MaxWave: Rapid Gravitational-Wave Reconstruction

• MaxWave is a gravitational-wave data-analysis framework that uses wavelet-based maximum-likelihood methods to reconstruct non-Gaussian features, including glitches and transient signals.
• It improves on the BayesWave initialization algorithm through enhancements such as a modified wavelet basis, TFτ transforms, and a downsampled heterodyned wavelet transform for faster computation.
• The framework’s coherent multi-detector extension enables real-time signal extraction and glitch rejection, improving event reconstruction and supporting machine-learning applications.

Searching arXiv for MaxWave and closely related gravitational-wave reconstruction papers. MaxWave is a gravitational-wave data-analysis framework for wavelet-based maximum-likelihood reconstruction of non-Gaussian structure in detector data. Introduced in "MaxWave: Rapid Maximum Likelihood Wavelet Reconstruction of Non-Gaussian features in Gravitational Wave Data" (Mathur et al., 18 Aug 2025), it is described as an improvement of the existing BayesWave initialization algorithm and as a rapid, low-latency approximate maximum likelihood method for reconstructing non-Gaussian features. Its stated scope includes de-noising long-duration signals, reconstructing glitches in forms useful for machine-learning analysis, and, in a later extension, coherent multi-detector signal reconstruction and glitch rejection in real time (Mathur et al., 22 Jun 2026).

1. Provenance, naming, and scope

MaxWave was published on 2025-08-18 as a gravitational-wave method centered on rapid maximum-likelihood wavelet reconstruction of non-Gaussian features (Mathur et al., 18 Aug 2025). A subsequent paper, "MaxWave Signal: Rapid, coherent maximum likelihood wavelet reconstruction of transient signals in gravitational wave data," published on 2026-06-22, extends the framework to coherent multi-detector signal reconstruction and glitch rejection (Mathur et al., 22 Jun 2026).

The term should be distinguished from "WaveMax: Radar Waveform Design via Convex Maximization of FrFT Phase Retrieval," which is a radar waveform-design method based on ambiguity-function phase retrieval and convex optimization rather than a gravitational-wave reconstruction algorithm (Pinilla et al., 24 Jan 2025). The similarity of names can obscure the fact that the two works address different inverse problems, different sensing modalities, and different mathematical objects.

Within gravitational-wave analysis, MaxWave is explicitly positioned relative to BayesWave: the 2025 paper states that it improves the existing BayesWave initialization algorithm (Mathur et al., 18 Aug 2025). The later coherent extension further states that it can complement existing burst search and reconstruction frameworks through a fundamentally distinct approach (Mathur et al., 22 Jun 2026).

2. Problem setting in gravitational-wave data analysis

The motivating context is the increase in transient-event rates driven by improved detector sensitivity. The 2025 MaxWave paper states that advancements in the sensitivity of gravitational-wave detectors have increased the detection rate of transient astrophysical signals, while the 2026 extension states that faster automated analysis is therefore required (Mathur et al., 18 Aug 2025).

MaxWave targets reconstruction of non-Gaussian features in detector data. In the gravitational-wave context, this category includes at least glitches and transient signal structure that departs from stationary Gaussian noise assumptions. The 2025 abstract emphasizes that the algorithm produces a rapid, low-latency approximate maximum likelihood solution for such features and can be used to de-noise long-duration signals, including the stochastic gravitational-wave background from numerous unresolved sources and continuous wave signals from isolated sources such as rotating neutron stars (Mathur et al., 18 Aug 2025).

A further stated use-case is machine-learning support. MaxWave is described as providing non-whitened, time and frequency domain, Gaussian-noise-free reconstructions of any glitch, with the intended benefit of aiding applications that classify and understand glitches (Mathur et al., 18 Aug 2025). This suggests a dual role: direct reconstruction for data conditioning and feature extraction, and indirect support for downstream discriminative or representation-learning pipelines.

3. Core methodological modifications in the original MaxWave formulation

The original MaxWave abstract identifies three specific enhancements over the existing BayesWave initialization algorithm (Mathur et al., 18 Aug 2025).

Enhancement	Stated purpose
Modified wavelet basis	Eliminate redundant inner product calculations
Shift from TFQ to TF$\tau$ transforms	Optimize wavelet subtractions
Downsampled heterodyned wavelet transform	Accelerate initial calculations

The first modification uses a modified wavelet basis to eliminate redundant inner product calculations (Mathur et al., 18 Aug 2025). In practical terms, this locates the efficiency gain in the linear-algebraic core of wavelet matching rather than only in search heuristics.

