We consider an uncertain linear inverse problem as follows. Given observation $\omega=Ax_*+\zeta$ where $A\in {\bf R}^{m\times p}$ and $\zeta\in {\bf R}^{m}$ is observation noise, we want to recover unknown signal $x_*$, known to belong to a convex set ${\cal X}\subset{\bf R}^{n}$. As opposed to the "standard" setting of such problem, we suppose that the model noise $\zeta$ is "corrupted" -- contains an uncertain (deterministic dense or singular) component. Specifically, we assume that $\zeta$ decomposes into $\zeta=N\nu_*+\xi$ where $\xi$ is the random noise and $N\nu_*$ is the "adversarial contamination" with known $\cal N\subset {\bf R}^n$ such that $\nu_*\in \cal N$ and $N\in {\bf R}^{m\times n}$. We consider two "uncertainty setups" in which $\cal N$ is either a convex bounded set or is the set of sparse vectors (with at most $s$ nonvanishing entries). We analyse the performance of "uncertainty-immunized" polyhedral estimates -- a particular class of nonlinear estimates as introduced in [15, 16] -- and show how "presumably good" estimates of the sort may be constructed in the situation where the signal set is an ellitope (essentially, a symmetric convex set delimited by quadratic surfaces) by means of efficient convex optimization routines.

翻译：我们考虑如下不确定线性逆问题。给定观测值 $\omega=Ax_*+\zeta$，其中 $A\in {\bf R}^{m\times p}$ 且 $\zeta\in {\bf R}^{m}$ 为观测噪声，我们希望恢复未知信号 $x_*$（已知其属于凸集 ${\cal X}\subset{\bf R}^{n}$）。与此类问题的"标准"设定不同，我们假设模型噪声 $\zeta$ 是"被污染"的——包含一个不确定的（确定性稠密或奇异）分量。具体而言，我们假设 $\zeta$ 可分解为 $\zeta=N\nu_*+\xi$，其中 $\xi$ 为随机噪声，$N\nu_*$ 为"对抗性污染"项，已知 $\cal N\subset {\bf R}^n$ 满足 $\nu_*\in \cal N$ 且 $N\in {\bf R}^{m\times n}$。我们考虑两种"不确定性设定"：$\cal N$ 要么是凸有界集，要么是稀疏向量集（至多包含 $s$ 个非零元素）。我们分析了"不确定性免疫"多面体估计量（即[15, 16]中提出的特定非线性估计量类别）的性能，并展示了当信号集为椭球拓扑集（本质上是由二次曲面限定的对称凸集）时，如何通过高效凸优化算法构建此类"可能最优"的估计量。