Matrix factorization

See also: Machine learning terms

Introduction

Matrix factorization is a technique in machine learning that aims to discover latent features underlying the interactions between two different kinds of entities. It has been widely used for tasks such as recommendation systems, dimensionality reduction, and data imputation. The primary goal of matrix factorization is to approximate a given matrix by factorizing it into two or more lower-dimensional matrices, which can then be used to reconstruct the original matrix with minimal loss of information.

Mathematical Background

Matrix factorization involves decomposing a given matrix A of dimensions m × n into two lower-dimensional matrices, P of dimensions m × k and Q of dimensions k × n, such that their product approximates the original matrix A. Mathematically, this can be represented as:

A ≈ P × Q

Here, k represents the number of latent features, which is a user-defined parameter. The main objective is to find the matrices P and Q that minimize the difference between the actual matrix A and the reconstructed matrix P × Q. This can be achieved by minimizing a loss function, typically the squared error between the original and reconstructed matrices.

Loss Function and Optimization

The most common loss function used in matrix factorization is the mean squared error (MSE), which calculates the squared difference between the original matrix and the product of the factorized matrices:

L(P, Q) = (1 / mn) ∑∑((A_ij - (P × Q)_ij)^2)

where L(P, Q) represents the loss function, and A_ij and (P × Q)_ij denote the elements of the original and reconstructed matrices, respectively.

To minimize the loss function, gradient descent or other optimization algorithms can be employed. These methods iteratively update the elements of P and Q by calculating the gradients of the loss function with respect to each element, and adjusting the element values accordingly.

Applications

Matrix factorization has found extensive applications in various domains, including:

Recommendation Systems

In recommender systems, matrix factorization has been widely used for collaborative filtering. User-item interactions, such as ratings, are organized into a sparse matrix, where each element represents the interaction between a user and an item. Matrix factorization is then used to discover latent features that explain the observed interactions, which can then be used to predict missing interactions and recommend items to users.

Dimensionality Reduction

Matrix factorization can also be applied for dimensionality reduction purposes, where high-dimensional data is projected onto a lower-dimensional space while preserving important information. Techniques such as principal component analysis (PCA) and non-negative matrix factorization (NMF) are examples of dimensionality reduction methods that use matrix factorization.

Data Imputation

Matrix factorization can be employed for data imputation, which involves filling in missing values in a dataset. By factorizing the incomplete matrix and reconstructing it using the latent features, the missing values can be estimated, allowing for further analysis or modeling on the imputed dataset.

Explain Like I'm 5 (ELI5)

Imagine you have a big LEGO castle built from many different LEGO pieces. Now, you want to take it apart and rebuild it using fewer pieces, but you still want the new castle to look as similar to the original one as possible. Matrix factorization does something like this, but with numbers in a table (called a matrix). It takes the big table and breaks it down into smaller tables, then puts them back together to get a table that's as close to the original one as possible. This can be useful for tasks like suggesting movies or songs, reducing the size