发表机构
School of Electrical Engineering, Iran University of Science and Technology（伊朗科学技术大学电气工程学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究从非线性观测中恢复一对稀疏向量的问题，提出结合Huber化数据保真度与广义折叠凹惩罚的正则化框架及NLD - PALM算法，推导估计误差界，实验显示该方法在精度和相变上优于基线，能成功分离特定信号。
AI 中文摘要
我们考虑从其叠加的有限数量非线性观测中恢复一对稀疏向量：$y_i = g(\langle\ba_i\rangle{\bPhi\bw^\ast+\bPsi\bz^\ast})+e_i$，$i = 1,\dots,m$，其中$m\ll n$，$\bPhi,\bPsi$为非相干正交基，$g$为标量链接，$e_i$为可能重尾或受污染的噪声。我们提出一个基于正则化的框架，结合了Huber化数据保真度与广义折叠凹惩罚（SCAD、MCP），以及带回溯的两块近端交替算法（NLD - PALM），其整个迭代序列在Kurdyka - Łojasiewicz性质下可证明收敛到临界点，具有局部线性速率。在统计方面，我们通过精确的符号正定分解建立了Huber化非线性损失的受限强凸性，并推导了在每个局部驻点处成立的阶为$\sigma\sqrt{s\log(n)/m}$的估计误差界，在β - 最小条件下无$\log n$和收缩偏差的神谕速率$\sigma\sqrt{s/m}$，以及通过线性替代和截断的Plan - Vershynin解耦对未知单调链接的共等恢复定理。该估计器不需要稀疏水平的知识，并且其保证在仅具有有限方差的对称噪声下成立。在$n = 512$下的实验表明，在冻结数据驱动的正则化规则下，比凸$\ell_1$分离和贪婪硬阈值基线有更早的相变，在$5\%$总异常值下比平方损失估计有$35$倍的精度优势，并且成功分离了通过饱和放大器观察到的尖峰加背景信号。
英文摘要
We consider the recovery of a pair of sparse vectors from a limited number of nonlinear observations of their superposition: $y_i=g(\inner{\ba_i}{\bPhi\bw^\ast+\bPsi\bz^\ast})+e_i$, $i=1,\dots,m$, with $m\ll n$, incoherent orthonormal bases $\bPhi,\bPsi$, a scalar link $g$, and noise $e_i$ that may be heavy-tailed or contaminated. We propose a regularization-based framework combining a Huberized data fidelity with generalized folded-concave penalties (SCAD, MCP), and a two-block proximal alternating algorithm with backtracking (NLD-PALM) whose whole iterate sequence provably converges to critical points under the Kurdyka--Łojasiewicz property, with local linear rates. On the statistical side we establish restricted strong convexity of the Huberized nonlinear loss through an exact sign-definite decomposition, and derive estimation error bounds of order $σ\sqrt{s\log(n)/m}$ that hold at \emph{every} localized stationary point, an oracle rate $σ\sqrt{s/m}$ free of $\log n$ and shrinkage bias under a beta-min condition, and a co-equal recovery theorem for \emph{unknown} monotone links via a linear surrogate and a clipped Plan--Vershynin decoupling. The estimator requires no knowledge of the sparsity levels, and its guarantees hold under symmetric noise with only finite variance. Experiments at $n=512$ under a frozen data-driven regularization rule show an earlier phase transition than convex $\ell_1$ demixing and greedy hard-thresholding baselines, a $35\times$ accuracy advantage over squared-loss estimation under $5\%$ gross outliers, and successful demixing of spike-plus-background signals observed through a saturating amplifier.
