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

• URL: http://arxiv.org/abs/2505.23552v1
• Date: Thu, 29 May 2025 15:28:46 GMT
• Title: Comparing the Moore-Penrose Pseudoinverse and Gradient Descent for Solving Linear Regression Problems: A Performance Analysis
• Authors: Alex Adams,
• Abstract summary: This paper investigates the comparative performance of two fundamental approaches to solving linear regression problems.<n>I review and discuss the theoretical underpinnings of both methods, analyze their computational complexity, and evaluate their empirical behavior on synthetic datasets.<n>My results delineate the conditions under which each method excels in terms of computational time, numerical stability, and predictive accuracy.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

Related papers

• A Simplified Analysis of SGD for Linear Regression with Weight Averaging [64.2393952273612]
  Recent work bycitetzou 2021benign provides sharp rates for SGD optimization in linear regression using constant learning rate.<n>We provide a simplified analysis recovering the same bias and variance bounds provided incitepzou 2021benign based on simple linear algebra tools.<n>We believe our work makes the analysis of gradient descent on linear regression very accessible and will be helpful in further analyzing mini-batching and learning rate scheduling.
  arXiv  Detail & Related papers  (2025-06-18T15:10:38Z)
• Comparative study of regression vs pairwise models for surrogate-based heuristic optimisation [1.2535250082638645]
  This paper addresses the formulation of surrogate problems as both regression models that approximate fitness (surface surrogate models) and a novel way to connect classification models (pairwise surrogate models) The performance of the overall search, when using online machine learning-based surrogate models, depends not only on the accuracy of the predictive model but also on the kind of bias towards positive or negative cases.
  arXiv  Detail & Related papers  (2024-10-04T13:19:06Z)
• Learning Graphical Factor Models with Riemannian Optimization [70.13748170371889]
  This paper proposes a flexible algorithmic framework for graph learning under low-rank structural constraints. The problem is expressed as penalized maximum likelihood estimation of an elliptical distribution. We leverage geometries of positive definite matrices and positive semi-definite matrices of fixed rank that are well suited to elliptical models.
  arXiv  Detail & Related papers  (2022-10-21T13:19:45Z)
• Sparse high-dimensional linear regression with a partitioned empirical Bayes ECM algorithm [62.997667081978825]
  We propose a computationally efficient and powerful Bayesian approach for sparse high-dimensional linear regression. Minimal prior assumptions on the parameters are used through the use of plug-in empirical Bayes estimates. The proposed approach is implemented in the R package probe.
  arXiv  Detail & Related papers  (2022-09-16T19:15:50Z)
• HyperImpute: Generalized Iterative Imputation with Automatic Model Selection [77.86861638371926]
  We propose a generalized iterative imputation framework for adaptively and automatically configuring column-wise models. We provide a concrete implementation with out-of-the-box learners, simulators, and interfaces.
  arXiv  Detail & Related papers  (2022-06-15T19:10:35Z)
• Robust Regularized Low-Rank Matrix Models for Regression and Classification [14.698622796774634]
  We propose a framework for matrix variate regression models based on a rank constraint, vector regularization (e.g., sparsity), and a general loss function. We show that the algorithm is guaranteed to converge, all accumulation points of the algorithm have estimation errors in the order of $O(sqrtn)$ally and substantially attaining the minimax rate.
  arXiv  Detail & Related papers  (2022-05-14T18:03:48Z)
• Extension of Dynamic Mode Decomposition for dynamic systems with incomplete information based on t-model of optimal prediction [69.81996031777717]
  The Dynamic Mode Decomposition has proved to be a very efficient technique to study dynamic data. The application of this approach becomes problematic if the available data is incomplete because some dimensions of smaller scale either missing or unmeasured. We consider a first-order approximation of the Mori-Zwanzig decomposition, state the corresponding optimization problem and solve it with the gradient-based optimization method.
  arXiv  Detail & Related papers  (2022-02-23T11:23:59Z)
• Linear regression with partially mismatched data: local search with theoretical guarantees [9.398989897176953]
  We study an important variant of linear regression in which the predictor-response pairs are partially mismatched. We use an optimization formulation to simultaneously learn the underlying regression coefficients and the permutation corresponding to the mismatches. We prove that our local search algorithm converges to a nearly-optimal solution at a linear rate.
  arXiv  Detail & Related papers  (2021-06-03T23:32:12Z)
• Piecewise linear regression and classification [0.20305676256390928]
  This paper proposes a method for solving multivariate regression and classification problems using piecewise linear predictors. A Python implementation of the algorithm described in this paper is available at http://cse.lab.imtlucca.it/bemporad/parc.
  arXiv  Detail & Related papers  (2021-03-10T17:07:57Z)
• Surrogate Models for Optimization of Dynamical Systems [0.0]
  This paper provides a smart data driven mechanism to construct low dimensional surrogate models. These surrogate models reduce the computational time for solution of the complex optimization problems.
  arXiv  Detail & Related papers  (2021-01-22T14:09:30Z)
• A Hypergradient Approach to Robust Regression without Correspondence [85.49775273716503]
  We consider a variant of regression problem, where the correspondence between input and output data is not available. Most existing methods are only applicable when the sample size is small. We propose a new computational framework -- ROBOT -- for the shuffled regression problem.
  arXiv  Detail & Related papers  (2020-11-30T21:47:38Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.