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 distribution-free testing model. Here, the distance between functions is measured with respect to an unknown distribution $D$ over $\mathbb{R}^n$ 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 poly(d/$\epsilon$) 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 $\epsilon$ 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.
This is a joint work with Arnab Bhattacharyya, Esty Kelman, Noah Fleming, and Yuichi Yoshida, and appeared in SODA'23.
Short Bio:
Vipul is a final year PhD student at Dept of CS, School of Computing, National University of Singapore, advised by Prof Arnab Bhattacharyya. His research interests lie in Complexity Theory, Combinatorics, Information Theory, and Theoretical CS, in general. His current work is focused in Property Testing, and Theoretical Machine Learning.
