ECCC-Report TR21-004https://eccc.weizmann.ac.il/report/2021/004Comments and Revisions published for TR21-004en-usSun, 10 Jan 2021 18:21:56 +0200
Paper TR21-004
| Junta Distance Approximation with Sub-Exponential Queries |
Vishnu Iyer,
Avishay Tal,
Michael Whitmeyer
https://eccc.weizmann.ac.il/report/2021/004Leveraging tools of De, Mossel, and Neeman [FOCS, 2019], we show two different results pertaining to the tolerant testing of juntas. Given black-box access to a Boolean function $f:\{\pm1\}^{n} \to \{\pm1\}$ we give a poly$(k, \frac{1}{\varepsilon})$ query algorithm that distinguishes between functions that are $\gamma$-close to $k$-juntas and $(\gamma+\varepsilon)$-far from $k'$-juntas, where $k' = O(\frac{k}{\varepsilon^2})$. In the non-relaxed setting, we extend our ideas to give a $2^{\tilde{O}(\sqrt{k}/\varepsilon)}$ (adaptive) query algorithm that distinguishes between functions that are $\gamma$-close to $k$-juntas and $(\gamma+\varepsilon)$-far from $k$-juntas. To the best of our knowledge, this is the first subexponential-in-$k$ query algorithm for approximating the distance of $f$ to being a $k$-junta (previous results of Blais, Canonne, Eden, Levi, and Ron [SODA, 2018] and De, Mossel, and Neeman [FOCS, 2019] required exponentially many queries in $k$). Our techniques are Fourier analytical and introduce the new notion of "normalized influences'' that might be of independent interest.Sun, 10 Jan 2021 18:21:56 +0200https://eccc.weizmann.ac.il/report/2021/004