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Estimating Euclidean Distance to Linearity

Authors Andrej Bogdanov , Lorenzo Taschin

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Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Theory of computation → Randomness, geometry and discrete structures
  • Theory of computation → Probabilistic computation
Keywords
  • sublinear-time algorithms
  • statistical estimation
  • analysis of boolean functions
  • property testing
  • regression

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Abstract

Given oracle access to a real-valued function on the n-dimensional Boolean cube, how many queries does it take to estimate the squared Euclidean distance to its closest linear function within ε? Our main result is that O(log³(1/ε) ⋅ 1/ε²) queries suffice. Not only is the query complexity independent of n but it is optimal up to the polylogarithmic factor.
Our estimator evaluates f on pairs correlated by noise rates chosen to cancel out the low-degree contributions to f while leaving the linear part intact. The query complexity is optimized when the noise rates are multiples of Chebyshev nodes.
In contrast, we show that the dependence on n is unavoidable in two closely related settings. For estimation from random samples, Θ(√n/ε + 1/ε²) samples are necessary and sufficient. For agnostically learning a linear approximation with ε mean-square regret under the uniform distribution, Ω(n/√ε) nonadaptively chosen queries are necessary, while O(n/ε) random samples are known to be sufficient (Linial, Mansour, and Nisan). 
Our upper bounds apply to functions with bounded 4-norm. Our lower bounds apply even to ± 1-valued functions.

Cite As Get BibTex

Andrej Bogdanov and Lorenzo Taschin. Estimating Euclidean Distance to Linearity. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.20

Author Details

Andrej Bogdanov
  • University of Ottawa, Canada
Lorenzo Taschin
  • EPFL, Lausanne, Switzerland

Funding

This work was supported by an NSERC discovery grant.

Acknowledgements

We thank Gautam Prakriya for insightful discussions and the anonymous ITCS reviewers for corrections and helpful comments.

References

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