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Estimating Multi-chirp Parameters using Curvature-guided Langevin Monte Carlo

Sattwik Basu, Debottam Dutta, Yu-Lin Wei, Romit Roy Choudhury

TL;DR

This work tackles estimating parameters of mixtures of high-order chirps from noisy observations by formulating a non-convex optimization problem over the phase parameters. It introduces Curvature-guided Langevin Monte Carlo (CG-LMC), which couples Langevin dynamics with an adaptive Gaussian smoothing controlled by the average curvature of the objective, enabling reliable convergence to the global minimum even at low SNR. The approach outperforms standard LMC and noise-annealed LMC baselines in synthetic experiments, maintains robustness to initialization, and includes a curvature-based mechanism to adjust the smoothing scale during optimization. The proposed method has practical implications for high-dimensional chirp parameter estimation in applications like radar, audio, and biomedical sensing, with future work pointing toward CRLB comparisons and source separation strategies.

Abstract

This paper considers the problem of estimating chirp parameters from a noisy mixture of chirps. While a rich body of work exists in this area, challenges remain when extending these techniques to chirps of higher order polynomials. We formulate this as a non-convex optimization problem and propose a modified Langevin Monte Carlo (LMC) sampler that exploits the average curvature of the objective function to reliably find the minimizer. Results show that our Curvature-guided LMC (CG-LMC) algorithm is robust and succeeds even in low SNR regimes, making it viable for practical applications.

Estimating Multi-chirp Parameters using Curvature-guided Langevin Monte Carlo

TL;DR

This work tackles estimating parameters of mixtures of high-order chirps from noisy observations by formulating a non-convex optimization problem over the phase parameters. It introduces Curvature-guided Langevin Monte Carlo (CG-LMC), which couples Langevin dynamics with an adaptive Gaussian smoothing controlled by the average curvature of the objective, enabling reliable convergence to the global minimum even at low SNR. The approach outperforms standard LMC and noise-annealed LMC baselines in synthetic experiments, maintains robustness to initialization, and includes a curvature-based mechanism to adjust the smoothing scale during optimization. The proposed method has practical implications for high-dimensional chirp parameter estimation in applications like radar, audio, and biomedical sensing, with future work pointing toward CRLB comparisons and source separation strategies.

Abstract

This paper considers the problem of estimating chirp parameters from a noisy mixture of chirps. While a rich body of work exists in this area, challenges remain when extending these techniques to chirps of higher order polynomials. We formulate this as a non-convex optimization problem and propose a modified Langevin Monte Carlo (LMC) sampler that exploits the average curvature of the objective function to reliably find the minimizer. Results show that our Curvature-guided LMC (CG-LMC) algorithm is robust and succeeds even in low SNR regimes, making it viable for practical applications.

Paper Structure

This paper contains 14 sections, 27 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Multi-component Chirps
  4. MAP estimation
  5. Nonlinear Least Squares
  6. Curvature-Guided Langevin Monte Carlo (CG-LMC)
  7. Background
  8. LMC for optimizing $J(\boldsymbol\varphi)$
  9. Gaussian-smoothed Langevin Monte Carlo
  10. Need for Curvature-guided Gaussian-smoothing
  11. Experiments and Results
  12. Conclusions and Future Work
  13. Acknowledgements
  14. Gradient of the Objective Function

Figures (3)

  • Figure 1: Real part of a damped chirp with 4-th order polynomial IP.
  • Figure 2: A slice of the non-convex 4-D objective function for a two-chirp mixture with $P=2$. Parameters $\varphi_{12}$, $\varphi_{22}$ are varied while others are fixed at optimum values.
  • Figure 3: (a) 9 sample runs in CG-LMC and 3 in NA-LMC, (b) Comparing $\sigma$ adaptation in CG-LMC vs. the fixed schedule in NA-LMC, (c) Visualizing $\mathrm{tr}\left\{\nabla_{\boldsymbol\varphi}^2 J\right\}$ over different CG-LMC runs.