Strong Inapproximability for a Promise Rank Problem

2026-05-12

AI Summary

This work investigates the computational hardness of finding low-rank matrices within a given linear subspace that is guaranteed to contain a rank‑1 matrix. Under the assumption that NP does not admit subexponential-time algorithms, it is shown that no polynomial-time algorithm can find a matrix of rank below $n^{o(1/\log\log n)}$, even when such a rank‑1 matrix exists. The paper introduces a novel combination of hypergraph consistency and moment-matrix techniques to establish strong inapproximability results without relying on the PCP framework, extending these results to arbitrary finite fields. Via two distinct reductions, the authors achieve a 1 vs. $k$ rank gap inapproximability within running times $n^{O(\log k)}$ and $n^{O(k)}$, respectively, substantially advancing the understanding of inapproximability in coding theory and lattice problems.

Abstract

Given a linear subspace of $n \times n$ matrices over $\mathbb F_{2^r}$ that is promised to contain a matrix of rank $1$, we prove that it is hard to find a matrix of rank $n^{o(1/\log \log n)}$, assuming NP doesn't have sub-exponential algorithms. In addition to being a basic problem, the hardness of this problem, even for the exact version, drove recent PCP-free inapproximability results for minimum distance and shortest vector problems concerning codes and lattices. The proof combines the concept of superposition soundness introduced by Khot and Saket with moment matrices. To produce a rank-gap of $1$ vs. $k$, the reduction runs in time $n^{O(\log k)}$. We also give another moment-matrix-based construction which runs in time $n^{O(k)}$ but works for any finite field $\mathbb F_q$.

Problem

• rank minimization
• inapproximability
• linear subspace
• matrix rank
• computational hardness

Innovation

• superposition soundness
• moment matrices
• rank inapproximability
• finite fields
• PCP-free hardness