Functional Multi-Target Detection via Bispectrum Inversion
This work provides a theoretical framework and practical algorithms for multi-target detection, which is important for signal processing applications where continuous, off-grid translations and correlated noise are present, extending beyond prior discrete models.
This paper addresses the problem of detecting multiple unknown translations of a compactly supported signal from a single noisy observation. The authors propose two uninitialized recovery algorithms that estimate the signal's bispectrum via a debiased third-order empirical autocorrelation and then recover the signal using either a functional frequency marching scheme or a Kotlarski-type deconvolution formula. They provide non-asymptotic recovery guarantees for both algorithms, demonstrating accurate recovery even in low-SNR regimes.
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.