How does matrix factorization work in recommender systems?

Matrix factorization is a technique used in recommender systems to predict user preferences by decomposing a user-item interaction matrix into lower-dimensional representations. The core idea is to break down a large, sparse matrix (where rows represent users, columns represent items, and entries represent interactions like ratings) into two smaller matrices: one representing users and their latent features, and the other representing items and their latent features. These latent features capture underlying patterns in the data, such as genre preferences in movies or stylistic traits in products. By multiplying the user and item matrices, the system approximates missing entries in the original matrix, enabling predictions about unobserved user-item interactions.

For example, consider a movie recommendation system with a user-item matrix containing ratings from 1 to 5. If User A rates Movie X as 4 but hasn’t rated Movie Y, matrix factorization might discover that User A has a latent feature value of 0.8 for “action movies,” while Movie X has a 0.9 for “action” and Movie Y has 0.7. Multiplying these values (0.8 * 0.7 = 0.56) could suggest a predicted rating of 3.5 for Movie Y, even though User A never rated it. The model is trained to minimize the difference between known ratings and predictions using optimization methods like stochastic gradient descent (SGD) or alternating least squares (ALS), often with regularization to avoid overfitting.

A practical implementation involves defining the number of latent factors (e.g., 10–100), initializing user and item matrices with random values, and iteratively updating them. For instance, in SGD, the algorithm loops through each observed rating, computes the prediction error, and adjusts the user and item vectors to reduce the error. Libraries like Surprise or TensorFlow provide tools to streamline this process. Challenges include selecting the right number of factors (too few may oversimplify; too many may overfit) and handling cold-start scenarios. Extensions like biased matrix factorization add user/item biases to account for systematic rating tendencies (e.g., some users rate higher on average). This approach balances interpretability and scalability, making it a foundational method for collaborative filtering.