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

arXiv:2603.15740 · astro-ph.HE, astro-ph.IM, gr-qc · Submitted 2026-03-16 · Read on arXiv

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Introduction to the show: ident: Astrophysics Radio. Generated commentary on the latest astrophysics papers.

Vera: Hi, I'm Vera.

Jocelyn: And I'm Jocelyn, here with Subrahmanyan.

Vera: We're looking at the paper: "Hankel low-rank matrix approximation for gravitational-wave data analysis".

Jocelyn: Here's the gist of the paper. The gist Hankel low-rank matrix approximation for gravitational-wave data analysis explores denoising techniques based on embedding time series into Hankel matrices to reduce signal extraction to a structured low-rank approximation problem, which is crucial for disentangling overlapping signals from instrumental noise in next-generation detectors like LISA.

Vera: The paper addresses the challenge of disentangling vast numbers of overlapping gravitational-wave signals from instrumental noise and one another in next-generation detectors such as LISA.

Jocelyn: This problem is framed as a structured low-rank approximation problem by embedding time series into Hankel matrices, where a superposition of n damped sinusoids corresponds to a matrix of rank 2n.

Vera: The authors benchmark three Hankel-based algorithms—ESPRIT, Cadzow iterations, and iteratively reweighted least squares (IRLS)—using synthetic data tailored to GW applications.

Title and authors: Vera: So we’re looking at this paper titled "Hankel low-rank matrix approximation for gravitational-wave data analysis". It sounds really technical, but what it's trying to do is tackle the huge problem of separating those overlapping gravitational-wave signals from the instrumental noise in detectors like LISA.

Jocelyn: Yeah, it’s about taking time series data and embedding them into a Hankel matrix so you can reduce that messy signal extraction problem down to finding a structured low-rank approximation. It’s basically turning a huge headache into a solvable math problem, right?

Subrahmanyan: From my side, the key idea is framing this as finding the best rank r matrix that approximates the noisy Hankel matrix H, where that rank is tied to the number of underlying signals. That sounds like it could be very powerful if we can actually find that low-rank structure in real data.

Vera: Exactly. The authors are benchmarking three specific algorithms—ESPRIT, Cadzow iterations, and iteratively reweighted least squares—using synthetic data designed specifically for gravitational-wave applications to see how well they perform against the theoretical limits.

Jocelyn: They’re showing that these methods can achieve performance consistent with what we expect from Fisher matrix bounds, which means they're doing a very good job of extracting the signal information even when it’s buried in noise.

Subrahmanyan: That consistency with those bounds is important because it tells us the algorithms are actually finding what they claim to be finding regarding the underlying physical parameters like frequencies and damping times.

Vera: So, before we get into how they did this, I want to make sure we're all on the same page about what’s actually happening here in terms of their approach.

The paper's summary: Vera: Well, the paper explains that they take a discrete time series h of length L and arrange it into a Hankel matrix H, which they call a Hankel matrix. If that original time series h is actually just a superposition of n damped sinusoids, then this Hankel matrix H has a rank equal to 2n <ref:2603.15740#pg1>.

Jocelyn: So the core idea is that if you know the signal is made up of these specific types of waves—monochromatic or damped—then finding the original signal boils down to finding a low-rank Hankel matrix H hat that approximates that noisy Hankel matrix H.

Subrahmanyan: The objective they state formally is minimizing the norm of the difference between H and a low-rank approximation, subject to the constraint that the rank of H has to be at most r, where r is related to n. That’s defining it as a structured low-rank approximation problem, which is what makes it mathematically tractable.

Vera: They test this approach in several scenarios: first they check it with just one monochromatic signal for a consistency check; then they move on to testing performance on overlapping signals and the global fit problem; and finally, they look at how well the method can separate two signals that are close together in frequency.

Jocelyn: And for those quasinormal modes, ESPRIT is used specifically to recover frequencies and damping times from the ringdown part of a gravitational-wave signal, which is pretty direct.

Subrahmanyan: That's interesting because recovering those QNM parameters directly from the ringdown suggests a way to get clean measurements of the black hole's properties even if the main signal is buried in noise.

