{
  "schema": "ml-prep/item@1",
  "item": {
    "id": "original-la-low-rank",
    "area": "mathematics",
    "topic": "linear-algebra",
    "origin": "original",
    "title": "Low-rank error and explained variance",
    "skills": [
      "svd",
      "low-rank-approximation",
      "pca-connection",
      "svd-pca"
    ],
    "priority": "core",
    "difficulty": "medium",
    "estimated_minutes": 8,
    "prerequisites": [
      "singular-values"
    ],
    "prompt": "A centered data matrix has singular values $5$, $2$, and $1$. Find the Frobenius error of its best rank-$1$ approximation and the fraction of total squared variation retained by the first principal component.\n",
    "answer": "The best rank-$1$ error is\n$$\n\\sqrt{2^2 + 1^2} = \\sqrt{5}.\n$$\nThe retained fraction is\n$$\n\\frac{5^2}{5^2 + 2^2 + 1^2} = \\frac{25}{30} = \\frac{5}{6}.\n$$\n",
    "check": {
      "kind": "numeric",
      "id": "original-la-low-rank",
      "values": [
        5,
        0.8333333333333334
      ]
    }
  }
}