Derandomized Non-Abelian Homomorphism Testing in Low Soundness Regime
Tushant Mittal, Sourya Roy
TL;DR
We address the problem of efficiently testing whether a matrix-valued function $f: G \to \mathbb{U}_t$ on a finite group $G$ correlates with a group homomorphism, in the challenging low-soundness setting. The authors introduce a derandomized Blum–Luby–Rubinfeld test using small-bias sets and develop a degree-2 expander mixing lemma to approximate the non-Abelian $U^2$-norm, enabling robust inference from test outcomes. They prove two main results: (i) a derandomized homomorphism testing theorem showing correlation with a clipped representation and the existence of a low-dimensional irrep with substantial Fourier mass; (ii) a derandomized BNP lemma bounding matrix-valued convolution over biased sets, with bounds that depend on the smallest nontrivial irrep dimension $D$. These results generalize prior full-randomness conclusions of Gowers–Hatami and Moore–Russell to near-optimal derandomized settings, achieving $(1+o(1))\log|G|$ randomness and revealing structural implications for $f$, including correlation to representations and low-dimension Fourier mass. The work advances derandomized testing for non-Abelian groups and has potential ramifications for PCPs, hardness of approximation, and quantum testing, by providing tools to certify proximity to homomorphisms under limited randomness. $${}$
Abstract
We give a randomness-efficient homomorphism test in the low soundness regime for functions, $f: G\to \mathbb{U}_t$, from an arbitrary finite group $G$ to $t\times t$ unitary matrices. We show that if such a function passes a derandomized Blum--Luby--Rubinfeld (BLR) test (using small-bias sets), then (i) it correlates with a function arising from a genuine homomorphism, and (ii) it has a non-trivial Fourier mass on a low-dimensional irreducible representation. In the full randomness regime, such a test for matrix-valued functions on finite groups implicitly appears in the works of Gowers and Hatami [Sbornik: Mathematics '17], and Moore and Russell [SIAM Journal on Discrete Mathematics '15]. Thus, our work can be seen as a near-optimal derandomization of their results. Our key technical contribution is a "degree-2 expander mixing lemma'' that shows that Gowers' $\mathrm{U}^2$ norm can be efficiently estimated by restricting it to a small-bias subset. Another corollary is a "derandomized'' version of a useful lemma due to Babai, Nikolov, and Pyber [SODA'08] and Gowers [Comb. Probab. Comput.'08].