21cmPSDenoiser: Denoising in 21-cm Cosmology

• The method introduces a score-based diffusion model that denoises a single noisy 2D 21-cm power spectrum to recover the underlying population mean, reducing sample variance by an order of magnitude.
• It leverages a U-Net convolutional architecture with time and noise-level conditioning and processes 21cmFAST simulation data through log-scaling and min–max normalization.
• The framework integrates seamlessly into Bayesian inference pipelines, yielding unbiased posteriors with approximately 50% narrower confidence intervals on astrophysical parameters.

Searching arXiv for the cited papers and closely related work on 21-cm denoising / foreground mitigation. 21cmPSDenoiser denotes, in the provided literature, two technically distinct denoising pipelines for 21-cm cosmology. In its explicit and primary recent usage, it is a score-based diffusion model that takes a single, forward-modelled realisation of the cylindrical $2$D 21-cm power spectrum and predicts the corresponding population mean during Bayesian inference, with the stated aim of mitigating sample variance in reionisation-era analyses (Breitman et al., 16 Jul 2025). In the same literature package, the label is also applied to the earlier RPCA + GILC pipeline for 21-cm signal recovery from foreground-contaminated intensity-mapping data, where denoising is formulated as a decomposition of the frequency–frequency covariance into a low-rank foreground term and a sparse HI term (Zuo et al., 2018). The shared designation therefore refers to a family resemblance at the level of inverse-problem structure rather than to a single invariant algorithm.

1. Definition, target quantity, and nomenclature

The 2025 formulation introduces \texttt{21cmPSDenoiser} as a score-based diffusion model for denoising a single, forward-modelled realisation of the $21$-cm $2$D power spectrum, with the output interpreted as the population-mean cylindrical power spectrum. The motivating problem is that state-of-the-art simulations of reionisation-era $21$-cm signal have limited volumes, generally orders of magnitude smaller than observations, so the Fourier modes in common between simulation and observation have limited overlap, especially in cylindrical $2$D $k$-space natural for $21$-cm interferometry. In that setting, sample variance is treated as the dominant stochastic contaminant to be removed from a single realisation (Breitman et al., 16 Jul 2025).

A recurrent misconception is to assimilate the method to a conventional emulator. The paper explicitly distinguishes it from emulators by stating that the denoiser is not tied to a particular model or simulator, since its input is a model-agnostic realisation of the $2$D $21$-cm power spectrum. It further reports that the model generalises to power spectra produced with a different $21$-cm simulator than those on which it was trained (Breitman et al., 16 Jul 2025).

The label is, however, not unique to this diffusion-based setting in the provided material. The same name is attached there to the RPCA + GILC workflow summarised from the 2018 foreground-subtraction study. That earlier method addresses a different inverse problem: recovering the cosmological $21$0-cm signal from intensity-mapping data dominated by coherent foreground radiation from the Milky Way and extragalactic radio sources. This suggests a useful terminological distinction between power-spectrum sample-variance denoising and covariance-structured signal recovery, even though both are described as denoising.

2. Diffusion-theoretic formulation

The theoretical framework in the 2025 method is a score-based diffusion model in which sample variance in a single $21$1D power-spectrum realisation is viewed as “noise” to be removed in order to recover the underlying population-mean power spectrum. The construction proceeds by specifying a forward stochastic diffusion process that gradually corrupts the clean mean power spectrum into Gaussian white noise, and then learning the time-dependent score function that enables reversal of that process (Breitman et al., 16 Jul 2025).

Let $21$2 denote a clean mean $21$3D power spectrum, and let $21$4 be its diffused counterpart at time $21$5. The forward variance-preserving SDE is

$21$6

with

$21$7

where $21$8 is a prescribed noise schedule and $21$9 is a standard Wiener process. As $2$0, $2$1.

