Let $A \in \mathbb{R}^{m \times n}$ be an arbitrary, known matrix and $e$ a $q$-sparse adversarial vector. Given $y = A x^\star + e$ and $q$, we seek the smallest robust solution set containing $x^\star$ that is uniformly recoverable from $y$ without knowing $e$. While exact recovery of $x^\star$ via strong (and often impractical) structural assumptions on $A$ or $x^\star$ (e.g., restricted isometry, sparsity) is well studied, recoverability for arbitrary $A$ and $x^\star$ remains open. Our main result shows that the smallest robust solution set is $x^\star + \ker(U)$, where $U$ is the unique projection matrix onto the intersection of rowspaces of all possible submatrices of $A$ obtained by deleting $2q$ rows. Moreover, we prove that every $x$ that minimizes the $\ell_0$-norm of $y - A x$ lies in $x^\star + \ker(U)$, which then gives a constructive approach to recover this set.

翻译：令 $A \in \mathbb{R}^{m \times n}$ 为任意已知矩阵，$e$ 为 $q$-稀疏对抗向量。给定 $y = A x^\star + e$ 和 $q$，我们寻求包含 $x^\star$ 的最小鲁棒解集，该解集可从 $y$ 中一致恢复，且无需知晓 $e$。尽管通过 $A$ 或 $x^\star$ 的强（且通常不实用）结构假设（例如限制等距性、稀疏性）来精确恢复 $x^\star$ 已得到充分研究，但针对任意 $A$ 和 $x^\star$ 的可恢复性问题仍然悬而未决。我们的主要结果表明，最小鲁棒解集为 $x^\star + \ker(U)$，其中 $U$ 是投影到通过删除 $2q$ 行得到的所有可能 $A$ 子矩阵行空间交集上的唯一投影矩阵。此外，我们证明每个最小化 $y - A x$ 的 $\ell_0$-范数的 $x$ 均位于 $x^\star + \ker(U)$ 中，这进而提供了一种恢复该解集的构造性方法。