ECCC-Report TR21-102https://eccc.weizmann.ac.il/report/2021/102Comments and Revisions published for TR21-102en-usMon, 19 Jul 2021 19:41:18 +0300
Revision 1
| Tight bounds on the Fourier growth of bounded functions on the hypercube |
Siddharth Iyer,
Anup Rao,
Victor Reis,
Thomas Rothvoss,
Amir Yehudayoff
https://eccc.weizmann.ac.il/report/2021/102#revision1We give tight bounds on the degree $\ell$ homogenous parts $f_\ell$ of a bounded function $f$ on the cube. We show that if $f: \{\pm 1\}^n \rightarrow [-1,1]$ has degree $d$, then $\| f_\ell \|_\infty$ is bounded by $d^\ell/\ell!$, and $\| \hat{f}_\ell \|_1$ is bounded by $d^\ell e^{{\ell+1 \choose 2}} n^{\frac{\ell-1}{2}}$. We describe applications to pseudorandomness and learning theory. We use similar methods to generalize the classical Pisier's inequality from convex analysis. Our analysis involves properties of real-rooted polynomials that may be useful elsewhere.Mon, 19 Jul 2021 19:41:18 +0300https://eccc.weizmann.ac.il/report/2021/102#revision1
Paper TR21-102
| Tight bounds on the Fourier growth of bounded functions on the hypercube |
Siddharth Iyer,
Anup Rao,
Victor Reis,
Thomas Rothvoss,
Amir Yehudayoff
https://eccc.weizmann.ac.il/report/2021/102We give tight bounds on the degree $\ell$ homogenous parts $f_\ell$ of a bounded function $f$ on the cube. We show that if $f: \{\pm 1\}^n \rightarrow [-1,1]$ has degree $d$, then $\| f_\ell \|_\infty$ is bounded by $d^\ell/\ell!$, and $\| \hat{f}_\ell \|_1$ is bounded by $d^\ell e^{{\ell+1 \choose 2}} n^{\frac{\ell-1}{2}}$. We describe applications to pseudorandomness and learning theory. We use similar methods to generalize the classical Pisier's inequality from convex analysis. Our analysis involves properties of real-rooted polynomials that may be useful elsewhere.Tue, 13 Jul 2021 20:55:40 +0300https://eccc.weizmann.ac.il/report/2021/102