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arxiv: 2606.20269 · v1 · pith:VP6JG5WPnew · submitted 2026-06-18 · 🌀 gr-qc · astro-ph.HE

An explicit and differentiable Wilson-Daubechies-Meyer transform for gravitational-wave data analysis

Avi Vajpeyi , Giorgio Mentasti , Quentin Baghi , Ollie Burke , Lorenzo Speri This is my paper

Pith reviewed 2026-06-26 16:34 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.HE
keywords Wilson-Daubechies-Meyer transformtime-frequency analysisgravitational-wave data analysisLISAwavelet packetJAXstationary noiselikelihood equivalence
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The pith

The Wilson-Daubechies-Meyer transform yields numerically equivalent likelihoods to the frequency domain for LISA galactic binaries under stationary noise.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper supplies a self-contained Python package implementing the Wilson-Daubechies-Meyer wavelet-packet time-frequency transform with explicit mathematical foundations. It provides both NumPy and JAX backends, with forward and inverse transforms validated to floating-point precision and the JAX version enabling fast GPU computation on million-point streams. The central demonstration shows that likelihoods computed in the WDM domain reproduce frequency-domain posteriors for a resolved LISA galactic binary when both use the same stationary noise model. This confirms that the two representations are interchangeable at the level of numerical precision in the tested case. The work supports later development of tiling choices suited to non-stationary noise and data gaps.

Core claim

We present wdm_transform, an open-source Python package implementing the WDM wavelet-packet time-frequency transform, and document its mathematical foundations, statistical properties, and practical implementation for gravitational-wave data analysis. The package supplies NumPy and JAX backends, both transforms validated to floating-point precision. As a worked example, we verify that the WDM-domain likelihood reproduces frequency-domain posteriors for a resolved LISA galactic binary under a shared stationary noise model, confirming numerical equivalence of the two representations in that controlled setting.

What carries the argument

The Wilson-Daubechies-Meyer wavelet-packet time-frequency transform, supplied with explicit forward and inverse formulas and a JAX-differentiable implementation.

If this is right

  • Systematic optimisation of WDM tilings becomes feasible for specific data-analysis tasks.
  • Direct comparisons with alternative time-frequency representations are enabled.
  • Analysis of non-stationary noise, stochastic backgrounds, and data gaps in future detectors is supported.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The JAX backend opens the possibility of embedding the transform inside gradient-based sampling or optimisation loops for more complex signal models.
  • The explicit formulation may allow direct derivation of statistical properties of the transform coefficients without reference to the frequency domain.
  • Extension to time-varying noise models could be tested by injecting controlled non-stationarity into the same LISA binary example.

Load-bearing premise

A single shared stationary noise model is sufficient to establish equivalence between the WDM and frequency-domain representations, with any implementation differences remaining below floating-point precision.

What would settle it

A side-by-side computation of WDM-domain and frequency-domain likelihood values or posterior samples for the same resolved LISA galactic binary under the shared stationary noise model that shows disagreement exceeding floating-point precision.

Figures

Figures reproduced from arXiv: 2606.20269 by Avi Vajpeyi, Giorgio Mentasti, Lorenzo Speri, Ollie Burke, Quentin Baghi.

Figure 1
Figure 1. Figure 1: Schematic illustration of the WDM forward transform. (a) Time-domain atoms [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Orthogonality structure and empirical decorrelation of the sampled WDM transform. Both panels use the small tiling Nt = Nf = 8, N = NtNf = 64, with packed atoms flattened by α = n(Nf + 1) + m. Panel (a) lives in the Nt(Nf + 1) = 72-dimensional packed coefficient space, while panel (b) lives in the N-dimensional signal space. The asymmetry GG† ∈ C 72×72 versus G†G ∈ C 64×64 is intrinsic to the rectangular e… view at source ↗
Figure 3
Figure 3. Figure 3: Architecture of the wdm transform package. The public API exposes the TimeSeries, FrequencySeries, and WDM datatypes with reversible conversions between them. The forward and inverse WDM transforms, together with the shared window machinery (windows.py), are dispatched to interchangeable NumPy, CuPy, and JAX backends, so the same high-level code runs on CPU, GPU, or TPU. § efficiency improves and, at N = 1… view at source ↗
Figure 4
Figure 4. Figure 4: Forward WDM runtime versus input length. All transforms use a fixed tiling with Nt = 1024 time bins and Nf = N/Nt frequency channels. Top: me￾dian single-transform wall-clock runtime over 7 runs af￾ter one warmup call. The thin solid black line shows the N log2 N reference scaling. Bottom: batched-execution speedup (WDM transform only) tserial/tbatch for a batch size of B = 3, where the serial baseline app… view at source ↗
Figure 5
Figure 5. Figure 5: Injected resolved galactic binary in both fre￾quency and WDM domains. (a): channel-A frequency￾domain data, the instrumental noise PSD, and the injected source, with the analysis band shaded. Bottom: zooms of the source band in the frequency domain (left (b)) and in the whitened WDM time-frequency plane (right (c)), where amplitude is shown in units of the per-pixel noise standard deviation. § the conditio… view at source ↗
Figure 7
Figure 7. Figure 7: Population PP plot. Empirical distribution of the injected truth’s posterior quantile across the 100 seeds, for each parameter, in the WDM domain (solid, lower opac￾ity) and frequency domain (dashed). Grey shading shows the nested 1/2/3σ pointwise confidence bands expected under the uniform null. The two domains track each other closely and every parameter remains within the 3σ band. The small￾est per-para… view at source ↗
read the original abstract

