Functional Multi-Target Detection via Bispectrum Inversion
2026-05-29 • Information Theory
Information Theory
AI summaryⓘ
The authors study how to find a hidden signal that appears many times in a noisy recording, even when the signal shifts by unknown amounts and the noise is complicated. They create two methods that use a special mathematical tool called the bispectrum to first estimate the signal’s patterns, then recover the signal itself. They prove these methods can reliably recover the signal under broad conditions without needing the signal to be limited in frequency. Their experiments show the methods work well even when the signal is very noisy.
multi-target detectionbispectrumautocorrelationGaussian noisefrequency marchingdeconvolutionnon-asymptotic guaranteessignal recoverycompact supportlow SNR
Authors
Anna Little, Daniel Sanz-Alonso, Mikhail Sweeney, Ruiyi Yang
Abstract
This paper develops a functional theory for multi-target detection, where a compactly supported signal is recovered from a single noisy observation containing many unknown translations of the signal. Our formulation allows continuous, off-grid translations and correlated stationary Gaussian process noise, extending beyond the discrete, grid-aligned, white-noise models common in prior work. We analyze two uninitialized recovery algorithms based on autocorrelation analysis; in particular, both algorithms first estimate the signal's bispectrum via a debiased third-order empirical autocorrelation. The signal is then recovered from the estimated bispectrum using either a functional frequency marching scheme or a Kotlarski-type deconvolution formula. For both algorithms, we prove non-asymptotic recovery guarantees for compactly supported signals without bandlimiting assumptions. The resulting error bounds depend on the smoothness of the signal and the accuracy of bispectrum estimation, with the latter governed by the noise characteristics and the number of signal occurrences. Numerical experiments validate our theory and demonstrate accurate recovery in low-SNR regimes.