Linear Algebra in Recommendation Systems of Machine Learning

Linear algebra in machine learning is one of the essential topics to learn and have an idea about, as it is applicable and plays a critical role in many machine learning applications. Recommendation systems are primary unsupervised machine learning applications industries and companies use to recommend particular things to their customers and audience.

In this article, we will discuss linear algebra in recommendation systems, how it is helpful for recommendations, and how it is used appropriately, with some examples. This article will help one to understand the application and role of linear algebra behind recommendation systems in machine learning. It will help answer interview questions related to the same as well.

Before directly jumping to the application of linear algebra in recommendation systems, let us try to understand a bit about the recommendation systems and how they work.

Table of Contents

  1. Working of Recommendation Systems
  2. Matrix Factorization using Linear Algebra
  3. Clustering Similar Users
  4. Matrix Imputations using Linear Algebra
  5. Key Takeaways
  6. Conclusion

Working on Recommendation Systems

As we discussed above, recommendation systems are intelligent systems or models trained on users’ data with the help of machine learning algorithms and techniques, which helps to recommend appropriate things to the user according to the behavior and interest of the user on a particular platform.

Mainly the recommendation systems can be divided into three categories.
1. Content-Based
2. Collaborative Filtering
3. Hybrid

Content-based recommendation systems are a type of model that recommend particular things to the user based on what the user like, what the user views, and all the things that the user may be interested in based on tags or categories that are given to particular events that user is watching or liking.

Collaborative filtering is a type of recommendation system that assigns specific similarity scores to the users and recommend particular things to the user based on what other user with the same similarity like. Here the users with the same behaviors or similarity scores are clustered into clusters or groups by using K means clustering technique or hierarchical clustering technique.

A hybrid type of recommendation system combines both of the above recommendation systems. For example, YouTuve recommends videos to users with a hybrid recommendation system.

Matric Factorization using Linear Algebra

In recommendation systems, our primary task is to recommend particulate things to the user based on the user’s activity and what the user likes. Here a matrix is created based on user history and interests, a large or significantly higher dimensional matrix based on the user’s data. Now if the matric is very large in dimensions, then it would be difficult to apply some of the algorithms and techniques to it, or it may take higher computational powers and time.

Here the matric factorization comes into the picture. We reduce or decompose the dimensionally of the matric and make it of lower dimension to make it more visual and solvable. Many matric factorization techniques are used to reduce the dimensionality of the matrix. Still, Singular Value Decomposition (SVD) is one of the most popular and accessible matrix factorization techniques.

Here in singular value decomposition, we divide the matric into three parts: left particular vector, suitable singular vector, and diagonal vector. Now with the reduced dimensionality of the matrix, we can easily handle the data and get the essential features that affect the most and create a reliable and accurate model.

Clustering Similar Users

As we discussed that there are multiple types of recommendation systems. One of the types is collaborative filtering-based systems, which recommends particular things to the users based on the activity of another user with a similar score.

Here basically, we calculate the similarity score between users and cluster the users with the same similarity. In clustering, we can use either K mean clustering or hierarchical clustering for clustering similar users. Now linear algebra plays a very crucial role while clustering the matching users. The similarity between users is either calculated by the distance between them, which may be a cosine distance, or the euclidian distance, where linear algebra greatly helps.

Matrix Imputations in Recommendation Systems

While recommending particular events, videos, or posts to the users, our main aim is to recommend the most relevant things based on the user’s history and similarity scores. Now here, in every dataset, we can not have a complete dataset or matrix with all the user ratings and history. Sometimes, we skip the particular user’s data, which can be a NaN value in the matric. Now, to recommend things to the users, we need complete data; for that, we need to impute the missing values in the matrix.

Low-Rank matrix imputation is a technique that is used to fill in the missing values in the matrix. We assume that the matrix we are filling is of a low-rank structure. The matrix with missing values is first decomposed in the lower dimensions using the singular value decomposition techniques. Once it is decomposed then, the missing value is imputed by multiplying appropriate rows and columns of the matrix with each other.

For example, if the first row of the lower dimensional matrix is a multiple of other rows, then the missing values in the other rows can be imputed by multiplying the values with appropriate rows and columns.

Key Takeaways

  1. Recommendation systems are one of the machine learning, which leverages linear algebra in many techniques.
  2. Linear algebra is used for matrix factorization in recommendation systems, where the matrix is decomposed into lower dimensions.
  3. In recommendation systems, linear algebra plays a crucial role in clustering similar users by calculating the appropriate distance between users in higher dimensions.
  4. Each time we may not have a complete matrix of users to fill the missing values, linear algebra can be helpful to where techniques like low-rank matrix and SVD are used.


In this article, we discussed the application and role of linear algebra in recommendation systems, how different recommendation systems work, and how linear algebra helps in many tasks of recommendations with examples. This article will help one to understand the importance of linear algebra in recommendation systems and will help answer interview questions related to the same as well.