Spherical ESPRIT by Paired Small Circles and Infinitesimal Rotations
Abstract
We study the recovery of a finite three-dimensional point cloud from its spherical Fourier signal subspace at a fixed wavenumber. Two ESPRIT constructions are analyzed. The first uses finite shifts between paired small circles; the second uses angular derivatives on the whole sphere. Both recover the same commuting coordinate matrices, either through their finite exponentials or directly from a coupled linear system. We prove that the continuous generator system is injective for every distinct point cloud and that one guard harmonic degree preserves the retained equations exactly. For s targets, the cutoff K≥ s gives generic finite identifiability, but not uniform conditioning. We also derive a finite-cutoff perturbation bound. The accuracy of the coordinate estimates is a separate issue. A one-target perturbation calculation shows that the original generator estimate can retain a nonzero bias as the wavenumber increases. To address this bias, we use the generator eigenbasis to separate approximate atoms and recover their locations from amplitude-normalized phases. Under pointwise atom dominance, the resulting error is O(κ−1); for a dominant two-atom mixture at fixed separation, it is O(κ−2). Experiments on structured point clouds with smooth perturbations compare the two constructions and their use as initializers for a common MUSIC refinement.
Related
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BibTeX
@misc{fannjiang2026spherical,
title = {{Spherical ESPRIT by Paired Small Circles and Infinitesimal Rotations}},
author = {Albert Fannjiang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33024},
eprint = {2609.33024},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33024},
}
RIS
TY - UNPB
TI - Spherical ESPRIT by Paired Small Circles and Infinitesimal Rotations
AU - Albert Fannjiang
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.33024
UR - https://arxiv.org/abs/2609.33024
AB - We study the recovery of a finite three-dimensional point cloud from its spherical Fourier signal subspace at a fixed wavenumber. Two ESPRIT constructions are analyzed. The first uses finite shifts between paired small circles; the second uses angular derivatives on the whole sphere. Both recover the same commuting coordinate matrices, either through their finite exponentials or directly from a coupled linear system. We prove that the continuous generator system is injective for every distinct point cloud and that one guard harmonic degree preserves the retained equations exactly. For s targets, the cutoff K≥ s gives generic finite identifiability, but not uniform conditioning. We also derive a finite-cutoff perturbation bound. The accuracy of the coordinate estimates is a separate issue. A one-target perturbation calculation shows that the original generator estimate can retain a nonzero bias as the wavenumber increases. To address this bias, we use the generator eigenbasis to separate approximate atoms and recover their locations from amplitude-normalized phases. Under pointwise atom dominance, the resulting error is O(κ^−1); for a dominant two-atom mixture at fixed separation, it is O(κ^−2). Experiments on structured point clouds with smooth perturbations compare the two constructions and their use as initializers for a common MUSIC refinement.
ER -