We consider the following basic, and very broad, statistical problem: Given a known high-dimensional distribution ${ \cal D}$ over $\mathbb{ R}^n$ and a collection of data points in $\mathbb{ R}^n$, distinguish between the two possibilities that (i) the data was drawn from ${ \cal D}$, versus (ii) the data was drawn from ${ \cal D}|_S$, i.e. from ${ \cal D}$ subject to truncation by an unknown truncation set $S \subseteq \mathbb{ R}^n$. We study this problem in the setting where ${ \cal D}$ is a high-dimensional i.i.d. product distribution and $S$ is an unknown degree-$d$ polynomial threshold function (one of the most well-studied types of Boolean-valued function over $\mathbb{ R}^n$). Our main results are an efficient algorithm when ${ \cal D}$ is a hypercontractive distribution, and a matching lower bound: $\bullet$ For any constant $d$, we give a polynomial-time algorithm which successfully distinguishes ${ \cal D}$ from ${ \cal D}|_S$ using $O(n^{ d/2})$ samples (subject to mild technical conditions on ${ \cal D}$ and $S$); $\bullet$ Even for the simplest case of ${ \cal D}$ being the uniform distribution over $\{ +1, -1\}^n$, we show that for any constant $d$, any distinguishing algorithm for degree-$d$ polynomial threshold functions must use $\Omega(n^{ d/2})$ samples.
Categories: cs.CC cs.DS