Low Degree Testing over the Reals

TL;DR

This paper introduces a distribution-free property tester for low-degree polynomials over the reals, capable of handling unknown distributions without finite support, with applications in robust polynomial testing.

Contribution

It presents the first distribution-free low-degree testing algorithm over the reals that works without finite support assumptions and under limited query precision.

Findings

The tester makes polynomially many queries in degree and error parameters.

It works under mild assumptions on query precision and rational point representation.

A new stability theorem for multivariate polynomials is proved.

Abstract

We 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 \emph{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.