WeightedMatrixFactorization Model for Recommender Systems

This repository aims to build a real Recommender System by using a Weighted Matrix Factorization model.

About data

The MovieLens dataset is a widely used benchmark dataset for recommendation systems research and evaluation. It contains user-item ratings collected from the MovieLens website, where users rate movies on a scale of 1 to 5.

For this project, it has been decided to use the MovieLens 100k: the smallest version of the dataset containing 100000 ratings. This version is commonly used for initial experimentation and prototyping due to its smaller size and faster processing capabilities.

Matrix Factorization Model

Let's consider a scenario with m users and items. Our objective with the recommendation system is to construct an m×n matrix, commonly known as the feedback matrix. This matrix represents ratings for each user-item pair. Note that, due to the limited number of available ratings, this matrix is very sparse.

Here's an example of a feedback matrix for a recommendation system with 3 users and 4 items (movies), where missing ratings are represented by '?' symbols:

	Item 1	Item 2	Item 3	Item 4
User 1	5	?	4	3
User 2	?	2	?	?
User 3	4	3	?	5

The goal of the Recommender system is to predict the missing ratings for each user-item pair marked with '?'.

By thinking of the feedback matrix as the product of two low-rank matrices, each with k latent factors, we simplify the essence of this problem. This simplification helps us frame the problem as a minimization task, where our goal is to optimize a function.

Our objective function is defined as follows:

$$\min_{U,V}{\sum_{(i,j) \in obs}{(A_{ij}-U_i\dot{V_j^{T}})^2+\lambda(||U_i||^2+||V_j||^2)}}$$

In this equation:

• obs represents the set of (i, j) pairs for which ratings are known (i.e., the observed ratings).
• Aij is the rating assigned to item j by user i.
• Ui and Vj denote the user and item latent factor matrices, respectively.
• λ serves as a regularization parameter to mitigate overfitting.

To tackle this minimization problem, two common optimization algorithms can be used:

• Stochastic Gradient Descent (SGD): This method iteratively adjusts the parameters (user and item vectors) by moving in the direction of the negative gradient of the objective function.
• Weighted Alternating Least Squares (WALS): An optimization algorithm that alternates between updating the user and item matrices while keeping the other fixed. This method can also be parallelized as updates for different users/items can be performed independently.

Both of these optimization techniques are implemented in this project.