Detecting Low-Degree Truncation
<|reference_start|>Detecting Low-Degree Truncation: 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.<|reference_end|>
@article{de2024detecting,
    title={Detecting Low-Degree Truncation},
    author={Anindya De and Huan Li and Shivam Nadimpalli and Rocco A. Servedio},
    journal={arXiv preprint arXiv:2402.08133},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.08133},
    primaryClass={cs.CC cs.DS}
}