发表机构
UC Davis（加州大学戴维斯分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对对抗子空间扰动下的三维定位问题，提出可验证球形MUSIC算法，推导其收敛性、充分条件及定位模，构造反例验证指数最优性。
AI 中文摘要
我们研究一种神谕子空间扰动模型，其中三维定位过程被提供一个s维子空间\\(\widetilde{\mathcal U}\\)和确定性误差界\\(\eps_{\rm sub}\\)，用于衡量\\(\widetilde{\mathcal U}\\)与波数为\\(\kappa\\)的远场模式的s维子空间\\(\cU\\)之间的正弦-θ距离。在明确的任意点云分离和条件假设下，我们证明了扰动后的球形MUSIC目标函数\\(\widetilde q(\bz) = 1-\\|P_{\widetilde{\mathcal U}}\varphi_\bz\\|_2^2\\)在每个球\\(B_{\gamma/\kappa}(x_j)\\)中都有一个唯一的强凸井，且该目标函数在可验证井的并集之外具有均匀的值间隙。步长为\\(h\asymp\kappa^{-2}\\)的固定步长梯度映射会使每个可验证井保持不变，并线性收敛到其唯一的极小值点。因此，对\\(O(\kappa^{-1})\\)网格进行阈值处理，接着从所有接受的网格点开始梯度下降并去除重复项，即可恢复所有相关的极小值点。任意点云框架分析通过绝对相干行和与Gershgorin定理给出了充分条件\\(\kappa\delta_X\gtrsim s^{2/3}\\)。我们还构造了低于\\(s^{1/6}\\)尺度的下框架反例、低于\\(s^{1/3}\\)尺度的上框架反例，以及表明指数\\(2/3\\)对于绝对行和论证是最优的示例。后者是证明方法的尖锐性结果，而非谱必要性主张。最后，对于包含均匀可容许单点位移路径的参数类，我们证明确定性神谕定位模为\\(\mathfrak R(\eps) \asymp \frac{\eps}{\kappa}\\)。
英文摘要
We study an oracle subspace-perturbation model for 3D localization. The procedure is given an $s$-dimensional subspace $\widetilde{\mathcal U}$ and a deterministic error bound \[ \varepsilon_{\mathrm{sub}} = \left\|P_{\widetilde{\mathcal U}}-P_{\mathcal U}\right\|_2, \] where $\mathcal{U}$ is an $s$-dimensional subspace of far-field patterns with wavenumber $κ$, and $P_{\mathcal{V}}$ denotes orthogonal projection onto a subspace $\mathcal{V}$. Under explicit separation and conditioning hypotheses for arbitrary point clouds, we prove that the perturbed spherical MUSIC objective \[ \widetilde{q}({\mathbf z}) = 1-\left\|P_{\widetilde{\mathcal{U}}}φ_{\mathbf z}\right\|_2^2 \] 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 $h\asympκ^{-2}$ 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 \[ \mathfrak{R}(\varepsilon_{\mathrm{sub}}) \lesssim \frac{\varepsilon_{\mathrm{sub}}}κ. \] The arbitrary-cloud frame analysis yields the sufficient condition \[ κδ_X\gtrsim s^{2/3} \] 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 \[ \mathfrak{R}(\varepsilon_{\mathrm{sub}}) \sim \frac{\varepsilon_{\mathrm{sub}}}κ. \]
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