Certified Spherical MUSIC for 3D Localization under Adversarial Subspace Perturbations
Abstract
We study an oracle subspace-perturbation model for 3D localization. The procedure is given an s-dimensional subspace 𝒰̃ and a deterministic error bound εsub = |P𝒰̃−P𝒰|2, where 𝒰 is an s-dimensional subspace of far-field patterns with wavenumber κ, and P𝒱 denotes orthogonal projection onto a subspace 𝒱. Under explicit separation and conditioning hypotheses for arbitrary point clouds, we prove that the perturbed spherical MUSIC objective q̃(z) = 1−|P𝒰̃φz|22 has a unique strongly convex well in every ball Bγ/κ(xj) and a uniform value gap outside the union of the certified wells. A fixed-step gradient map with hasympκ−2 leaves each well invariant and converges linearly to its unique minimizer. Consequently, thresholding on an O(κ−1)-mesh, followed by gradient descent and duplicate removal, recovers all relevant minima with localization bound R(εsub) ≲ εsub/κ. The arbitrary-cloud frame analysis yields the sufficient condition κδX≳ s2/3 through an absolute coherence row sum and Gershgorin's theorem. We construct lower-frame counterexamples below the s1/6 scale, upper-frame counterexamples below the s1/3 scale, and examples showing that 2/3 is optimal for the absolute-row-sum argument. Finally, for parameter classes containing a uniformly admissible one-point displacement path, we prove R(εsub) ∼ εsub/κ.
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BibTeX
@misc{fannjiang2026certified,
title = {{Certified Spherical MUSIC for 3D Localization under Adversarial Subspace Perturbations}},
author = {Albert Fannjiang and Chuong Nguyen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03264},
eprint = {2609.03264},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03264},
}
RIS
TY - UNPB
TI - Certified Spherical MUSIC for 3D Localization under Adversarial Subspace Perturbations
AU - Albert Fannjiang
AU - Chuong Nguyen
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.03264
UR - https://arxiv.org/abs/2609.03264
AB - We study an oracle subspace-perturbation model for 3D localization. The procedure is given an s-dimensional subspace 𝒰̃ and a deterministic error bound ε_sub = |P_𝒰̃−P_𝒰|_2, where 𝒰 is an s-dimensional subspace of far-field patterns with wavenumber κ, and P_𝒱 denotes orthogonal projection onto a subspace 𝒱. Under explicit separation and conditioning hypotheses for arbitrary point clouds, we prove that the perturbed spherical MUSIC objective q̃(z) = 1−|P_𝒰̃φ_z|_2^2 has a unique strongly convex well in every ball B_γ/κ(x_j) and a uniform value gap outside the union of the certified wells. A fixed-step gradient map with hasympκ^−2 leaves each well invariant and converges linearly to its unique minimizer. Consequently, thresholding on an O(κ^−1)-mesh, followed by gradient descent and duplicate removal, recovers all relevant minima with localization bound R(ε_sub) ≲ ε_sub/κ. The arbitrary-cloud frame analysis yields the sufficient condition κδ_X≳ s^2/3 through an absolute coherence row sum and Gershgorin's theorem. We construct lower-frame counterexamples below the s^1/6 scale, upper-frame counterexamples below the s^1/3 scale, and examples showing that 2/3 is optimal for the absolute-row-sum argument. Finally, for parameter classes containing a uniformly admissible one-point displacement path, we prove R(ε_sub) ∼ ε_sub/κ.
ER -