**6 a] Briefly explain alternating Least squares methods.**

**Alternating Least Squares (ALS)** is an optimization technique often used for solving matrix factorization problems in recommendation engines, such as predicting user preferences for items.

### The ALS Algorithm Explained:

**Objective**: The goal is to find two matrices, ( U ) and ( V ), such that their product approximates the original user-item matrix ( X ). You aim to minimize the difference between the actual values in ( X ) and the predicted values from ( U \cdot V^T ), while also keeping the values of ( U ) and ( V ) small.**Iterative Process**: ALS solves the optimization problem by alternating between fixing one matrix (either ( U ) or ( V )) and solving for the other in a series of iterations:

**Fix ( V )**and solve for ( U ).**Fix ( U )**and solve for ( V ).- Repeat these steps until the changes in ( U ) and ( V ) become insignificant, indicating that the algorithm has converged.

**Closed-form solution**: While the overall optimization problem does not have a simple closed-form solution, each step (optimizing ( U ) or ( V ) while the other is fixed) reduces to a least squares problem, which does have a closed-form solution.**Parallelization**: Because each user or item can be updated independently based on their specific ratings, ALS can be easily parallelized, making it computationally efficient for large datasets.**Regularization**: The algorithm can include regularization terms to ensure that the problem remains convex and to avoid overfitting.

##### Example Steps:

**Start with a random matrix ( V )**.**Optimize ( U ) while keeping ( V ) fixed**: For each user, solve a least squares problem to find the optimal user vector ( u_i ).**Optimize ( V ) while keeping ( U ) fixed**: For each item, solve a least squares problem to find the optimal item vector ( v_j ).**Repeat until convergence**: The process is repeated until the changes in ( U ) and ( V ) are minimal.

ALS is widely used in recommendation systems due to its simplicity, scalability, and ease of parallelization, making it suitable for large-scale applications.