Recovery from linear measurements under sparse adversarial corruption is typically formulated as an exact-recovery problem: one seeks structural conditions on $A$ (e.g., the restricted isometry property) that guarantee unique recovery of $x^\star$ from $y = A x^\star + e$ with $\left\lVert e \right\rVert_0 \leq q$. However, in practice, these conditions are rarely met and are hard to verify, and so the existing guarantees provide no guidance once exact recovery fails. This limitation obscures even simple robustness phenomena -- for instance, repeated rows in $A$ can preserve nontrivial information about $x^\star$ under sparse corruption. In this paper, we address the more general question: for arbitrary $A \in \mathbb{R}^{m \times n}$, what information about $x^\star$ remains robust in $y$ despite any $q$-sparse adversarial corruption $e$? We show that the robust information is precisely $x^\star + \ker(U)$, where $U$ is the orthogonal projection onto the intersection of rowspaces of all submatrices of $A$ obtained by deleting $2q$ rows. This characterization clarifies, for each sparsity level $q$, how the row structure of $A$ determines whether a $q$-sparse $e$ allows exact, partial, or only trivial recovery, thereby extending the standard exact-recovery framework. We further prove that every $x$ that minimizes $\left\lVert y - A x \right\rVert_0$ belongs to $x^\star + \ker(U)$, yielding a constructive approach to recover this set. For i.i.d. Gaussian $A$, we show a sharp phase transition: depending on $m$, $n$, and $q$, either exact recovery holds or no nontrivial recovery is possible. We sketch two applications: robust network tomography and signal reconstruction from oversampled DCT measurements.

翻译：在稀疏对抗扰动下从线性测量中恢复通常被表述为精确恢复问题：即在$A$上寻找保证从$y = A x^\star + e$（其中$\left\lVert e \right\rVert_0 \leq q$）唯一恢复$x^\star$的结构条件（如限定等距性质）。然而在实践中，这些条件很少被满足且难以验证，因此现有保证在精确恢复失效时无法提供任何指导。这种局限性甚至掩盖了简单的鲁棒性现象——例如，$A$中的重复行在稀疏扰动下仍能保留关于$x^\star$的非平凡信息。本文针对更一般的问题展开研究：对于任意$A \in \mathbb{R}^{m \times n}$，尽管存在任意$q$稀疏对抗扰动$e$，$y$中关于$x^\star$的哪些信息保持鲁棒？我们证明，鲁棒信息恰好是$x^\star + \ker(U)$，其中$U$是投射到$A$删除$2q$行后所有子矩阵行空间交集上的正交投影。该刻画阐明了对于每个稀疏度$q$，$A$的行结构如何决定$q$稀疏扰动$e$是允许精确恢复、部分恢复还是仅平凡恢复，从而扩展了标准精确恢复框架。我们进一步证明，每个最小化$\left\lVert y - A x \right\rVert_0$的$x$都属于$x^\star + \ker(U)$，从而提供了恢复该集合的构造性方法。对于独立同分布的高斯矩阵$A$，我们发现尖锐相变现象：取决于$m,n,q$，要么实现精确恢复，要么无法进行任何非平凡恢复。最后简述两个应用：鲁棒网络层析成像与过采样DCT测量下的信号重建。