The paper's improvements: Vera: Now, the authors aren’t just presenting one method; they are comparing ESPRIT, Cadzow iterations, and Iteratively Reweighted Least Squares. They’re showing that these different approaches have varying levels of success depending on what you are trying to extract from the data.

Jocelyn: ESPRIT is non-iterative; it tries to estimate frequencies and amplitudes in two constraint sets directly using a two-step projection method, and they found it can be really effective at extracting a few of the loudest QNMs simultaneously.

Subrahmanyan: But the authors note that ESPRIT might not lead to the absolute optimal solutions for that structured low-rank approximation problem because it’s non-iterative, which makes me wonder if an iterative approach has to be better in certain cases.

Vera: That’s where Cadzow iterations come in; this is an iterative method that alternates between projecting onto the space of rank r matrices and the subspace of Hankel matrices. It's a bit trickier because the set of rank r matrices isn't convex, so it’s not guaranteed to find the global minimizer.

Jocelyn: However, they show that when you combine ESPRIT with Cadzow, that combination is most effective at simultaneously extracting a few of the loudest QNMs in a signal.

Subrahmanyan: I'm also looking at Iteratively Reweighted Least Squares, which uses a continuously differentiable surrogate rank function to minimize a quadratic approximation of that rank function around the Hankel estimate. This method has potential because it’s iterative and involves optimizing those non-convex but continuous rank surrogates with decreasing smoothing parameter.

Vera: And they’ve included some practical details on efficiency, noting that using FFT for operations like multiplication by a vector can make the Hankel-based algorithms scale in O(rLlog L) time, which is pretty good for long time series.

Jocelyn: They also discuss noise handling; they assume white Gaussian noise, but they show how you can extend this to general Gaussian noise using a whitening procedure or by using a weighted matrix norm instead of the standard Frobenius norm in the optimization problem.

Conclusion: Vera: So, to wrap up the Hankel low-rank matrix approximation for gravitational-wave data analysis paper, what we have here is a set of tools—ESPRIT, Cadzow, and IRLS—to tackle that complex signal extraction problem by framing it as a structured low-rank approximation.

Jocelyn: The main implication is that these methods can provide parameter estimates for signals with performance consistent with Fisher matrix bounds, which means we get reliable measurements of frequencies and amplitudes even when the data is messy.

Subrahmanyan: For the bigger picture, this suggests that by using these techniques, we are setting a path toward extracting cleaner information from LISA data, which is crucial because those detectors are going to be seeing an overwhelming number of overlapping sources.

Vera: They also give us a way to estimate the true number of components in a signal mixture by looking for an “elbow” in the mean-square residual plot as you try different trial numbers, which helps when you don't know how many signals there are beforehand.

Jocelyn: And for those who are struggling with closely spaced signals, they show that Cadzow has some "super-resolution" capabilities, meaning it can resolve frequency differences below half the size of a Fourier frequency bin.

Subrahmanyan: Ultimately, this paper on Hankel low-rank matrix approximation for gravitational-wave data analysis shows how to use structured approximations to make sense of the vast amounts of overlapping signals we expect from future detectors.

Department of Applied Mathematics and Statistics, Johns Hopkins University · Department of Physics and Astronomy, West Virginia University · Center for Gravitational Waves and Cosmology (GWAC), West Virginia University · William H. Miller III Department of Physics & Astronomy, Johns Hopkins University · Department of Mathematics, University of Central Florida

astro-ph.HE, astro-ph.IM, gr-qc

Submitted: 2026-03-16

Updated: 2026-10-21

Comments: 18 pages, 5 figures, 2 tables; expanded discussion, new summary table; for associated implementation, see https://github.com/cosmovlad/lisa-hankel

Journal ref: Phys. Rev. D 114, 064079 (2026)

DOI: 10.1103/j9j3-fn3q

Code: https://github.com/cosmovlad/lisa-hankel

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 80/100

The gist: The gist Hankel low-rank matrix approximation for gravitational-wave data analysis explores denoising techniques based on embedding time series into Hankel matrices to reduce signal extraction to a