The reverse-time dynamics are written as

$2$2

where $2$3 is the marginal density at time $2$4. The equivalent probability-flow ODE,

$2$5

is deterministic and has the same marginal laws. The score is parametrised by a neural network $2$6, trained with a continuous-time denoising score-matching objective. One form given in the summary is

$2$7

with the equivalent form

$2$8

In this formulation, the denoiser is not estimating instrument noise directly; it is learning the score of the diffused distribution over mean power spectra so that a noisy single-realisation input can be transported toward the corresponding population mean. That distinction is central to interpreting the method’s scope.

3. Network architecture, preprocessing, and training corpus

The neural backbone is a U-Net convolutional autoencoder with time- and noise-level conditioning. Its input is a single noisy $2$9D power-spectrum realisation $21$0 together with the scalar diffusion time $21$1, and its output is the estimated score $21$2 (Breitman et al., 16 Jul 2025).

The reported hyperparameters are: diffusion time horizon $21$3; effectively infinite numbers of training timesteps because the loss is continuous-time; Adam optimizer with learning rate $21$4; batch size $21$5 $21$6D power-spectrum patches; and weight decay $21$7. Training is stated to converge in $21$8 epochs on $21$9 M samples (Breitman et al., 16 Jul 2025).

The training database is derived from $2$0cmFAST v3 coeval boxes of side length $2$1 cMpc and cell size $2$2 cMpc. It spans $2$3 parameter combinations with $2$4 IC realisations each, for $2$5 light-cone boxes. For each box, the pipeline slices into $2$6 redshift bins and computes cylindrical power spectra on $2$7 log-spaced $2$8 $2$9 bins, giving $k$0 M $k$1D power spectra in total. Preprocessing consists of taking $k$2 and applying min–max scaling to $k$3 (Breitman et al., 16 Jul 2025).

The astrophysical training distribution is also constrained rather than purely ad hoc. The summary states that $k$4 are drawn from a posterior conditioned on UV luminosity functions, CMB $k$5, and Ly$k$6 dark fraction, while the X-ray parameters are flat in $k$7 and $k$8 eV. This means that the denoiser is trained on a broad but structured astrophysical ensemble rather than on a single fiducial history.

4. Embedding in Bayesian inference

The method is designed to be inserted into an explicit Gaussian-likelihood Bayesian inference pipeline, such as $k$9CMMC, immediately after forward-modelling the $21$0D power spectrum from a single simulation realisation. The resulting workflow can be summarised as follows (Breitman et al., 16 Jul 2025):

1. Propose $21$1 and generate one $21$2D $21$3-cm light cone, then compute the cylindrical $21$4D power-spectrum realisation $21$5.
2. Optionally crop to the experiment’s EoR window in $21$6.
3. If the denoiser is used, compute

$21$7

otherwise set $21$8.

1. Spherically average $21$9.
2. Apply the instrument window $2$0 to both model and data.
3. Evaluate the Gaussian likelihood

$2$1

with total variance $2$2.

Two choices are given for the forward-model variance. In the “Sample variance” baseline,

$2$3

while with the denoiser,

$2$4

where $2$5 is measured on the validation set. This is methodologically important: the denoiser changes both the mean model entering the likelihood and the treatment of forward-model uncertainty.

The computational-cost claim is similarly specific. Denoising one $2$6D power spectrum requires $2$7 s on a V100 GPU, quoted as the median over $2$8 probability-flow ODE draws. Fixing & Pairing, by contrast, requires a second full simulation per likelihood step, i.e. $2$9 cost, whereas \texttt{21cmPSDenoiser} adds $21$0 overhead to a typical $21$1 s emulator-based likelihood (Breitman et al., 16 Jul 2025).

5. Performance metrics and reported results

Performance is quantified with a fractional error metric

$21$2

where $21$3 is the mean power spectrum from $21$4 IC realisations at each $21$5 (Breitman et al., 16 Jul 2025).

