A $d^{1/2+o(1)}$ Monotonicity Tester for Boolean Functions on $d$-Dimensional Hypergrids
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866913842886344704 |
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| author | Black, Hadley Chakrabarty, Deeparnab Seshadhri, C. |
| author_facet | Black, Hadley Chakrabarty, Deeparnab Seshadhri, C. |
| contents | Monotonicity testing of Boolean functions on the hypergrid, $f:[n]^d \to \{0,1\}$, is a classic topic in property testing. Determining the non-adaptive complexity of this problem is an important open question. For arbitrary $n$, [Black-Chakrabarty-Seshadhri, SODA 2020] describe a tester with query complexity $\widetilde{O}(\varepsilon^{-4/3}d^{5/6})$. This complexity is independent of $n$, but has a suboptimal dependence on $d$. Recently, [Braverman-Khot-Kindler-Minzer, ITCS 2023] and [Black-Chakrabarty-Seshadhri, STOC 2023] describe $\widetilde{O}(\varepsilon^{-2} n^3\sqrt{d})$ and $\widetilde{O}(\varepsilon^{-2} n\sqrt{d})$-query testers, respectively. These testers have an almost optimal dependence on $d$, but a suboptimal polynomial dependence on $n$.
In this paper, we describe a non-adaptive, one-sided monotonicity tester with query complexity $O(\varepsilon^{-2} d^{1/2 + o(1)})$, independent of $n$. Up to the $d^{o(1)}$-factors, our result resolves the non-adaptive complexity of monotonicity testing for Boolean functions on hypergrids. The independence of $n$ yields a non-adaptive, one-sided $O(\varepsilon^{-2} d^{1/2 + o(1)})$-query monotonicity tester for Boolean functions $f:\mathbb{R}^d \to \{0,1\}$ associated with an arbitrary product measure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_01416 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A $d^{1/2+o(1)}$ Monotonicity Tester for Boolean Functions on $d$-Dimensional Hypergrids Black, Hadley Chakrabarty, Deeparnab Seshadhri, C. Data Structures and Algorithms Monotonicity testing of Boolean functions on the hypergrid, $f:[n]^d \to \{0,1\}$, is a classic topic in property testing. Determining the non-adaptive complexity of this problem is an important open question. For arbitrary $n$, [Black-Chakrabarty-Seshadhri, SODA 2020] describe a tester with query complexity $\widetilde{O}(\varepsilon^{-4/3}d^{5/6})$. This complexity is independent of $n$, but has a suboptimal dependence on $d$. Recently, [Braverman-Khot-Kindler-Minzer, ITCS 2023] and [Black-Chakrabarty-Seshadhri, STOC 2023] describe $\widetilde{O}(\varepsilon^{-2} n^3\sqrt{d})$ and $\widetilde{O}(\varepsilon^{-2} n\sqrt{d})$-query testers, respectively. These testers have an almost optimal dependence on $d$, but a suboptimal polynomial dependence on $n$. In this paper, we describe a non-adaptive, one-sided monotonicity tester with query complexity $O(\varepsilon^{-2} d^{1/2 + o(1)})$, independent of $n$. Up to the $d^{o(1)}$-factors, our result resolves the non-adaptive complexity of monotonicity testing for Boolean functions on hypergrids. The independence of $n$ yields a non-adaptive, one-sided $O(\varepsilon^{-2} d^{1/2 + o(1)})$-query monotonicity tester for Boolean functions $f:\mathbb{R}^d \to \{0,1\}$ associated with an arbitrary product measure. |
| title | A $d^{1/2+o(1)}$ Monotonicity Tester for Boolean Functions on $d$-Dimensional Hypergrids |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2304.01416 |