Key concepts

Hankel Matrix: A special matrix constructed from a time series where the entries are arranged along diagonals. In this context, it captures the temporal structure of gravitational-wave data, allowing complex signals to be represented by a much smaller set of parameters (low rank).
Low-Rank Approximation: The goal is to find a matrix that closely resembles the original noisy Hankel matrix but has a very low rank. This simplification reduces the complexity of analyzing massive datasets, making it easier to isolate the underlying physical signal from random noise.
ESPRIT Algorithm: A non-iterative method used to estimate frequencies and amplitudes within a structured low-rank problem. It leverages rotational invariance properties to efficiently solve for parameters, proving effective for extracting key features like Quasinormal Modes (QNMs) in gravitational wave signals.
Quasinormal Modes (QNMs): These are characteristic frequencies and damping times associated with the ringdown phase of a gravitational wave signal, particularly from black hole mergers. ESPRIT is used to recover these specific physical properties from the noisy data.

Terminology

Benchmark — 1× in this paper · explained in 2 episodes

Summary

The gist Hankel low-rank matrix approximation for gravitational-wave data analysis explores denoising techniques based on embedding time series into Hankel matrices to reduce signal extraction to a structured low-rank approximation problem, which is crucial for disentangling overlapping signals from instrumental noise in next-generation detectors like LISA.

Introduction and Motivation

The paper addresses the challenge of disentangling vast numbers of overlapping gravitational-wave signals from instrumental noise and one another in next-generation detectors such as LISA. This problem is framed as a structured low-rank approximation problem by embedding time series into Hankel matrices, where a superposition of n damped sinusoids corresponds to a matrix of rank 2n. The authors benchmark three Hankel-based algorithms—ESPRIT, Cadzow iterations, and iteratively reweighted least squares (IRLS)—using synthetic data tailored to GW applications. These methods achieve near-optimal performance consistent with Fisher matrix bounds, evidenced by an inverse-square scaling of the mismatch with the signal-to-noise ratio.

How it works

The core methodology involves embedding a time series into a Hankel matrix H, where the relationship between entries is given by hl = hˆl + wl = Xn k=1 a k e − γk l sin(2πfkl + ϕk). The problem of extracting the signal reduces to finding a low-rank Hankel matrix Hˆ which best approximates the original “noisy” Hankel matrix H. The objective is formally stated as minimizing H − H(h)F s.t. rank(H) ≤ r, H ∈ Hd1×d2.

The paper tests three specific scenarios for signal extraction:

One monochromatic signal: This serves as a consistency check and approximates a slowly evolving GW source.
A superposition of multiple monochromatic signals: This tests performance on overlapping signals, including the global fit problem.
Frequency separation: This tests how well algorithms resolve two signals with a small frequency separation.
Quasinormal modes (QNMs): The ESPRIT algorithm is used to recover QNM frequencies and damping times from the ringdown part of a GW signal.

Algorithms Tested

The paper evaluates three distinct low-rank Hankel denoising algorithms:

