear function and softmax normalization, aggregates a global visual vector, refines region features with a denoising autoencoder, and outputs enhanced visual features $$Y \approx\left(\sum_{j=0}^{\ell_n} n_j \varphi_j \right)\Bigg/\left( 1-\sum\limits_{j=1}^{\ell_d} d_j \varphi_j\right),$$0 for cross-modal fusion. The GonF loss combines a squared reconstruction error with an entropy-style regularizer over attention weights, and $$Y \approx\left(\sum_{j=0}^{\ell_n} n_j \varphi_j \right)\Bigg/\left( 1-\sum\limits_{j=1}^{\ell_d} d_j \varphi_j\right),$$1 is typically in $$Y \approx\left(\sum_{j=0}^{\ell_n} n_j \varphi_j \right)\Bigg/\left( 1-\sum\limits_{j=1}^{\ell_d} d_j \varphi_j\right),$$2. In ablation, the baseline without GonF or AMA has AP 38.77 and AF 2.90, GonF alone reaches AP 41.75 and AF 2.14, and full MacVQA reaches AP 43.38 and AF 2.32 on standard tasks; the abstract also reports 42.53% average accuracy and 3.60% average forgetting on novel composition tasks (Li et al., 5 Jan 2026).

In deep reinforcement learning, Automatic Noise Filtering is implemented as dynamic sparse training of the input layer only. The method targets the extremely noisy environment (ENE), where up to 99% of input features are pure noise. ANF initializes sparse input layers with sparsity $$Y \approx\left(\sum_{j=0}^{\ell_n} n_j \varphi_j \right)\Bigg/\left( 1-\sum\limits_{j=1}^{\ell_d} d_j \varphi_j\right),$$3, prunes a drop fraction $$Y \approx\left(\sum_{j=0}^{\ell_n} n_j \varphi_j \right)\Bigg/\left( 1-\sum\limits_{j=1}^{\ell_d} d_j \varphi_j\right),$$4 every $$Y \approx\left(\sum_{j=0}^{\ell_n} n_j \varphi_j \right)\Bigg/\left( 1-\sum\limits_{j=1}^{\ell_d} d_j \varphi_j\right),$$5 environment steps, and regrows the same number of random connections at zero initialization. On Humanoid-v3, ANF-TD3 reaches about 4968.3 return with 262,400 parameters, while dense TD3 reaches 817.3 with 1,032,448 parameters; in sparser configurations, 95% global sparsity still yields 4806.4 return with 51,622 parameters. The paper summarizes that ANF variants can outperform standard SAC and TD3 while using up to 95% fewer weights, and that ANF adapts faster after feature permutations in the PENE transfer-learning setting (Grooten et al., 2023).

LiDAR SLAM supplies a third formulation. ADA-DPM treats ANF as learned point filtering: a Dynamic Segmentation Head removes descriptors predicted as dynamic, a Global Importance Scoring Head down-weights weak or noisy correspondences, and GLI-GCN fuses multi-scale neighborhoods to improve descriptor discrimination before filtering. The weighted alignment term is

$$Y \approx\left(\sum_{j=0}^{\ell_n} n_j \varphi_j \right)\Bigg/\left( 1-\sum\limits_{j=1}^{\ell_d} d_j \varphi_j\right),$$6

with $$Y \approx\left(\sum_{j=0}^{\ell_n} n_j \varphi_j \right)\Bigg/\left( 1-\sum\limits_{j=1}^{\ell_d} d_j \varphi_j\right),$$7, $$Y \approx\left(\sum_{j=0}^{\ell_n} n_j \varphi_j \right)\Bigg/\left( 1-\sum\limits_{j=1}^{\ell_d} d_j \varphi_j\right),$$8, and $$Y \approx\left(\sum_{j=0}^{\ell_n} n_j \varphi_j \right)\Bigg/\left( 1-\sum\limits_{j=1}^{\ell_d} d_j \varphi_j\right),$$9. On benchmark trajectories, ADA-DPM achieves APE 0.46 on SemanticKITTI sequence 00 and 1.62 on MulRan K03, and remains usable under noise injected into up to 50% of the point cloud: mean APE increases from 3.14 without noise to 4.18 at 50% noise. The ablations attribute gains separately to dynamic-point removal, importance scoring, and GLI-GCN (Shao et al., 22 Jun 2025).

