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

2025-10-28

AI Summary

This work addresses the recoverability of an unknown ground-truth signal $x^*$ from linear observations $y = Ax^* + e$ corrupted by sparse adversarial noise $e$ with unknown support but known sparsity level $q$. Specifically, it asks: given only $q$, what is the minimal consistent recovery set—the smallest set containing $x^*$ that carries maximal informative structure? Prior approaches rely on strong structural assumptions (e.g., RIP or signal sparsity), whereas this paper provides, for the first time, an exact characterization valid for *any* sensing matrix $A$: the minimal consistent recovery set is $x^* + ker(U)$, where $U$ is the restriction of $A$ to the intersection of its row space and a particular projection subspace. This reveals that the kernel structure of $U$ fundamentally governs recovery limits. Moreover, all $ell_0$-minimizing solutions necessarily lie within this set. The result establishes a unified, assumption-free recovery theory and yields a computationally tractable recovery framework.

Technology Category

• Machine Learning: Kernel Methods
• Game Theory and Economic Paradigms: Adversarial Learning
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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

Problem

• Recovering signal from sparse adversarial corruption
• Finding minimal recoverable set without structural assumptions
• Characterizing recoverable set via kernel of projection matrix

Innovation

• Recovers signal plus kernel of projection matrix
• Minimizes L0-norm of measurement residuals
• Uses rowspace intersection of all submatrices