ECCC-Report TR22-051https://eccc.weizmann.ac.il/report/2022/051Comments and Revisions published for TR22-051en-usMon, 18 Apr 2022 21:34:56 +0300
Paper TR22-051
| Low Degree Testing over the Reals |
Noah Fleming,
Vipul Arora,
Arnab Bhattacharyya,
Esty Kelman,
Yuichi Yoshida
https://eccc.weizmann.ac.il/report/2022/051We study the problem of testing whether a function $f: \mathbb{R}^n \to \mathbb{R}$ is a polynomial of degree at most $d$ in the distribution-free testing model. Here, the distance between functions is measured with respect to an unknown distribution $\mathcal{D}$ over $\mathbb{R}^n$ from which we can draw samples. In contrast to previous work, we do not assume that $\mathcal{D}$ has finite support.
We design a tester that given query access to $f$, and sample access to $\mathcal{D}$, makes $(d/\varepsilon)^{O(1)}$ many queries to $f$, accepts with probability $1$ if $f$ is a polynomial of degree $d$, and rejects with probability at least $2/3$ if every degree-$d$ polynomial $P$ disagrees with $f$ on a set of mass at least $\varepsilon$ with respect to $\mathcal{D}$. Our result also holds under mild assumptions when we receive only a polynomial number of bits of precision for each query to $f$, or when $f$ can only be queried on rational points representable using a logarithmic number of bits. Along the way, we prove a new stability theorem for multivariate polynomials that may be of independent interest. Mon, 18 Apr 2022 21:34:56 +0300https://eccc.weizmann.ac.il/report/2022/051