Item-Based Collaborative Filtering

User-based filtering asks "who is like you?". Item-based filtering turns it around: to predict how you'd rate an item, find the items most similar to it that you have already rated, and average your own ratings of those.

Task: write predict_item_cf(ratings, user, item, k) returning the prediction, rounded to 4 decimal places.

The steps:

1. Candidates — every other item the target user has already rated.
2. Similarity — for each candidate, cosine similarity between its column and the target item's column, over only the users who rated both. No shared users, or a zero magnitude, means a similarity of 0.
3. Neighbours — the k most similar candidates. Ties go to the lower item index.
4. Predict — the weighted average of the user's raw ratings of those neighbours, with the absolute similarities as the divisor.

• ratings is a list of rows (users) by columns (items), with None for unrated. zip(*ratings) gives you the columns.
• If the similarities all come to 0, return 0.0.
• Notice there's no mean-centering here, unlike the user-based version. Every rating in this average came from the same user, so there's no difference in generosity to correct for — the thing that made centring necessary there is absent by construction.

Item-based is the version that actually got deployed at scale, and the reason is practical rather than mathematical: item-item similarities are far more stable than user-user ones. A film's relationship to other films barely moves from week to week, so the similarity matrix can be computed nightly and cached, while a user's taste profile shifts with every click. Amazon's recommender was built on exactly this asymmetry.