Matrix Factorization

Overview

Matrix factorization decomposes the user-item interaction matrix R (m×n) into two low-rank matrices: U (m×k) and V (n×k), where k << min(m,n). Predicted rating: r̂ᵢⱼ = uᵢ · vⱼ. Trains in O(k × nnz × iterations) where nnz = non-zero entries.

When to Use

Trigger conditions:

• Scaling CF beyond pairwise similarity (millions of users/items)
• Discovering latent factors that explain user-item interactions
• Predicting ratings for unobserved user-item pairs

When NOT to use:

• When interaction data is extremely sparse (< 0.1% fill) — insufficient for learning
• When you need real-time updates (retraining is expensive)

Algorithm

IRON LAW: Rank k Controls Bias-Variance Trade-Off
- Too LOW k: underfits, misses nuanced preferences (high bias)
- Too HIGH k: overfits to noise, poor generalization (high variance)
- Typical k: 20-200. Select via cross-validation on held-out ratings.
- Always add regularization (λ) to prevent overfitting.

Phase 1: Input Validation

Load sparse interaction matrix. Split into train/validation/test. Check minimum density. Gate: Train matrix has sufficient entries per user and item.

Phase 2: Core Algorithm

ALS (Alternating Least Squares):

1. Initialize U, V randomly (or with SVD warm-start)
2. Fix V, solve for U: minimize ||R - UV^T||² + λ(||U||² + ||V||²)
3. Fix U, solve for V using same objective
4. Alternate until convergence (RMSE change < ε)

SGD alternative: Update u_i, v_j incrementally for each observed rating using gradient descent.

Phase 3: Verification

Compute RMSE on held-out validation set. Compare against baseline (global mean, user mean). Gate: Validation RMSE significantly below baseline.

Phase 4: Output

Return top-N predictions per user with predicted scores.

Output Format

{
  "recommendations": [{"user_id": "u1", "items": [{"item_id": "i5", "predicted_rating": 4.3}]}],
  "metadata": {"rank_k": 50, "regularization": 0.01, "iterations": 20, "train_rmse": 0.82, "val_rmse": 0.91}
}

Examples

Sample I/O

Input: 3×3 rating matrix R (0 = unobserved), k=1

R = [[5, 3, 0],
     [4, 0, 2],
     [0, 1, 1]]

Expected: After ALS with k=1 (one latent factor, λ=0.01, 50 iterations), approximate factorization:

U ≈ [[2.24], [1.84], [0.53]]
V ≈ [[2.23], [1.06], [0.98]]
R_hat ≈ [[4.99, 2.37, 2.20],
         [4.10, 1.95, 1.80],
         [1.18, 0.56, 0.52]]

Verify: R_hat ≈ R on observed entries (within 0.2 RMSE). U[0] >> U[2] correctly captures user 0's higher ratings.

Edge Cases

Input	Expected	Why
User with 1 rating	Poor predictions for that user	Insufficient data to learn user factors
Highly popular item	Predicted near average	Dominant first latent factor captures popularity
All ratings = 5	Trivial factorization	No variance to learn from

Gotchas

• Implicit data needs different loss: For clicks/views (no explicit ratings), use weighted matrix factorization (Hu et al. 2008) with confidence weighting, not RMSE.
• Cold start remains: New users/items have no entries in R. MF can't factorize what doesn't exist. Use side features or hybrid approaches.
• Negative sampling: For implicit feedback, you must sample negative examples (unobserved ≠ disliked). Random negative sampling works but biased sampling is better.
• Initialization matters: Random initialization can converge to poor local optima. SVD-based warm-start often helps.
• Bias terms: Add user bias bᵢ and item bias bⱼ: r̂ᵢⱼ = μ + bᵢ + bⱼ + uᵢ·vⱼ. This captures systematic rating tendencies.

References

• For ALS vs SGD comparison, see references/optimization-comparison.md
• For implicit feedback matrix factorization, see references/implicit-mf.md