# Matrix Factorization

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Matrix factorization is a technique for dimensionality reduction and latent factor analysis.
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### How is it work?

It works by decomposing a matrix into two or more lower-dimensional matrices. The original matrix can then be reconstructed by multiplying the two lower-dimensional matrices together.

### Working process

The process of matrix factorization ***involves minimizing the error*** between ***the reconstructed matrix*** and the ***original matrix*** by *<mark style="color:red;">**adjusting**</mark>* the values in the factor matrices. This can be achieved through various optimization algorithms, such as *<mark style="color:red;">**gradient descent**</mark>* or *<mark style="color:red;">**alternating least squares**</mark>*.

### How to implement it?

It can be performed using different methods, including [SVD](/wiki/ai-techniques/adaptation/lora/svd.md), NMF, and PMF. Each method has its own characteristics and is suitable for different scenarios.

### The most common application of Matrix factorization

#### Collaborative Filtering

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It is used to make personalized recommendations based on the preferences and behavior of users.
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In collaborative filtering, a matrix is constructed with users as rows, items as columns, and the entries representing user-item interactions (such as ratings or purchase history). Matrix factorization aims to approximate this original matrix by finding two (or more) lower-rank matrices whose product closely matches the original matrix.

#### Recommender Systems

Matrix factorization has proven to be effective in recommender systems because it can ***handle sparse data***, ***capture latent factors***, and provide personalized recommendations even for new or unrated items.


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