Outlier Robust Multivariate Polynomial Regression

• URL: http://arxiv.org/abs/2403.09465v1
• Date: Thu, 14 Mar 2024 15:04:45 GMT
• Title: Outlier Robust Multivariate Polynomial Regression
• Authors: Vipul Arora, Arnab Bhattacharyya, Mathews Boban, Venkatesan Guruswami, Esty Kelman,
• Abstract summary: We are given a set of random samples $(mathbfx_i,y_i) in [-1,1]n times mathbbR$ that are noisy versions of $(mathbfx_i,p(mathbfx_i)$. The goal is to output a $hatp$, within an $ell_in$-distance of at most $O(sigma)$ from $p$.
• Score: 27.03423421704806
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study the problem of robust multivariate polynomial regression: let $p\colon\mathbb{R}^n\to\mathbb{R}$ be an unknown $n$-variate polynomial of degree at most $d$ in each variable. We are given as input a set of random samples $(\mathbf{x}_i,y_i) \in [-1,1]^n \times \mathbb{R}$ that are noisy versions of $(\mathbf{x}_i,p(\mathbf{x}_i))$. More precisely, each $\mathbf{x}_i$ is sampled independently from some distribution $\chi$ on $[-1,1]^n$, and for each $i$ independently, $y_i$ is arbitrary (i.e., an outlier) with probability at most $\rho < 1/2$, and otherwise satisfies $|y_i-p(\mathbf{x}_i)|\leq\sigma$. The goal is to output a polynomial $\hat{p}$, of degree at most $d$ in each variable, within an $\ell_\infty$-distance of at most $O(\sigma)$ from $p$. Kane, Karmalkar, and Price [FOCS'17] solved this problem for $n=1$. We generalize their results to the $n$-variate setting, showing an algorithm that achieves a sample complexity of $O_n(d^n\log d)$, where the hidden constant depends on $n$, if $\chi$ is the $n$-dimensional Chebyshev distribution. The sample complexity is $O_n(d^{2n}\log d)$, if the samples are drawn from the uniform distribution instead. The approximation error is guaranteed to be at most $O(\sigma)$, and the run-time depends on $\log(1/\sigma)$. In the setting where each $\mathbf{x}_i$ and $y_i$ are known up to $N$ bits of precision, the run-time's dependence on $N$ is linear. We also show that our sample complexities are optimal in terms of $d^n$. Furthermore, we show that it is possible to have the run-time be independent of $1/\sigma$, at the cost of a higher sample complexity.

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