Tensor Factorization

Matrix factorization asks a two-part question: how much does this user like this item. But the same user wants different things on a Monday commute than on a Saturday evening, and a matrix has nowhere to put that.

Adding a third mode
Tensor factorization extends latent factors to a three-dimensional array — user × item × context — where the context axis holds whatever situational variable matters: time of day, location, device, who the user is with. Each slice is one context's user-item matrix, mostly empty. The CP decomposition learns one factor vector per user, per item and per context value and predicts by summing their elementwise product; Tucker decomposition adds a small core tensor that lets the three factor spaces interact with learned weights, the form Multiverse Recommendation uses.

Why it matters
The alternatives throw information away. Pre-filtering trains a separate model per context and starves each of data; post-filtering reorders a context-blind list and can only demote, never discover. Tensor factorization learns the context's effect as a shared representation, so an evening-viewing pattern learned across many users transfers to a user with few evening interactions.

What goes wrong
Sparsity compounds: splitting an already-sparse matrix across context values leaves slices with almost nothing in them. Continuous context has to be bucketed, and the buckets are a modelling decision — too fine and every slice is empty, too coarse and the signal disappears.