alsabsolute

alsabsolute is a variant of the Alternating Least Squares (ALS) algorithm used in matrix factorization for recommender systems. It replaces the traditional L2 loss with an L1 (absolute-value) loss on the reconstruction error, aiming to improve robustness to outliers and heavy-tailed noise in observed ratings.

Definition and objective: Given a partially observed user-item rating matrix R, with factorization R ≈ U V^T, alsabsolute minimizes sum over observed entries |r_ui - u_i^T v_j| + regularization terms such as lambda (||U||^2 + ||V||^2). Because the L1 loss is not differentiable at zero, subproblems are solved via methods such as linear programming, coordinate descent, or iteratively reweighted least squares (IRLS). The algorithm alternates updating user factors U while holding V fixed, then updating V while holding U fixed, repeating until convergence.

Advantages and challenges: The L1 loss reduces sensitivity to outliers and can yield sparser residuals, which can improve recommendation quality in noisy datasets. However, optimization is typically more computationally intensive than standard ALS, and convergence properties depend on the solver and regularization. Hyperparameters include the regularization strength and the specific method used for the subproblems.

History and context: The approach is discussed in the literature on robust matrix factorization and recommender systems as a counterpart to L2-based ALS. It has been explored in settings with explicit feedback and with varying sparsity patterns, sometimes in combination with other robust loss functions such as the Huber loss.

See also: Alternating Least Squares, L1 loss, robust regression, recommender systems, IRLS, matrix factorization.