TY - GEN

T1 - Low Degree Testing over the Reals

AU - Arora, Vipul

AU - Bhattacharyya, Arnab

AU - Fleming, Noah

AU - Kelman, Esty

AU - Yoshida, Yuichi

N1 - Publisher Copyright:
Copyright © 2023 by SIAM.

PY - 2023

Y1 - 2023

N2 - We study the problem of testing whether a function f : Rn → 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 Rn 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 poly(d/ε) 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.

AB - We study the problem of testing whether a function f : Rn → 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 Rn 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 poly(d/ε) 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.

UR - http://www.scopus.com/inward/record.url?scp=85170092452&partnerID=8YFLogxK

M3 - ???researchoutput.researchoutputtypes.contributiontobookanthology.conference???

AN - SCOPUS:85170092452

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 738

EP - 792

BT - 34th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2023

PB - Association for Computing Machinery

T2 - 34th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2023

Y2 - 22 January 2023 through 25 January 2023

ER -