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Low Degree Testing over the Reals

Series: Bangalore Theory Seminars

Speaker: Vipul Arora National University of Singapore

Date/Time: Jan 13 17:45:00

Location: CSA Seminar Hall (Room No. 254, First Floor)

Abstract:
We study the problem of testing whether a function $f: reals^n to reals$ 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 $reals^n$ 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 $poly(d/eps)$ many queries to $f$, accepts with probability $1$ if $f$ is a polynomial of degree $d$, and rejects with probability at least $mathfrac{2}{3}$ if every degree-$d$ polynomial $P$ disagrees with $f$ on a set of mass at least $eps$ 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.


This is a joint work with Arnab Bhattacharyya, Esty Kelman, Noah Fleming, and Yuichi Yoshida, and will appear in SODA23.

Microsoft teams link:

Link


We are grateful to the Kirani family for generously supporting the theory seminar series


Hosts: Aditya Subramanian, Aditya Abhay Lonkar, Rahul Madhavan & Rameesh Paul