What Can Be Recovered Under Sparse Adversarial Corruption? Assumption-Free Theory for Linear Measurements

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 set containing $x^\star$--hence the one conveying maximal information about $x^\star$--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 best that one can hope to recover 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.