On the test set, after cropping to $21$6, the reported summary statistics are:

Method	Median FE	68% CL
Baseline (single realisation)	$21$7	$21$8
Fix & Pair	$21$9	$21$0
21cmPSDenoiser	$21$1	$21$2

The same study states that individual samples of $21$3D Fourier amplitudes of wave modes relevant to current $21$4-cm observations can deviate from the mean by over $21$5 for $21$6 cMpc simulations, even when only considering stochasticity due to sampling of Gaussian initial conditions, and that \texttt{21cmPSDenoiser} reduces this deviation by an order of magnitude. It is further reported to outperform Fixing & Pairing by a factor of few at almost no additional computational cost (Breitman et al., 16 Jul 2025).

For parameter inference, the HERA-mock experiment isolates the practical consequence of denoising. The traditional pipeline with no $21$7 cut and a single realisation yields highly biased posteriors, described as $21$8 off and overconfident. Applying only the $21$9 cut makes the posteriors unbiased but much wider because sample variance dominates. Combining the denoiser with $21$00 yields unbiased posteriors with $21$01 narrower $21$02 intervals on each astrophysical parameter, and the abstract summarises this as an unbiased posterior that is $21$03 narrower (Breitman et al., 16 Jul 2025).

6. Relation to the RPCA + GILC pipeline and broader methodological scope

The 2018 foreground-subtraction method provides a distinct denoising paradigm that is also associated with the name in the provided materials. There, the observation model for intensity mapping is

$21$04

with frequency–frequency covariance

$21$05

where $21$06 is the low-rank foreground covariance and $21$07 is the sparse $21$08-cm covariance. Renaming $21$09, the method seeks

$21$10

with $21$11 and $21$12 (Zuo et al., 2018).

The ideal RPCA formulation is

$21$13

and its convex relaxation, Principal Component Pursuit, is

$21$14

The augmented Lagrangian is solved with Inexact ALM, alternating singular-value thresholding for $21$15, soft-thresholding for $21$16, and dual updates for $21$17, with stopping criterion

$21$18

The universal choice

$21$19

is stated to work well without tuning, with $21$20 in this application (Zuo et al., 2018).

The significance of this earlier method lies in the structural assumptions it exploits: frequency coherence makes the foreground covariance low-rank, whereas the $21$21-cm signal covariance is assumed sparse and dominant on or near the diagonal. The summary explicitly notes that sparsity of the frequency covariance of the $21$22-cm signal is first explored there. In simulations over $21$23 MHz with $21$24 channels and $21$25 pixels, the RPCA + GILC pipeline recovers $21$26 to relative error

$21$27

stated as an order of magnitude better than classic PCA. Angular power spectra and cross-spectra coincide at the $21$28–$21$29 level over $21$30, the transfer function is within a few percent of unity, and the $21$31D line-of-sight power spectrum is also well recovered, especially at high $21$32 (Zuo et al., 2018).

The distinction between the 2018 and 2025 pipelines is therefore substantive. The RPCA + GILC method denoises covariance structure in foreground-contaminated map-space data, whereas the diffusion model denoises sample variance in cylindrical power-spectrum space. Their assumptions and failure modes are correspondingly different. For the RPCA approach, violations of low-rank foreground covariance or sparse signal covariance—such as calibration artifacts or polarization leakage—may break the separation, motivating extensions like Stable PCP with $21$33 and the constraint $21$34, partial-SVD solvers for large $21$35, and GNILC for local foreground-rank estimation in space and scale (Zuo et al., 2018). For the diffusion model, the decisive assumptions are those implicit in the training distribution, the score-learning objective, and the use of validation-set network uncertainty within the likelihood (Breitman et al., 16 Jul 2025).

Taken together, the two usages define a broader methodological pattern: both versions of 21cmPSDenoiser exploit strong structural priors to transform a single contaminated or stochastic realisation into an estimator of an underlying cosmological quantity, but they do so on different observables and against different sources of error.