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This is a medium difficulty Machine Learning question, commonly asked during Take-home Project rounds at Databricks.

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This question is commonly asked for Data Scientist candidates at Databricks during technical interviews.

Explain Linear Regression Feature Transformation Equivalence

Last updated: Mar 29, 2026

Quick Overview

This question evaluates understanding of linear regression feature representations, linear-algebraic equivalence under ordinary least squares, and the statistical implications of feature transformations, testing competency in feature engineering, model identifiability, and numerical aspects of estimation.

Databricks

Aug 4, 2025, 10:55 AM

Data Scientist

Take-home Project

Machine Learning

Linear Regression Feature Representations and High-Dimensional Modelling

Context

You are evaluating two linear regression specifications that use different feature representations derived from the same two original predictors x1 and x2.

Questions

Equivalence of feature representations
- Model A: linear in the original features x1 and x2.
- Model B: linear in the transformed features z1 = x1 + x2 and z2 = x1 − x2.
Are Model A and Model B equivalent in terms of the functions they can represent and the fitted predictions under ordinary least squares (OLS)? Provide a mathematical explanation.
High-dimensional linear modeling
- If you have more than 1,000 predictors and want to fit a linear model, what problems might occur, and how would you mitigate them?

Solution

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