The Purest Understanding of Linear Regression

2026-07-09 · Source: Machine Learning on Medium · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & Analytics, Mathematics & Computational Sciences · Depth: Advanced, medium

Summary

This article presents a "pure" algebraic understanding of linear regression, centering on the fundamental equation y = Xβ where β is the linear function's configuration and X is the input. It argues that everything else, including the canonical inversion form β = X⁻¹y for exactly determined systems and the least squares solution β* = X⁺y for overdetermined cases, is derived purely through algebra. The author critiques the common geometric interpretation (scatter plots) as misleading, emphasizing that "linear" refers to the algebraic transformation, not a fitted line or hyperplane. The piece also highlights the distinction between scalar algebra and vector linear algebra, and briefly connects linear regression's analytical solution to iterative gradient descent in neural networks.

Key takeaway

For machine learning engineers or data scientists building linear models, understanding the algebraic purity of y = Xβ is crucial. This perspective clarifies that geometric interpretations can be misleading, and that the pseudoinverse X⁺ is a direct algebraic solution to the least squares problem. You should prioritize the algebraic formulation to avoid common conceptual pitfalls and better grasp the underlying mechanics of both linear regression and single-layer neural networks.

Key insights

The core of linear regression is finding the configuration β in y = Xβ through algebraic inversion or least squares optimization.

Principles

• Linear regression is fundamentally algebraic, not geometric.
• y = Xβ defines the canonical production form.
• The pseudoinverse X⁺ is an algebraic byproduct, not a goal.

Method

For exactly determined systems, solve β = X⁻¹y. For overdetermined systems, minimize ||y - Xβ*||² to find β* = (XᵀX)⁻¹Xᵀy.

In practice

• Focus on y = Xβ as the core conceptual model.
• Understand X's column of 1s as an intercept placeholder.
• Recognize gradient descent as an iterative X⁺ approximation.

Topics

• Linear Regression
• Linear Algebra
• Pseudoinverse
• Least Squares
• Neural Networks
• Matrix Inversion

Best for: Research Scientist, AI Scientist, Machine Learning Engineer, Data Scientist

Related on AIssential

• The Geometry of Linear Equations and Linear Regression — Geometric understanding of linear algebra clarifies solutions and underpins linear regression.
• Neural-Guided Domain Restriction to Accelerate Pseudospectra Computation for Structured Non-normal Banded Matrices — A neural network predicts sensitive regions to accelerate pseudospectra computation for non-normal matrices.
• The Essence of Linear Regression — Linear regression fits a line to data, using least squares to minimize prediction errors and R-squared/P-value to quantify confidence.
• Understanding Linear Regression and Various Mathematical Functions in Machine Learning — Mathematics underpins machine learning, enabling algorithms to analyze data and make predictions.
• Ridge Regression Is Just a Diagonal — Ridge regression stabilizes OLS by adding a diagonal "ridge" to the covariance matrix, preventing singularity and shrinking coefficients.
• quadratic form derivative #maths #machinelearning #datascience #mathematics — The derivative of a quadratic form x transpose Ax with respect to x is 2Ax.

See Counsel's argued verdicts on the open AI decisions leaders are weighing →

Open in AIssential →

Editorial summary, takeaway, and curation by AIssential. Original article published by Machine Learning on Medium.