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New Direct Sum Tests

Authors Alek Westover , Edward Yu , Kai Zhe Zheng

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LIPIcs.ITCS.2025.94.pdf
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Author Details

Alek Westover
  • MIT, Cambridge, MA, USA
Edward Yu
  • MIT, Cambridge, MA, USA
Kai Zhe Zheng
  • MIT, Cambridge, MA, USA

Funding

  • Westover, Alek: Supported by MIT SPUR.
  • Yu, Edward: Supported by MIT SPUR.
  • Zheng, Kai Zhe: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, USA. Supported by the NSF GRFP DGE-2141064.

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Alek Westover, Edward Yu, and Kai Zhe Zheng. New Direct Sum Tests. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 94:1-94:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.94

Abstract

A function f:[n]^{d} → 𝔽₂ is a direct sum if there are functions L_i:[n] → 𝔽₂ such that f(x) = ∑_i L_i(x_i). In this work we give multiple results related to the property testing of direct sums. 
Our first result concerns a test proposed by Dinur and Golubev in [Dinur and Golubev, 2019]. We call their test the Diamond test and show that it is indeed a direct sum tester. More specifically, we show that if a function f is ε-far from being a direct sum function, then the Diamond test rejects f with probability at least Ω_{n,ε}(1). Even in the case of n = 2, the Diamond test is, to the best of our knowledge, novel and yields a new tester for the classic property of affinity. 
Apart from the Diamond test, we also analyze a broad family of direct sum tests, which at a high level, run an arbitrary affinity test on the restriction of f to a random hypercube inside of [n]^d. This family of tests includes the direct sum test analyzed in [Dinur and Golubev, 2019], but does not include the Diamond test. As an application of our result, we obtain a direct sum test which works in the online adversary model of [Iden Kalemaj et al., 2022].
Finally, we also discuss a Fourier analytic interpretation of the diamond tester in the n = 2 case, as well as prove local correction results for direct sum as conjectured by [Dinur and Golubev, 2019].

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Linearity testing
  • Direct sum
  • Grids

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