{
  "id": "pairwise-distances",
  "title": "Vectorized pairwise distances",
  "kind": "implement",
  "summary": "Euclidean distance and cosine similarity matrices from one matrix product, with a clip at 0 and a zero-vector rule.",
  "difficulty": "medium",
  "tags": ["distances", "vectorization", "cosine-similarity", "numerical-stability"],
  "lesson": "phases/01-math-foundations/14-norms-and-distances",
  "cheatsheet": "math-for-ml",
  "minutes": 20,
  "entry": ["pairwise_sq_distances", "pairwise_distances", "cosine_similarity_matrix"],
  "collections": ["math-foundations"],
  "time_limit_s": 10,
  "mutants": {
    "cosine_norm_axis0": "Takes the norms of B over axis 0, so it divides by column norms and not row norms.",
    "cosine_zero_nan": "Divides by the norm product with no zero rule, so a zero vector gives NaN.",
    "no_clip": "Does not clip the squared distances at 0, so rounding gives a negative value and the square root gives NaN.",
    "plus_cross_term": "Adds 2ab to the squared norms and does not subtract it.",
    "square_of_sum": "Squares the sum of each row and not the sum of the squares."
  }
}