Hankel low-rank matrix approximation for gravitational-wave data analysis

Abstract

Next-generation gravitational-wave (GW) detectors, such as the Laser Interferometer Space Antenna (LISA), will observe vast numbers of overlapping signals. Disentangling these signals from instrumental noise and from one another constitutes a significant data analysis challenge. We explore a denoising technique based on embedding time series into Hankel matrices: a superposition of $n$ (damped) sinusoids corresponds to a matrix of rank $2n$. Thus, the problem of signal extraction is reduced to a structured low-rank approximation problem. Using synthetic data tailored to GW applications, we benchmark three Hankel-based algorithms: ESPRIT, Cadzow iterations, and iteratively reweighted least squares (IRLS). Our test scenarios include isolated and multi-component monochromatic signals, the resolution of sources with closely spaced frequencies, and the recovery of black hole quasinormal modes (QNM). All three algorithms achieve near-optimal performance consistent with Fisher matrix bounds, evidenced by an inverse-square scaling of the mismatch with the signal-to-noise ratio. Furthermore, a proof-of-concept application to numerical relativity waveforms validates the ability of these algorithms to extract QNM frequencies from ringdown signals. Hankel low-rank approximation therefore offers a transparent, computationally efficient avenue for preprocessing GW time series.

Figures (5)

• Figure 1: Mismatch $\mathcal{M}$ vs. SNR $\rho$ for the single monochromatic signal experiment. Top: Combined results for ESPRIT (stars), Cadzow (crosses), and IRLS (dots). Bottom: Individual results for each algorithm, where data points are the medians over 50 noise realizations and error bars span the 16th-84th percentiles. The solid lines represent the best-fit power law obtained from an MCMC procedure (see Eq. (17) of Hogg:2010yz), with faint lines indicating samples from the posterior. Outliers identified by this procedure (posterior probability $>0.5$) are marked in black using the respective algorithm's symbol.
• Figure 2: Mismatch $\mathcal{M}$ vs. the normalized SNR $\overline{\rho} = \rho/\sqrt{n}$ for the multiple-signal experiment with a known number of signals. This plot combines results for superpositions of $n=3$, $5$ and $7$ signals, denoised with ESPRIT (blue stars), Cadzow (orange crosses), and IRLS (green dots). The fact that all data points follow the universal trend $\mathcal{M} \propto \overline{\rho}\,{}^{-2}$ confirms that the denoising performance for all three algorithms scales as expected. Qualitatively, the scatter is smallest for IRLS and largest for ESPRIT. This scatter, as well as the outliers, typically corresponds to superpositions where some signals are significantly weaker than the others.
• Figure 3: Mismatch $\mathcal{M}$ (right $y$-axis, dashed lines) and the mean square (MS) of the residual (left $y$-axis, solid lines) as a function of the trial number of components $n_{\text{trial}}$, for experiments with an unknown number of signals. The three panels correspond to experiments with a true number of signals $n_{\text{true}} = 3, 5,$ and $7$ (top to bottom). For all algorithms, the mismatch reaches a minimum when $n_{\text{trial}}$ approximately matches the ground truth (vertical dashed line). The residual MS exhibits an "elbow" at this point before continuing to decrease, which is a signature of overfitting. Interestingly, in the middle panel, the minima and elbows correctly indicate only four components, as one of the five signals in this particular realization was too weak for the algorithms to resolve. The normalized SNRs of randomly generated superpositions are indicated in each panel.
• Figure 4: Denoising performance as a function of the relative frequency separation $\delta = |f_2-f_1|/f_1$. Top: The median relative r.m.s. error in frequency reconstruction $\sigma_f$ for ESPRIT (blue), Cadzow (orange), and IRLS (green) algorithms across runs with low, moderate, and high SNRs (from left to right). The horizontal dashed line marks a threshold $\sigma=0.5$. The approximate SNR indicated in the top left refers to both panels in a column. Bottom: The scaled mismatch $\mathcal{M}\overline{\rho}^2$ for the same set of runs and algorithms. Points represent the median over 50 noise realizations, and error bars span the 16th-84th percentiles. In all panels, the position of the Fourier resolution limit is marked by a black arrow on the horizontal axis.
• Figure 5: QNM spectroscopy of the SXS waveform SXS:BBH:0305 in the absence of synthetic noise. The left and right columns show results for the spherical harmonic modes $(2,2)$ and $(3,2)$, respectively. The blue and orange tracks represent frequencies extracted by ESPRIT (top) and Cadzow iterations (bottom) across varying start times. The color shading indicates the start time offset, with darker shades corresponding to later start times and shorter ringdown tails. The open markers indicate the theoretical complex QNM frequencies of the fundamental $(\ell, m, n_{\rm tone})=(220)$ mode (circles) and of the $(320)$ mode (squares).