The second modification replaces traditional time-frequency-quality factor wavelet transforms with time-frequency-time extent transforms, written as TF$\tau$, in order to optimize wavelet subtractions (Mathur et al., 18 Aug 2025). Because the abstract pairs TFQ and TF$\tau$ directly, the intended emphasis is not merely a change of parameterization but a change aimed at more efficient subtraction behavior during reconstruction.

The third modification is a downsampled heterodyned wavelet transform used to accelerate initial calculations (Mathur et al., 18 Aug 2025). Together with the other two changes, this places MaxWave in the class of reconstruction methods that seek latency reduction by restructuring the transform and basis evaluation stages rather than by abandoning likelihood-based modeling.

4. Maximum-likelihood wavelet modeling in the MaxWave lineage

The 2025 abstract identifies MaxWave as an approximate maximum-likelihood reconstruction method but does not, in the material summarized here, provide the explicit objective function. A later extension, MaxWave Signal, makes the statistical structure explicit for the coherent multi-detector case by modeling each detector time series as

$D_i(t)=h_i(t)+n_i(t),$

with zero-mean, stationary Gaussian noise characterized by one-sided power spectral density $S_i(f)$, and likelihood

$$\mathcal{L}(h\,|\,D)\propto \exp\!\left[-\frac12\sum_{i=1}^K \langle D_i-h_i \mid D_i-h_i\rangle_i\right].$$

The corresponding log-likelihood is written as

$$\ell(h)= -\frac12\sum_i \int_0^\infty \frac{|D_i(f)-h_i(f)|^2}{S_i(f)}\,df+\mathrm{const.}$$

(Mathur et al., 22 Jun 2026)

In that extension, the waveform is represented as a sparse superposition of complex wavelets,

$h(t)=\sum_{k=1}^{N_w} A_k\,\psi_k(t;\theta_k),$

with parameters

$\theta_k=\{t_0^k,f_0^k,Q^k,\phi_0^k\},$

and a frequency-domain sine-Gaussian form

$$\Psi_k(f)=A_k \exp\!\left[-\frac{Q_k^2}{4}\left(\frac{f-f_0^k}{f_0^k}\right)^2\right] e^{-2\pi i f t_0^k+i\phi_0^k},$$

together with $\tau$0 (Mathur et al., 22 Jun 2026). At fixed wavelet morphology, maximizing the likelihood over amplitudes yields the linear system

$\tau$1

so that

$\tau$2

These equations come from the coherent extension rather than the 2025 abstract itself, but they clarify what the designation "maximum likelihood wavelet reconstruction" means within the MaxWave program: sparse wavelet morphology, noise-weighted inner products, and analytic or semi-analytic amplitude estimation embedded inside a nonlinear search over wavelet parameters (Mathur et al., 22 Jun 2026).

5. Coherent multi-detector extension and glitch rejection

MaxWave Signal extends the framework from rapid single-stream reconstruction to coherent network analysis. Its procedure begins by running the single-detector MaxWave glitch model on each detector and selecting the detector with largest SNR as the dominant reconstruction; the other detectors are aligned to it using the FINDCHIRP $\tau$3-statistic

$\tau$4

whose magnitude peak determines the time delay $\tau$5, whose argument gives the phase shift $\tau$6, and whose normalized magnitude gives the amplitude scale $\tau$7 (Mathur et al., 22 Jun 2026).

The aligned detector streams are then combined through adaptive noise weighting based on the geometric mean PSD

$\tau$8

with effective strain

$\tau$9

so that the network inner product reduces to a single-detector form with effective PSD $\tau$0 (Mathur et al., 22 Jun 2026). The paper states that this synthetic detector amplifies non-Gaussian features, down-weights noisy detectors, and preserves Gaussian noise statistics.

Glitch rejection is implemented in two stages. First, only detectors satisfying the inter-detector light-travel-time condition of approximately $\tau$1 and SNR $\tau$2 enter the coherent combination. Second, after reconstructing the coherent signal on the synthetic detector, the method shifts and rescales the coherent reconstruction back to each detector, forms residuals, and re-runs MaxWave; if the residual reconstruction has more than one wavelet, the event is flagged as a coincident-event non-removal, i.e. glitch (Mathur et al., 22 Jun 2026).

