Certified Spherical MUSIC for 3D Localization under Adversarial Subspace Perturbations

Abstract: We study an oracle subspace-perturbation model in which the 3D localization procedure is given an (s)-dimensional subspace (\widetilde{\mathcal U}) and a deterministic error bound $\eps_{\rm sub}$ measuring the sine-theta distance between $\widetilde{\mathcal U}$ and an $s$-dimensional subspace $\cU$ of far-field patterns with wavenumber $κ$. Under explicit arbitrary-cloud separation and conditioning hypotheses, we prove that the perturbed spherical MUSIC objective [ \widetilde q(\bz) = 1-|P_{\widetilde{\mathcal U}}\varphi_\bz|2^2 ] has a unique strongly convex well in each ball (B{γ/κ}(x_j)) and the objective has a uniform value gap outside the union of the certified wells. A fixed-step gradient map with (h\asympκ^{-2}) leaves every certified well invariant and converges linearly to its unique minimizer. Consequently, thresholding on an (O(κ^{-1}))-mesh, followed by gradient descent from all accepted grid points and duplicate removal, recovers all relevant minima. The arbitrary-cloud frame analysis gives the sufficient condition [ κδ_X\gtrsim s^{2/3} ] through an absolute coherence row sum and Gershgorin's theorem. We also construct lower-frame counterexamples below the (s^{1/6}) scale, upper-frame counterexamples below the (s^{1/3}) scale, and examples showing that the exponent (2/3) is optimal for the absolute-row-sum argument. The latter is a sharpness result for the proof method and is not a spectral necessity claim. Finally, for parameter classes containing a uniformly admissible one-point displacement path, we prove that the deterministic oracle localization modulus is [ \mathfrak R(\eps) \asymp \frac{\eps}κ. ]