Low Degree Testing over the Reals

04/18/2022
by   Vipul Arora, et al.
0

We study the problem of testing whether a function f: ℝ^n →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 𝒟 over ℝ^n from which we can draw samples. In contrast to previous work, we do not assume that 𝒟 has finite support. We design a tester that given query access to f, and sample access to 𝒟, makes (d/ε)^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 ε with respect to 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.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
09/08/2019

Distribution-Free Testing of Linear Functions on R^n

We study the problem of testing whether a function f:R^n->R is linear (i...
research
01/01/2019

Almost Optimal Distribution-free Junta Testing

We consider the problem of testing whether an unknown n-variable Boolean...
research
05/08/2023

Low-Degree Testing Over Grids

We study the question of local testability of low (constant) degree func...
research
03/25/2021

Random restrictions and PRGs for PTFs in Gaussian Space

A polynomial threshold function (PTF) f:ℝ^n →ℝ is a function of the form...
research
12/22/2022

Multivariate Polynomial Regression of Euclidean Degree Extends the Stability for Fast Approximations of Trefethen Functions

We address classic multivariate polynomial regression tasks from a novel...
research
07/30/2023

On Approximability of Satisfiable k-CSPs: IV

We prove a stability result for general 3-wise correlations over distrib...
research
06/28/2023

Refining the Adaptivity Notion in the Huge Object Model

The Huge Object model for distribution testing, first defined by Goldrei...

Please sign up or login with your details

Forgot password? Click here to reset