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Revision #1 to TR18-187 | 9th April 2019 06:14

#### Domain Reduction for Monotonicity Testing: A $o(d)$ Tester for Boolean Functions in $d$-Dimensions

Revision #1
Accepted on: 9th April 2019 06:14
Keywords:

Abstract:

We describe a $\tilde{O}(d^{5/6})$-query monotonicity tester for Boolean functions $f:[n]^d \to \{0,1\}$ on the $n$-hypergrid. This is the first $o(d)$ monotonicity tester with query complexity independent of $n$. Motivated by this independence of $n$, we initiate the study of monotonicity testing of measurable Boolean functions $f:\mathbb{R}^d \to \{0,1\}$ over the continuous domain, where the distance is measured with respect to a product distribution over $\mathbb{R}^d$. We give a $\tilde{O}(d^{5/6})$-query monotonicity tester for such functions.
Our main technical result is a domain reduction theorem for monotonicity. For any function $f:[n]^d \to \{0,1\}$, let $\epsilon_f$ be its distance to monotonicity. Consider the restriction $\hat{f}$ of the function on a random $[k]^d$ sub-hypergrid of the original domain. We show that for $k = \text{poly}(d/\epsilon)$, the expected distance of the restriction is $\mathbb{E}[\epsilon_{\hat{f}}] = \Omega(\epsilon_f)$. Previously, such a result was only known for $d=1$ (Berman-Raskhodnikova-Yaroslavtsev, STOC 2014). Our result for testing Boolean functions over $[n]^d$ then follows by applying the $d^{5/6}\cdot \text{poly}(1/\epsilon,\log n, \log d)$-query hypergrid tester of Black-Chakrabarty-Seshadhri (SODA 2018).
To obtain the result for testing Boolean functions over $\mathbb{R}^d$, we use standard measure theoretic tools to reduce monotonicity testing of a measurable function $f$ to monotonicity testing of a discretized version of $f$ over a hypergrid domain $[N]^d$ for large, but finite, $N$ (that may depend on $f$). The independence of $N$ in the hypergrid tester is crucial to getting the final tester over $\mathbb{R}^d$.

### Paper:

TR18-187 | 4th November 2018 22:19

#### Domain Reduction for Monotonicity Testing: A $o(d)$ Tester for Boolean Functions on Hypergrids

TR18-187
Publication: 6th November 2018 20:45
Keywords:

Abstract:

Testing monotonicity of Boolean functions over the hypergrid, $f:[n]^d \to \{0,1\}$, is a classic problem in property testing. When the range is real-valued, there are $\Theta(d\log n)$-query testers and this is tight. In contrast, the Boolean range qualitatively differs in two ways:
(1) Independence of $n$: There are testers with query complexity independent of $n$ [Dodis et al. (RANDOM 1999); Berman et al. (STOC 2014)], with linear dependence on $d$.
(2) Sublinear in $d$: For the $n=2$ hypercube case, there are testers with $o(d)$ query complexity [Chakrabarty, Seshadhri (STOC 2013); Khot et al. (FOCS 2015)].
It was open whether one could obtain both properties simultaneously. This paper answers this question in the affirmative. We describe a $\tilde{O}(d^{5/6})$-query monotonicity tester for $f:[n]^d \to \{0,1\}$.
Our main technical result is a domain reduction theorem for monotonicity. For any function $f$, let $\epsilon_f$ be its distance to monotonicity. Consider the restriction $\hat{f}$ of the function on a random $[k]^d$ sub-hypergrid of the original domain. We show that for $k = \text{poly}(d/\epsilon)$, the expected distance of the restriction $\mathbb{E}[\epsilon_{\hat{f}}] = \Omega(\epsilon_f)$. Therefore, for monotonicity testing in $d$ dimensions, we can restrict to testing over $[n]^d$, where $n = \text{poly}(d/\epsilon)$. Our result follows by applying the $d^{5/6}\cdot \text{poly}(1/\epsilon,\log n, \log d)$-query hypergrid tester of Black-Chakrabarty-Seshadhri (SODA 2018).

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