What Can Be Recovered Under Sparse Adversarial Corruption? Assumption-Free Theory for Linear Measurements
Vishal Halder, Alexandre Reiffers-Masson, Abdeldjalil Aïssa-El-Bey, Gugan Thoppe
TL;DR
The paper addresses robust recovery from linear measurements under unknown $q$-sparse adversarial corruption by deriving a universal characterization of the recoverable information. It shows that the best one can hope to recover is the affine subspace $x^\star + \ker(U)$, where $U$ is the orthogonal projector onto the intersection of rowspaces of all submatrices $A_T$ formed by deleting $2q$ rows, i.e., $\mathcal{R} = \bigcap_{|T|=m-2q} \operatorname{rowspan}(A_T)$. A linear, $(A,q)$-robust function is given by $\mathscr{G}^*(x) = U x$, and $\ker(U)$ equals the span of the ambiguity set $S_q^{A}$, enabling a constructive recovery via $\ell_0$-decoding: any minimizer of $\|y - A x\|_0$ satisfies $U \hat{x} = U x^\star$. An algorithm is proposed to compute $U$ by aggregating information from all $A_T$, though it is exponential in the number of rows; empirical results illustrate the theory in small-scale network tomography settings. The work provides a universal, exact, assumption-free framework for adversarial robustness and lays the groundwork for scalable relaxations and randomized approaches.
Abstract
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.