** ESPRIT Algorithm: This is a non-iterative method that estimates frequencies and amplitudes in the two constraint sets of the structured low-rank approximation problem. It involves a two-step projection method onto the set of matrices with rank at most r and then uses rotational invariance properties to extract parameters. The algorithm is presented in Algorithm 1. It is noted that ESPRIT might not lead to optimal solutions of the underlying SLRA problem because it is inherently non-iterative. The algorithm can be used to obtain frequencies and amplitudes by solving a convex optimization problem after the frequencies are known. It is also noted that ESPRIT and Cadzow iterations are most effective at simultaneously extracting a few of the loudest QNMs in a signal. The implementation of ESPRIT can be done in O(rLlog L) time. 2. Cadzow Iterations: This is an iterative approach that alternatingly projects onto the space of rank r matrices and onto the subspace of Hankel matrices. The process starts with H(0) = H(h) and iteratively computes H(t) by projecting He (t-1) to the closest Hankel matrix via antidiagonal averaging. A drawback is that due to the non-convex nature of the set of matrices of rank at most r, it is generally unclear whether Algorithm 2 returns a global minimizer of (16). 3. Iteratively Reweighted Least Squares (IRLS): This method adapts the IRLS framework to the denoising problem by using a continuously differentiable, but similarly non-convex surrogate rank function Jε(H). It minimizes a quadratic surrogate of Jε(H) centered about the Hankelized estimate H(t) of the time series vector h(t). The algorithm is formulated as Algorithm 3. It includes an adaptive scheduler for the regularization parameter λ to drive the algorithm towards a meaningful solution. In this refinement step, the spectral tail ratio βspec is monitored to increase λ if the solution meets tolerance but βspec remains below a threshold close to unity. The IRLS method has the potential for finding better solutions of the underlying structured low-rank approximation problem through its iterative nature involving the optimization of non-convex, but continuous rank surrogates with decreasing smoothing parameter. 4. ESPRIT and Cadzow combined: The authors demonstrate that ESPRIT and the combined Cadzow-ESPRIT algorithms are most effective at simultaneously extracting a few of the loudest QNMs in a signal. The sequential extraction approach for QNM spectroscopy resulted in poorer recovery of the remaining modes. 5. Computational Efficiency: Core operations like multiplication by a vector can be implemented using the Fast Fourier Transform (FFT), resulting in an efficient scaling of the Hankel-based algorithms with the length of the time series, L. This allows for computing step 1 of Algorithm 1 or step 5 of Algorithm 3 in O(rLlog L) time. The implementation of IRLS is based on NumPy and SciPy and makes use of SciPy’s LinearOperator class for computational efficiency. 6. Noise Handling: The analysis assumes signals embedded in stationary, white Gaussian noise, with a straightforward extension to Gaussian noise with an arbitrary PSD via a whitening procedure. This can be done by applying a weighted projection onto the Hankel subspace or by using a weighted matrix norm in the optimization problem instead of the standard Frobenius norm. 7. Parameter Estimation: Postprocessing resulting denoised time series hb with steps 2-5 of Algorithm 1 can be used to obtain frequencies and amplitudes relevant for the time series at hand. The results for the one-signal experiment show that all methods clearly exhibit the expected inverse-square relationship, M ∝ ρ −2. The best-fit power-law exponents are −2.02+0.04−0.04 for ESPRIT, −2.09+0.04−- - - 1.95+ - 2. The results for multiple signals show that the scatter in the IRLS data points (green dots) is visibly smaller than that for Cadzow (orange crosses), which in turn is smaller than for ESPRIT (blue stars). The location of the “elbow” in the mean-square residual plot can serve as a heuristic for estimating the number of signals in real data.

Improvements for AI systems

High-speed denoising for gravitational-wave data streams can be achieved by embedding time series into Hankel matrices and solving a structured low-rank approximation problem. This allows AI systems to reduce noisy detector signals to a superposition of n damped sinusoids by finding the best rank r matrix, as described by the problem in Eq. 16.
Parameter estimation for gravitational-wave sources can be improved using ESPRIT or Cadzow iterations, which are non-iterative and iterative algorithms respectively. This enables AI systems to recover frequencies and amplitudes of signals from noisy data with performance consistent with Fisher matrix bounds, as shown in the results section of the paper.
The system can estimate the true number of components in a signal mixture by identifying an “elbow” in the mean-square residual as a function of trial number. This feature allows AI systems to determine how many underlying physical signals are present when they are not known beforehand, which is crucial for tackling complex global fit problems.
The system can perform frequency separation of closely spaced signals with high accuracy by using iterative algorithms like Cadzow, which demonstrate super-resolution capabilities. This means the AI system can resolve frequency differences below half the size of a Fourier frequency bin, improving the ability to disentangle overlapping signals in LISA data.
The system can recover black hole quasinormal modes from ringdown signals by applying ESPRIT directly or after Cadzow denoising. This allows AI systems to extract the complex frequencies and damping times of gravitational waves from simulations with high fidelity, serving as a proof-of-concept for signal extraction in numerical relativity waveforms.