Outlier Robust Multivariate Polynomial Regression
arxiv(2024)
摘要
We study the problem of robust multivariate polynomial regression: let
pℝ^n→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 (𝐱_i,y_i) ∈ [-1,1]^n ×ℝ that are noisy
versions of (𝐱_i,p(𝐱_i)). More precisely, each
𝐱_i is sampled independently from some distribution χ on
[-1,1]^n, and for each i independently, y_i is arbitrary (i.e., an
outlier) with probability at most ρ < 1/2, and otherwise satisfies
|y_i-p(𝐱_i)|≤σ. The goal is to output a polynomial
p̂, of degree at most d in each variable, within an
ℓ_∞-distance of at most O(σ) 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^nlog d), where the hidden constant
depends on n, if χ is the n-dimensional Chebyshev distribution. The
sample complexity is O_n(d^2nlog d), if the samples are drawn from the
uniform distribution instead. The approximation error is guaranteed to be at
most O(σ), and the run-time depends on log(1/σ). In the setting
where each 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/σ, at the cost of a
higher sample complexity.
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