bibkey: bendory2017bispectrum
authors: Tamir Bendory, Nicolas Boumal, Chao Ma, Zhizhen Zhao, Amit Singer
year: 2017
title: Bispectrum Inversion with Application to Multireference Alignment
doi: 10.1109/TSP.2017.2775591
url: https://arxiv.org/abs/1705.00641v2
claim: Proposition II.2 gives source-dependent local stable bispectrum inversion for real or complex signals with nonvanishing DFT, using Frobenius error on a bispectrum estimator.
strata_touched:
  - D5/S3/Fourier/DCDominantFullBispectrumStability
license: citation-only
triage: anchor

Stable inversion and conventions

Proposition II.2 states that an estimator exists such that, for each real or complex signal x with nonvanishing DFT, there are constants delta(x)>0 and L(x)<infinity. If an estimated bispectrum differs from the true bispectrum by at most delta(x) in Frobenius norm, the estimator’s signal error, minimized over circular shifts in counting Euclidean norm, is at most L(x) times that Frobenius error. These constants depend on the source signal. This is an uncalibrated local inverse result, so removal of independent DC calibration is not claimed as new here.

Equation II.3 uses the unnormalized negative-exponent DFT: translating x to x(n-s) multiplies its kth coefficient by exp(-2 pi i k s/N). Equation II.5 writes the bispectrum as y(k1) conjugate(y(k2)) y(k2-k1). The bijective index change (k1,k2)=(k,k+l) gives y(k)y(l)conjugate(y(k+l)), the convention used by DCDominantFullBispectrumStability. This permutation preserves the set of entries, the uniform entrywise error, and the Frobenius norm. Uniform entrywise error epsilon on N squared entries implies Frobenius error at most N epsilon; the two error hypotheses are not interchangeable with a dimension-independent constant.

The repository theorem compares the true full bispectra of two actual complex signals, assumes a positive source spectral lower bound and source DC dominance, and gives the explicit radius 2 epsilon/m squared, including signal-ball uniqueness. It does not construct the paper’s estimator on arbitrary perturbed matrices. The explicit estimate is a repository synthesis using existing phase correction; no literature novelty claim is attached to it.

Verified locator

• DOI: https://doi.org/10.1109/TSP.2017.2775591
• Versioned source: https://arxiv.org/abs/1705.00641v2
• Full text: https://ar5iv.labs.arxiv.org/html/1705.00641; Section II, equations II.3 and II.5, and Proposition II.2. The arXiv record lists the original posting as 2017-05-01 and version 2 as 2017-10-06.