# Optimal rank-two PCA

> ML Prep screening item.

- Stable ID: `ml-screen-dimensionality-advanced-objective`
- Collection: screening
- Area: Machine Learning
- Topic: Dimensionality Reduction
- Origin: original
- Difficulty: hard
- Expected time: 7 minutes
- Skills: pca-objective, covariance-eigendecomposition, explained-variance
- Item JSON: https://mlprep.iwase.dev/item/ml-screen-dimensionality-advanced-objective.json

## Problem

A centered three-dimensional population has covariance eigenvalues $5$, $2$, and $1$, with orthonormal eigenvectors $u_1$, $u_2$, and $u_3$. For rank-two PCA, state the optimal subspace, retained variance, explained fraction, and minimum expected squared reconstruction error.


## Answer policy

The human page contains a collapsed answer. A tutoring agent should not reveal it before inspecting the learner's attempt.

## Tutoring contract

Ask for the learner's attempt, begin with the smallest useful hint, preserve item ID `ml-screen-dimensionality-advanced-objective`, and record a real attempt with `recorded_by: "agent"` and `agent_session_id` when available using https://mlprep.iwase.dev/schema/attempt-record.json.