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Spectral Norm, Economical Sieve, and Linear Invariance Testing of Boolean Functions

Swarnalipa Datta, Arijit Ghosh, Chandrima Kayal, Manaswi Paraashar, Manmatha Roy

TL;DR

This work studies tolerant linear isomorphism testing of Boolean functions in the known--unknown model, distinguishing when an unknown function $f$ can be linearly transformed to a known function $g$ within a tolerance $\epsilon$ or is at least $\epsilon+\omega$ far. The authors develop a non-adaptive randomized tester with query complexity $\widetilde{O}((m/\omega)^4)$, where $m$ bounds the spectral norm of $g$, significantly improving the prior $\widetilde{O}((m/\omega)^{24})$ bound. A central technical contribution is the Economical Sieve, a query-efficient local list-corrector that isolates and leverages heavy Fourier coefficients without explicit coefficient identification. They also prove a near-matching lower bound $\Omega(m^2)$ (for constant $\omega$) via a reduction from the Approximate Matrix Rank problem using Maiorana--McFarland constructions, strengthening prior lower bounds. The results nearly close the gap between upper and lower bounds for tolerant linear isomorphism testing and introduce tools (Economical Sieve) that may find broader use in invariant property testing and Fourier-analytic analysis of Boolean functions.

Abstract

Given Boolean functions \( f, g : \mathbb{F}_2^n \to \{-1,+1\} \), we say they are {\em linearly isomorphic} if there exists \( A \in \mathrm{GL}_n(\mathbb{F}_2) \) such that \( f(x)=g(Ax) \) for all \( x \). We study this problem in the tolerant property testing framework under the known--unknown model, where \( g \) is given explicitly and \( f \) is accessible only via oracle queries, meaning the algorithm may adaptively request the value of \( f(x) \) for inputs \( x \in \mathbb{F}_2^n \) of its choice. Given parameters \( ε\ge 0 \) and \( ω>0 \), the goal is to distinguish whether there exists \( A \in \mathrm{GL}_n(\mathbb{F}_{2})\) such that the normalized Hamming distance between \( f \) and \( g(Ax) \) is at most \( ε\), or whether for every \( A \in \mathrm{GL}_n(\mathbb{F}_2) \) the distance is at least \( ε+ω\). Our main result is a tolerant tester making \( \widetilde{O} \left( \left( m/ω\right)^4 \right) \) queries to \( f \), where \( m \) is an upper bound on the spectral norm of \( g \), improving the previous \( \widetilde{O} \left( \left( m/ω\right)^{24} \right) \) bound of Wimmer and Yoshida. We complement this with a nearly matching lower bound of \( Ω(m^2) \) for constant \( ω\) (for example, \( ω=1/4 \)), improving the prior \( Ω(\log m) \) lower bound of Grigorescu, Wimmer and Xie. A key technical ingredient on the algorithmic side is a query-efficient local list corrector. For the lower bound, we give a reduction from communication complexity using a novel subclass of Maiorana--McFarland functions from symmetric-key cryptography.

Spectral Norm, Economical Sieve, and Linear Invariance Testing of Boolean Functions

TL;DR

This work studies tolerant linear isomorphism testing of Boolean functions in the known--unknown model, distinguishing when an unknown function can be linearly transformed to a known function within a tolerance or is at least far. The authors develop a non-adaptive randomized tester with query complexity , where bounds the spectral norm of , significantly improving the prior bound. A central technical contribution is the Economical Sieve, a query-efficient local list-corrector that isolates and leverages heavy Fourier coefficients without explicit coefficient identification. They also prove a near-matching lower bound (for constant ) via a reduction from the Approximate Matrix Rank problem using Maiorana--McFarland constructions, strengthening prior lower bounds. The results nearly close the gap between upper and lower bounds for tolerant linear isomorphism testing and introduce tools (Economical Sieve) that may find broader use in invariant property testing and Fourier-analytic analysis of Boolean functions.

Abstract

Given Boolean functions , we say they are {\em linearly isomorphic} if there exists \( A \in \mathrm{GL}_n(\mathbb{F}_2) \) such that \( f(x)=g(Ax) \) for all . We study this problem in the tolerant property testing framework under the known--unknown model, where is given explicitly and is accessible only via oracle queries, meaning the algorithm may adaptively request the value of \( f(x) \) for inputs of its choice. Given parameters and , the goal is to distinguish whether there exists \( A \in \mathrm{GL}_n(\mathbb{F}_{2})\) such that the normalized Hamming distance between and \( g(Ax) \) is at most , or whether for every \( A \in \mathrm{GL}_n(\mathbb{F}_2) \) the distance is at least . Our main result is a tolerant tester making \( \widetilde{O} \left( \left( m/ω\right)^4 \right) \) queries to , where is an upper bound on the spectral norm of , improving the previous \( \widetilde{O} \left( \left( m/ω\right)^{24} \right) \) bound of Wimmer and Yoshida. We complement this with a nearly matching lower bound of \( Ω(m^2) \) for constant (for example, ), improving the prior \( Ω(\log m) \) lower bound of Grigorescu, Wimmer and Xie. A key technical ingredient on the algorithmic side is a query-efficient local list corrector. For the lower bound, we give a reduction from communication complexity using a novel subclass of Maiorana--McFarland functions from symmetric-key cryptography.
Paper Structure (13 sections, 5 theorems, 62 equations, 1 table, 2 algorithms)

This paper contains 13 sections, 5 theorems, 62 equations, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Our Contribution
  4. Proof ideas
  5. Proof idea of Theorem \ref{['theorem: lineariso testing upperbound']}
  6. Proof idea of Theorem \ref{['theorem: lineariso testing lowerbound']}
  7. Background
  8. A query-efficient tester for the linear isomorphism problem
  9. Proof of Theorem \ref{['theorem: lineariso testing upperbound']}
  10. Proof of Lemma \ref{['lemma: economicsieve']}
  11. A lower bound for testing linear isomorphism
  12. Proof of Theorem \ref{['theorem: lineariso testing lowerbound']}
  13. Conclusion

Key Result

Theorem 1.2

Let $\epsilon \ge 0$ and $\omega > 0$. There exists a non-adaptive randomized algorithm (see Algorithm algo: tester for lineariso) that solves the $(\epsilon, \omega)$-Tolerant Linear Isomorphism Testing problem for a known function $g : \mathbb{F}_2^n \to \{-1, +1\}$ with spectral norm at most $m$

Theorems & Definitions (29)

  • Definition 1.1: $(\epsilon,\omega)$-Tolerant Linear Isomorphism Testing
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Lemma 4.1: Economical Sieve
  • Theorem 4.1
  • proof
  • Claim 4.2
  • proof
  • Claim 4.3
  • ...and 19 more