6. Adaptive noise modeling in biological and scientific inference

Some of the most conceptually distinct ANF formulations do not denoise signals directly but adapt noise models, scaling laws, or feedback dynamics. In adaptive sensory networks, high-frequency noise is filtered by output degradation through time-averaging, whereas low-frequency noise is damped by adaptation through negative feedback. For the generic three-node adaptive network, linearization gives

$$\frac{|\vec{w}-\vec{w}^*|}{|\vec{w}|}\le \frac{\langle \varepsilon^2\rangle \nu}{A}, \qquad \mathrm{RMSE}\le \frac{\langle \varepsilon^2\rangle \nu}{A\sqrt{\ell_n+\ell_d+1}}.$$0

This transfer function suppresses high-frequency input noise as $$\frac{|\vec{w}-\vec{w}^*|}{|\vec{w}|}\le \frac{\langle \varepsilon^2\rangle \nu}{A}, \qquad \mathrm{RMSE}\le \frac{\langle \varepsilon^2\rangle \nu}{A\sqrt{\ell_n+\ell_d+1}}.$$1 and low-frequency noise as $$\frac{|\vec{w}-\vec{w}^*|}{|\vec{w}|}\le \frac{\langle \varepsilon^2\rangle \nu}{A}, \qquad \mathrm{RMSE}\le \frac{\langle \varepsilon^2\rangle \nu}{A\sqrt{\ell_n+\ell_d+1}}.$$2. Applied to E. coli chemotaxis, the paper identifies CheY-P dephosphorylation, not receptor adaptation, as the Berg–Purcell time-averaging mechanism, with $$\frac{|\vec{w}-\vec{w}^*|}{|\vec{w}|}\le \frac{\langle \varepsilon^2\rangle \nu}{A}, \qquad \mathrm{RMSE}\le \frac{\langle \varepsilon^2\rangle \nu}{A\sqrt{\ell_n+\ell_d+1}}.$$3 and $$\frac{|\vec{w}-\vec{w}^*|}{|\vec{w}|}\le \frac{\langle \varepsilon^2\rangle \nu}{A}, \qquad \mathrm{RMSE}\le \frac{\langle \varepsilon^2\rangle \nu}{A\sqrt{\ell_n+\ell_d+1}}.$$4. It also identifies a previously unstudied low-frequency noise source caused by random walk in a ligand gradient, which is rendered finite only by receptor adaptation (Sartori et al., 2011).

In orbit determination, ANF appears as online process-noise covariance estimation for Kalman filtering. ASNC and ADMC fuse covariance matching with state noise compensation or dynamic model compensation, while constraining the discrete covariance to remain consistent with the underlying continuous-time stochastic acceleration model. Unlike raw covariance matching, these methods do not require ad hoc PSD repair and extrapolate correctly over outages. In a spacecraft-navigation case study around asteroid 433 Eros, CM produced 12 filter divergences in the imperfect-maneuver case, whereas ASNC, ADMC, and IMM remained consistent. In the no-maneuver case, ASNC reduced position error by 47% and velocity error by 51% versus CM. Runtime overheads were also reported: for ETS-UKF, CM adds 0.0034%, ASNC 0.01%, ADMC 47%, and IMM 100% (Stacey et al., 2019).

Quantum error mitigation supplies a measurement-filtering analogue. Noise-adaptive folding sets the maximum noise-amplification factor by the measured initial error strength,

$$\frac{|\vec{w}-\vec{w}^*|}{|\vec{w}|}\le \frac{\langle \varepsilon^2\rangle \nu}{A}, \qquad \mathrm{RMSE}\le \frac{\langle \varepsilon^2\rangle \nu}{A\sqrt{\ell_n+\ell_d+1}}.$$5

with $$\frac{|\vec{w}-\vec{w}^*|}{|\vec{w}|}\le \frac{\langle \varepsilon^2\rangle \nu}{A}, \qquad \mathrm{RMSE}\le \frac{\langle \varepsilon^2\rangle \nu}{A\sqrt{\ell_n+\ell_d+1}}.$$6, then constructs exponentially spaced target factors and filters unreliable runs or points using Gaussian-mixture criteria. Entire runs are rejected when $$\frac{|\vec{w}-\vec{w}^*|}{|\vec{w}|}\le \frac{\langle \varepsilon^2\rangle \nu}{A}, \qquad \mathrm{RMSE}\le \frac{\langle \varepsilon^2\rangle \nu}{A\sqrt{\ell_n+\ell_d+1}}.$$7; pointwise observable filtering uses $$\frac{|\vec{w}-\vec{w}^*|}{|\vec{w}|}\le \frac{\langle \varepsilon^2\rangle \nu}{A}, \qquad \mathrm{RMSE}\le \frac{\langle \varepsilon^2\rangle \nu}{A\sqrt{\ell_n+\ell_d+1}}.$$8. On Grover, HHL, and Hadamard-Ladder circuits executed on ibmq_ehningen, the strongest reported improvement is 29.81% RMSE reduction for Hadamard-Ladder with standard ZNE plus $$\frac{|\vec{w}-\vec{w}^*|}{|\vec{w}|}\le \frac{\langle \varepsilon^2\rangle \nu}{A}, \qquad \mathrm{RMSE}\le \frac{\langle \varepsilon^2\rangle \nu}{A\sqrt{\ell_n+\ell_d+1}}.$$9-filtering (Koenig et al., 7 May 2025).

A plausible implication of these examples is that ANF is best understood not as a narrow denoising primitive but as a broader adaptive signal-selection doctrine. In some domains the adaptive object is a filter kernel; in others it is a covariance, a scaling factor, a relevance score, or a feedback timescale. The common technical commitment is to let the system infer, from data or dynamics, which fluctuations should be preserved as informative structure and which should be attenuated as noise.