quiz.json

{
  "questions": [
    {
      "stage": "pre",
      "question": "What advantage does SVD have over eigendecomposition?",
      "options": ["SVD is always faster to compute", "SVD works on any matrix of any shape, while eigendecomposition requires square matrices with full eigenvector sets", "SVD produces smaller output matrices", "SVD does not require numerical computation"],
      "correct": 1,
      "explanation": "Eigendecomposition only works on square matrices that have n linearly independent eigenvectors. SVD decomposes ANY m x n matrix into U * Sigma * V^T with no restrictions on shape or rank."
    },
    {
      "stage": "pre",
      "question": "What does 'low-rank approximation' mean?",
      "options": ["Removing rows with low values from a matrix", "Approximating a matrix by keeping only its most important components, producing a simpler matrix with fewer independent directions", "Converting a matrix to a lower precision data type", "Sorting the rows of a matrix by their magnitude"],
      "correct": 1,
      "explanation": "A rank-k approximation keeps only the top k singular values and their vectors, discarding the rest. The Eckart-Young theorem proves this is the BEST possible approximation of that rank."
    },
    {
      "stage": "post",
      "question": "In SVD A = U * Sigma * V^T, what geometric operation does each factor represent?",
      "options": ["U scales, Sigma rotates, V^T translates", "V^T rotates in input space, Sigma scales along principal axes, U rotates into output space", "U compresses, Sigma expands, V^T normalizes", "All three factors perform the same operation: rotation"],
      "correct": 1,
      "explanation": "SVD reveals that every matrix performs: (1) V^T rotates inputs to align with principal directions, (2) Sigma stretches/compresses along each axis, (3) U rotates the result into the output space. Rotate, scale, rotate."
    },
    {
      "stage": "post",
      "question": "Why does sklearn implement PCA using SVD instead of eigendecomposition of the covariance matrix?",
      "options": ["SVD produces different results that are more accurate for ML", "SVD works directly on the data matrix without forming the covariance matrix, avoiding squaring the condition number and improving numerical stability", "SVD is easier to parallelize on GPUs", "SVD does not require centering the data"],
      "correct": 1,
      "explanation": "Forming A^T*A squares the singular values (and condition number). If A has singular values [1000, 0.001], A^T*A has eigenvalues [10^6, 10^-6] -- 6 digits of precision lost. SVD avoids this by working directly on A."
    },
    {
      "stage": "post",
      "question": "How does truncated SVD enable recommendation systems to predict missing ratings?",
      "options": ["It fills missing entries with the average rating", "It decomposes the ratings matrix into latent user and movie profiles; the dot product of a user profile with a movie profile predicts the missing rating", "It clusters similar users together and copies their ratings", "It trains a neural network on the observed ratings"],
      "correct": 1,
      "explanation": "SVD decomposes the ratings matrix into user profiles (U), latent factor importance (Sigma), and movie profiles (V^T). The low-rank reconstruction fills in missing entries based on the latent factors (genre, era, style) that explain user preferences."
    }
  ]
}