The Wilson-Daubechies-Meyer (WDM) time-frequency transform has been widely used in gravitational-wave astronomy, yet a self-contained, mathematically explicit reference for practitioners remains lacking. This is especially true for those wishing to adopt the transform in modern Python and JAX inference workflows. We present wdm_transform, an open-source Python package implementing the WDM wavelet-packet time-frequency transform, and document its mathematical foundations, statistical properties, and practical implementation for gravitational-wave data analysis. The package supplies NumPy and JAX backends, both transforms (forward and inverse) validated to floating-point precision, with the JAX backend enabling GPU-accelerated transforms of million-point data streams in tens of milliseconds. As a worked example, we verify that the WDM-domain likelihood reproduces frequency-domain posteriors for a resolved LISA galactic binary under a shared stationary noise model, confirming numerical equivalence of the two representations in that controlled setting. This work paves the way for systematic optimisation of WDM tilings, a particularly promising direction for the non-stationary noise, stochastic backgrounds, and data gaps anticipated in future detectors, and for direct comparisons with alternative time-frequency representations needed to meet the challenges of future gravitational-wave data analysis.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript presents the open-source wdm_transform Python package implementing the Wilson-Daubechies-Meyer (WDM) wavelet-packet time-frequency transform, with NumPy and JAX backends. It supplies explicit mathematical foundations and statistical properties, validates both forward and inverse transforms to floating-point precision, and demonstrates via a worked example that the WDM-domain likelihood reproduces frequency-domain posteriors for one resolved LISA galactic binary under a shared stationary noise model.

Significance. If the numerical validations hold, the package supplies a reproducible, differentiable tool for time-frequency GW analysis that is particularly suited to non-stationary noise, stochastic backgrounds, and data gaps. The explicit documentation, JAX GPU acceleration for million-point streams, and open-source release with both backends are concrete strengths that lower the barrier for adoption in modern inference pipelines.

major comments (2)
  1. [Validation subsection] Validation subsection: the abstract and main text claim that forward and inverse transforms are validated to floating-point precision, yet no quantitative error metrics (maximum absolute difference, relative L2 error, or per-bin statistics) are reported for the tested data streams. This directly supports the central numerical-equivalence claim and must be supplied.
  2. [Likelihood comparison example] Likelihood comparison example: the demonstration that WDM-domain and frequency-domain posteriors match is restricted to a single resolved galactic binary under one explicitly shared stationary noise model. The text should state the precise data length, frequency range, and any exclusion rules applied to the likelihood, as these choices are load-bearing for the reported numerical equivalence.
minor comments (2)
  1. [Implementation section] The description of the JAX backend performance (tens of milliseconds for million-point streams) would benefit from explicit hardware specifications and a comparison table against the NumPy backend.
  2. [Mathematical foundations] Notation for the Meyer wavelet scaling function and the associated packet coefficients should be cross-referenced to a single equation number throughout the mathematical foundations section to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive feedback and positive recommendation for minor revision. We address each major comment below and will update the manuscript accordingly.

read point-by-point responses

  1. Referee: [Validation subsection] Validation subsection: the abstract and main text claim that forward and inverse transforms are validated to floating-point precision, yet no quantitative error metrics (maximum absolute difference, relative L2 error, or per-bin statistics) are reported for the tested data streams. This directly supports the central numerical-equivalence claim and must be supplied.

    Authors: We agree that quantitative error metrics are required to fully support the floating-point precision claim. In the revised manuscript we will add a table (or expanded subsection) reporting the maximum absolute difference, relative L2 error, and per-bin statistics for both the forward and inverse transforms on the tested streams. revision: yes

  2. Referee: [Likelihood comparison example] Likelihood comparison example: the demonstration that WDM-domain and frequency-domain posteriors match is restricted to a single resolved galactic binary under one explicitly shared stationary noise model. The text should state the precise data length, frequency range, and any exclusion rules applied to the likelihood, as these choices are load-bearing for the reported numerical equivalence.

