Lasso Regression: Why the Solution Lives on a Diamond

2026-04-23 · Source: Towards Data Science · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & Analytics · Depth: Intermediate, extended

Summary

This article explains Lasso Regression using vector and projection concepts, contrasting it with traditional calculus-based explanations. It demonstrates how standard linear regression can overfit, particularly with many features, by perfectly fitting training data but failing on new, unseen data. The author uses a house price prediction example with features like size and age, showing how a linear regression model achieves zero error on training data (coefficients β₀ = 1, β₁ = 2, β₂ = 1) but predicts 24 for a new house with an actual price of 5.5. Lasso addresses this by introducing a constraint on the sum of absolute coefficients, forcing some to shrink towards or to zero, thereby performing feature selection and improving generalization. The article details how centering data simplifies the problem by removing the intercept's influence during coefficient calculation, and how the Lasso constraint forms a diamond shape in coefficient space, reducing the problem to a projection onto a line.

Key takeaway

For Machine Learning Engineers building predictive models, understanding Lasso Regression's geometric interpretation is crucial for mitigating overfitting. You should consider implementing Lasso, especially when dealing with high-dimensional datasets, to improve model generalization. By centering your data and applying a coefficient constraint, you can force less impactful features' coefficients to zero, leading to a more stable and interpretable model. Experiment with different constraint strengths via cross-validation to find the optimal balance between bias and variance for your specific problem.

Key insights

Lasso Regression uses vector projections and coefficient constraints to prevent overfitting by shrinking feature coefficients.

Principles

• Overfitting occurs when models memorize data, not patterns.
• Lasso improves generalization by limiting total coefficient contribution.
• Centering data simplifies regression by isolating feature effects.

Method

Lasso regression involves centering feature and target vectors, applying a constraint on the sum of absolute coefficients (e.g., |β₁| + |β₂| ≤ 2), and then projecting the target onto the constrained solution space to find optimal, shrunken coefficients.

In practice

• Use Lasso when features outnumber observations.
• Center data before applying Lasso for clearer feature impact.
• Employ cross-validation to select the optimal Lasso constraint.

Topics

• Lasso Regression
• Linear Regression
• Overfitting
• Feature Selection
• Vector Projections

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

Related on AIssential

• Lasso Regression is a Laplace Prior — Lasso regression uses an L1 penalty to achieve automatic feature selection by driving less important coefficients to zero.
• Overfitting vs Underfitting: A Simple Explanation — Balancing model complexity is crucial to avoid underfitting (too simple) and overfitting (too complex) for effective generalization.
• Linear Regression Is Actually a Projection Problem, Part 1: The Geometric Intuition — Understanding vectors, dot products, and projections provides a geometric intuition for linear regression.
• Machine Learning | Ridge Regression-L2 — Ridge Regression uses L2 regularization to stabilize coefficients, reduce overfitting, and manage multicollinearity in linear models.
• Generalization in Nonlinear Least Squares via Learned Feature Geometry — New generalization bounds for nonlinear least-squares models depend on learned feature geometry and data-dependent effective dimension, not just parameter count.
• Elastic Net = Ridge + Lasso — Elastic Net blends Ridge and Lasso penalties to achieve both sparsity and stable grouping of correlated features.

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Editorial summary, takeaway, and curation by AIssential. Original article published by Towards Data Science.