Problems

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Baseline Predictor

Recommender Systems

Hard

A collaborative-filtering baseline separates ratings into a global average, a user bias, and an item bias. A zero in ratings_matrix means that no rating was observed and must be excluded from all averages.

Compute each user's bias:

b_u = \overline{r}_u - \mu

Compute each item's bias:

b_i = \overline{r}_i - \mu

Predict a requested user-item pair:

\widehat{r}_{ui} = \mu + b_u + b_i

The global average is the mean of every nonzero rating. A user's observed-rating mean determines the user bias, while an item's observed-rating mean determines the item bias. Use a bias of zero for a user or item with no observed ratings. Return predictions for target_pairs in their original order.

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Examples

Input: ratings_matrix = [[5, 3, 0], [4, 0, 1], [0, 1, 5]], target_pairs = [[0, 2], [1, 1], [2, 0]]

Output: [3.833333, 1.333333, 4.333333]

Explanation: The global mean is 19 / 6. Each prediction adds that mean to the corresponding user and item deviations.

Input: ratings_matrix = [[5, 0], [0, 3]], target_pairs = [[0, 1], [1, 0]]

Output: [4.0, 4.0]

Hint 1

Build user means from rows and item means from columns while skipping zeros.

Hint 2

For each pair, add the global mean, its user bias, and its item bias.

Requirements

Compute the global mean from nonzero ratings only.
Compute user and item biases from their observed ratings only.
Use zero bias when a user or item has no observed ratings.
Return one prediction for each pair in target_pairs.

Constraints

ratings_matrix is rectangular and contains at least one nonzero rating.
A zero entry represents a missing rating.
Every target pair contains valid user and item indices.
Time limit: 300 ms.

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