ECCC-Report TR19-087https://eccc.weizmann.ac.il/report/2019/087Comments and Revisions published for TR19-087en-usSat, 29 Aug 2020 19:35:44 +0300
Revision 1
| Coin Theorems and the Fourier Expansion |
Rohit Agrawal
https://eccc.weizmann.ac.il/report/2019/087#revision1In this note we compare two measures of the complexity of a class $\mathcal F$ of Boolean functions studied in (unconditional) pseudorandomness: $\mathcal F$'s ability to distinguish between biased and uniform coins (the coin problem), and the norms of the different levels of the Fourier expansion of functions in $\mathcal F$ (the Fourier growth). We show that for coins with low bias $\varepsilon = o(1/n)$, a function's distinguishing advantage in the coin problem is essentially equivalent to $\varepsilon$ times the sum of its level $1$ Fourier coefficients, which in particular shows that known level $1$ and total influence bounds for some classes of interest (such as constant-width read-once branching programs) in fact follow as a black-box from the corresponding coin theorems, thereby simplifying the proofs of some known results in the literature. For higher levels, it is well-known that Fourier growth bounds on all levels of the Fourier spectrum imply coin theorems, even for large $\varepsilon$, and we discuss here the possibility of a converse.Sat, 29 Aug 2020 19:35:44 +0300https://eccc.weizmann.ac.il/report/2019/087#revision1
Paper TR19-087
| Coin Theorems and the Fourier Expansion |
Rohit Agrawal
https://eccc.weizmann.ac.il/report/2019/087In this note we compare two measures of the complexity of a class $\mathcal F$ of Boolean functions studied in (unconditional) pseudorandomness: $\mathcal F$'s ability to distinguish between biased and uniform coins (the coin problem), and the norms of the different levels of the Fourier expansion of functions in $\mathcal F$ (the Fourier growth). We show that for coins with low bias $\varepsilon = o(1/n)$, a function's distinguishing advantage in the coin problem is essentially equivalent to $\varepsilon$ times the sum of its level $1$ Fourier coefficients, which in particular shows that known level $1$ and total influence bounds for some classes of interest (such as constant-width read-once branching programs) in fact follow as a black-box from the corresponding coin theorems, thereby simplifying the proofs of some known results in the literature. For higher levels, it is well-known that Fourier growth bounds on all levels of the Fourier spectrum imply coin theorems, even for large $\varepsilon$, and we discuss here the possibility of a converse.Tue, 11 Jun 2019 17:34:45 +0300https://eccc.weizmann.ac.il/report/2019/087