    Authors: We agree that the precise parameters of the demonstration must be stated explicitly for reproducibility. In the revised manuscript we will add the data length (sample count and duration), frequency range, and any bin-exclusion rules used in the LISA galactic-binary likelihood comparison. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central contribution is an open-source implementation of the WDM transform (forward and inverse) with explicit documentation of its mathematical foundations and validation to floating-point precision. The worked example verifies numerical equivalence of WDM-domain and frequency-domain likelihoods for one LISA galactic binary under an explicitly shared stationary noise model. This is an external implementation check against an independent frequency-domain representation, not a derivation that reduces to fitted parameters or self-citations by construction. No load-bearing steps match the enumerated circularity patterns; the result is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are introduced in the abstract; the work rests on the pre-existing mathematical definition of the WDM transform and standard assumptions of stationary Gaussian noise.

pith-pipeline@v0.9.1-grok · 5763 in / 1066 out tokens · 28959 ms · 2026-06-26T16:34:16.454975+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

50 extracted references · 1 canonical work pages

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    coefficient layout explicitly, spell out the DC and Nyquist channel treatment, and set the normalization, phase, and indexing conventions used by the implemen- tation. These choices are then validated through ex- act round-trip reconstruction tests – applying the for- ward transform followed by its inverse – and empiri- cal checks of coefficient decorrela...

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    Interior channels(0< m < N f) Starting from Eq. (10) with the interior-channel basis element from Eq. (11), the forward transform can be 3 Our definitions of the Wilson basis functions (11) carefully in- clude them= 0 andm=N f in order for orthogonality to hold. evaluated in the Fourier domain using wnm = N/2−1X l=−N/2 ˜x[l] ˜g∗ nm[l],(17) For a real-valu...

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    DC channel(m= 0) The DC edge channel has a doubled time-shift ex- ponent and noC nm factor in the basis definition (11). 7 Following the same Fourier-domain evaluation yields wn0 = √ 2ℜ   Nt/2−1X l=1 ˜x[l]e4πiln/Nt ˜φ[l]   + 1√ 2 ˜x[0] ˜φ[0]./github (21) Note the factor of 4πin the exponent (versus 2πfor interior channels): the DC basis element oscill...

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    Nyquist channel(m=N f) The Nyquist edge channel is handled analogously to the DC channel but centered at the Nyquist frequency fmax. Its structure mirrors the DC case with appropri- ate frequency shifts. In practice, the Nyquist channel is computed using the same windowed-FFT machinery as the interior channels, with the frequency shift set to mNt/2 =N/2. ...

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    Now we have to consider three cases

    The second (quasi)normality condition We want to evaluate X l ˜gnm[l]˜g∗ pq[l] (A10) we immediately note that, since the window support in frequency isN t/2 we are forced to haveq=m. Now we have to consider three cases. Casem= 0, for which we have X l ˜gn0[l]˜g∗ p0[l] = 1 2 N/2−1X l=−N/2 e−4πil(n−p)/Nt ˜φ2[l] = 1 2 Nt/2−1X l=−Nt/2 e−4πil(n−p)/Nt ˜φ2[l] = ...

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    Interior channels(0< m < N f) Inserting the interior-channel basis element from Eq. (11) and splitting the sum over positive and negative frequencies gives wnm = 1√ 2 N/2−1X l=0 ˜x[l]e2πiln/Nt C ∗ nm ˜φ l− mNt 2 +C nm ˜φ l+ mNt 2 + 1√ 2 −1X l=−N/2 ˜x[l]e2πiln/Nt C ∗ nm ˜φ l− mNt 2 +C nm ˜φ l+ mNt 2 .(B2) Using the reality condition ˜x[−l] = ˜x[l]∗ togethe...

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    DC channel(m= 0) For the DC edge channel, the basis element has a doubled time-shift exponent and noC nm factor. The same positive/negative frequency splitting yields wn0 = 1√ 2 N/2−1X l=0 ˜x[l]e4πiln/Nt ˜φ[l] +1√ 2 −1X l=−N/2 ˜x[l]e4πiln/Nt ˜φ[l] = √ 2ℜ   Nt/2−1X l=1 ˜x[l]e4πiln/Nt ˜φ[l]   + 1√ 2 ˜x[0] ˜φ[0],(B4) which is Eq. (21) of the main text

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    Nyquist channel(m=N f) wnm = 1√ 2 N/2−1X l=0 ˜x[l]e4πiln/Nt ˜φ l− N 2 + ˜φ l+ N 2 + 1√ 2 −1X l=−N/2 ˜x[l]e4πiln/Nt ˜φ l− N 2 + ˜φ l+ N 2 = 1√ 2 N/2−1X l=0 ˜x[l]e4πiln/Nt ˜φ l− N 2 + ˜φ l+ N 2 + 1√ 2 N/2X l=1 ˜x[−l]e−4πiln/Nt ˜φ −l− N 2 + ˜φ −l+ N 2 .(B5) Thel= 0 term in the first summation vanishes, and thel=N/2 term is treated separately: wnm = √ 2ℜ N/2−...

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