A near-optimal Quadratic Goldreich-Levin algorithm
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913846780755968 |
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| author | Briët, Jop Castro-Silva, Davi |
| author_facet | Briët, Jop Castro-Silva, Davi |
| contents | In this paper, we give a quadratic Goldreich-Levin algorithm that is close to optimal in the following ways. Given a bounded function $f$ on the Boolean hypercube $\mathbb{F}_2^n$ and any $\varepsilon>0$, the algorithm returns a quadratic polynomial $q: \mathbb{F}_2^n \to \mathbb{F}_2$ so that the correlation of $f$ with the function $(-1)^q$ is within an additive $\varepsilon$ of the maximum possible correlation with a quadratic phase function. The algorithm runs in $O_\varepsilon(n^3)$ time and makes $O_\varepsilon(n^2\log n)$ queries to $f$, which matches the information-theoretic lower bound of $Ω(n^2)$ queries up to a logarithmic factor.
As a result, we obtain a number of corollaries:
- A near-optimal self-corrector of quadratic Reed-Muller codes, which makes $O_\varepsilon(n^2\log n)$ queries to a Boolean function $f$ and returns a quadratic polynomial $q$ whose relative Hamming distance to $f$ is within $\varepsilon$ of the minimum distance.
- An algorithmic polynomial inverse theorem for the order-3 Gowers uniformity norm.
- An algorithm that makes a polynomial number of queries to a bounded function $f$ and decomposes $f$ as a sum of poly$(1/\varepsilon)$ quadratic phase functions and error terms of order $\varepsilon$.
Our algorithm is obtained using ideas from recent work on quantum learning theory. Its construction deviates from previous approaches based on algorithmic proofs of the inverse theorem for the order-3 uniformity norm (and in particular does not rely on the recent resolution of the polynomial Fre\uıman-Ruzsa conjecture). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_13134 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A near-optimal Quadratic Goldreich-Levin algorithm Briët, Jop Castro-Silva, Davi Computational Complexity Combinatorics In this paper, we give a quadratic Goldreich-Levin algorithm that is close to optimal in the following ways. Given a bounded function $f$ on the Boolean hypercube $\mathbb{F}_2^n$ and any $\varepsilon>0$, the algorithm returns a quadratic polynomial $q: \mathbb{F}_2^n \to \mathbb{F}_2$ so that the correlation of $f$ with the function $(-1)^q$ is within an additive $\varepsilon$ of the maximum possible correlation with a quadratic phase function. The algorithm runs in $O_\varepsilon(n^3)$ time and makes $O_\varepsilon(n^2\log n)$ queries to $f$, which matches the information-theoretic lower bound of $Ω(n^2)$ queries up to a logarithmic factor. As a result, we obtain a number of corollaries: - A near-optimal self-corrector of quadratic Reed-Muller codes, which makes $O_\varepsilon(n^2\log n)$ queries to a Boolean function $f$ and returns a quadratic polynomial $q$ whose relative Hamming distance to $f$ is within $\varepsilon$ of the minimum distance. - An algorithmic polynomial inverse theorem for the order-3 Gowers uniformity norm. - An algorithm that makes a polynomial number of queries to a bounded function $f$ and decomposes $f$ as a sum of poly$(1/\varepsilon)$ quadratic phase functions and error terms of order $\varepsilon$. Our algorithm is obtained using ideas from recent work on quantum learning theory. Its construction deviates from previous approaches based on algorithmic proofs of the inverse theorem for the order-3 uniformity norm (and in particular does not rely on the recent resolution of the polynomial Fre\uıman-Ruzsa conjecture). |
| title | A near-optimal Quadratic Goldreich-Levin algorithm |
| topic | Computational Complexity Combinatorics |
| url | https://arxiv.org/abs/2505.13134 |