The extension reports quantitative performance. For GW150914, the coherent detector Q-scan shows clear amplification of signal power, and the time-domain match with the NR template improves from $\tau$3 in H1 alone and $\tau$4 in L1 alone to $\tau$5 after coherence. In injections in real LIGO O3 noise, the coherent match improves by $\tau$6 for network SNR $\tau$7 and by $\tau$8 for SNR $\tau$9. The $D_i(t)=h_i(t)+n_i(t),$0-statistic test alone rejects $D_i(t)=h_i(t)+n_i(t),$1 of $D_i(t)=h_i(t)+n_i(t),$2 artificial simultaneous glitches; adding the residuals test raises rejection to $D_i(t)=h_i(t)+n_i(t),$3, while cross-type glitches reject at $D_i(t)=h_i(t)+n_i(t),$4. In a real $D_i(t)=h_i(t)+n_i(t),$5 O3 run with H1+L1, the method recovered GW190521_030229 at SNR $D_i(t)=h_i(t)+n_i(t),$6 and GW190521_074359 at SNR $D_i(t)=h_i(t)+n_i(t),$7 at the correct GPS times, with no false alarms in a $D_i(t)=h_i(t)+n_i(t),$8 time-shift analysis, corresponding to $D_i(t)=h_i(t)+n_i(t),$9, reported as less than $S_i(f)$0/day. Runtime scaling on a single CPU core for $S_i(f)$1 is given as $S_i(f)$2--$S_i(f)$3 for $S_i(f)$4 detectors and $S_i(f)$5 wavelet layers, $S_i(f)$6--$S_i(f)$7 for $S_i(f)$8 detectors, and $S_i(f)$9--$$\mathcal{L}(h\,|\,D)\propto \exp\!\left[-\frac12\sum_{i=1}^K \langle D_i-h_i \mid D_i-h_i\rangle_i\right].$$0 for $$\mathcal{L}(h\,|\,D)\propto \exp\!\left[-\frac12\sum_{i=1}^K \langle D_i-h_i \mid D_i-h_i\rangle_i\right].$$1 detectors; even with $$\mathcal{L}(h\,|\,D)\propto \exp\!\left[-\frac12\sum_{i=1}^K \langle D_i-h_i \mid D_i-h_i\rangle_i\right].$$2 layers, runtime remains below $$\mathcal{L}(h\,|\,D)\propto \exp\!\left[-\frac12\sum_{i=1}^K \langle D_i-h_i \mid D_i-h_i\rangle_i\right].$$3 for $$\mathcal{L}(h\,|\,D)\propto \exp\!\left[-\frac12\sum_{i=1}^K \langle D_i-h_i \mid D_i-h_i\rangle_i\right].$$4 detectors and below $$\mathcal{L}(h\,|\,D)\propto \exp\!\left[-\frac12\sum_{i=1}^K \langle D_i-h_i \mid D_i-h_i\rangle_i\right].$$5 for $$\mathcal{L}(h\,|\,D)\propto \exp\!\left[-\frac12\sum_{i=1}^K \langle D_i-h_i \mid D_i-h_i\rangle_i\right].$$6 detectors (Mathur et al., 22 Jun 2026).

6. Applications, significance, and relation to adjacent work

The original MaxWave abstract identifies three application domains. First, it can de-noise long-duration signals, specifically including the stochastic gravitational-wave background from numerous unresolved sources and continuous wave signals from isolated sources such as rotating neutron stars. Second, it reconstructs non-Gaussian features in a form suitable for low-latency analysis. Third, it can aid machine-learning applications by providing non-whitened, time and frequency domain, Gaussian-noise-free reconstructions of any glitch (Mathur et al., 18 Aug 2025).

The coherent extension adds a fourth application domain: real-time, model-independent signal reconstruction that strengthens detection confidence and improves sensitivity to diverse signal morphologies while safely denoising gravitational-wave data without removing transient signals (Mathur et al., 22 Jun 2026). In that sense, MaxWave occupies a methodological position between fully Bayesian, potentially higher-latency reconstruction pipelines and purely discriminative trigger-generation systems.

A common misconception is to conflate MaxWave with WaveMax because of the near-anagrammatic titles. The distinction is categorical. MaxWave concerns wavelet reconstruction of non-Gaussian features in gravitational-wave detector data and coherent multi-detector signal extraction (Mathur et al., 18 Aug 2025, Mathur et al., 22 Jun 2026). WaveMax, by contrast, addresses radar waveform recovery from ambiguity-function magnitudes using the fractional Fourier transform and a convex low-rank formulation (Pinilla et al., 24 Jan 2025).

The present public summary of the original 2025 MaxWave paper is abstract-level: it specifies the algorithm’s stated enhancements, target use-cases, and low-latency objective, while the later 2026 extension provides the most explicit equations, workflow, and performance figures available in the material summarized here (Mathur et al., 18 Aug 2025, Mathur et al., 22 Jun 2026). This suggests that the MaxWave research program is best understood as a fast maximum-likelihood wavelet-reconstruction family whose defining themes are computationally efficient basis evaluation, low-latency subtraction and denoising, and increasingly coherent exploitation of multi-detector information.