Outlier Robust Multivariate Polynomial Regression Study

Core Concepts

Algorithm achieves robust multivariate polynomial regression with optimal sample complexity.

Abstract

The study focuses on robust multivariate polynomial regression, extending univariate results. It introduces a Chebyshev partition for the cube Cn and proposes an algorithm for median-based recovery. The algorithm iteratively refines estimates to achieve an additive error of η. Theorems provide bounds on approximation errors and sample complexities, optimizing for different distributions and precision settings. Lower bounds are also discussed, showing the optimality of the proposed algorithms.

Stats

Sample complexity: On(dn log d)
Approximation error: O(σ)
Run-time: poly(log ∥p∥C,∞, M, log(1/η))

Quotes

"The goal is to output a polynomial bp within an ℓ∞-distance of at most O(σ) from p."
"Our algorithm almost solves the Robust Multivariate Polynomial Regression Problem up to an additive error of η."

Key Insights Distilled From

Outlier Robust Multivariate Polynomial Regression

by Vipul Arora,... at arxiv.org 03-15-2024

https://arxiv.org/pdf/2403.09465.pdf

Deeper Inquiries

How does the study's approach compare to existing methods in robust regression

The study's approach in robust regression builds upon existing methods by generalizing the results from univariate to multivariate settings. While previous works focused on solving the problem for one variable, this study extends the solution to multiple variables, providing a more comprehensive and versatile solution. By considering a tensorization of Chebyshev partitions and adapting median-based recovery techniques, the algorithm achieves robust polynomial regression with optimal sample complexity.

What implications do the results have for practical applications in machine learning

The results of this study have significant implications for practical applications in machine learning. Robust multivariate polynomial regression is a fundamental problem with various real-world applications such as computer vision, object boundary modeling, and curve fitting. The algorithm's ability to handle noisy data with outliers provides a reliable method for fitting polynomials to complex datasets while maintaining accuracy even in the presence of noise.
In practical machine learning scenarios where data may contain outliers or noise, having an efficient and accurate method for robust regression can improve model performance and reliability. The sample complexities provided by this study offer insights into how many samples are needed to achieve a certain level of approximation error, guiding practitioners in designing experiments and collecting data effectively.
Furthermore, the concept of Chebyshev partitions allows for structured division of multidimensional spaces into cells based on extremal points derived from Chebyshev polynomials. This partitioning technique can be leveraged in various mathematical research areas beyond robust regression, such as numerical analysis, optimization problems involving grids or meshes, computational geometry algorithms like Voronoi diagrams construction or Delaunay triangulation.