Matrix Factorization

Picture a table with one row per user and one column per movie, filled with star ratings. Almost all cells are empty: nobody has seen more than a tiny fraction of the catalog. A recommender system has to guess the empty cells.

Matrix factorization assumes the table has a simple hidden structure. Each movie is described by a few numbers (say, how much action, romance, comedy it has) and each user by how much they like each of those. A predicted rating is just the user’s numbers times the movie’s numbers, added up: a dot product of two embeddings.

The model is never told what the factors mean. It learns them by fitting the ratings that do exist, with regularization to stop it from memorizing. Afterwards, the factors often line up with recognizable genres.

The math

With user vectors $p_u$, item vectors $q_i$ and biases, minimize over the observed ratings only:

$$\sum_{(u,i) \text{ observed}} \left( r_{ui} - \mu - b_u - b_i - p_u^\top q_i \right)^2 + \lambda \left( \lVert p_u \rVert^2 + \lVert q_i \rVert^2 + b_u^2 + b_i^2 \right)$$

Solve with stochastic gradient descent or alternating least squares (fix $P$, solve for $Q$ exactly, swap).

The approach became famous during the Netflix Prize, which shaped streaming recommendations. Modern systems replace the dot product with deep networks and add many more signals, but the core idea of user and item embeddings remains.