Comparing the Moore-Penrose Pseudoinverse and Gradient Descent for Solving Linear Regression Problems: A Performance Analysis

TL;DR

This paper compares the Moore-Penrose pseudoinverse and gradient descent for linear regression, analyzing their efficiency, stability, and accuracy through theoretical discussion and empirical testing to guide method selection.

Contribution

It provides a comprehensive performance analysis of both methods, highlighting their strengths and optimal use cases in linear regression problems.

Findings

Pseudoinverse is faster for small datasets with stable solutions.

Gradient descent scales better with large, high-dimensional data.

Each method has specific conditions where it outperforms the other.

Abstract

This paper investigates the comparative performance of two fundamental approaches to solving linear regression problems: the closed-form Moore-Penrose pseudoinverse and the iterative gradient descent method. Linear regression is a cornerstone of predictive modeling, and the choice of solver can significantly impact efficiency and accuracy. I review and discuss the theoretical underpinnings of both methods, analyze their computational complexity, and evaluate their empirical behavior on synthetic datasets with controlled characteristics, as well as on established real-world datasets. My results delineate the conditions under which each method excels in terms of computational time, numerical stability, and predictive accuracy. This work aims to provide practical guidance for researchers and practitioners in machine learning when selecting between direct, exact solutions and iterative, approximate solutions for linear regression tasks.