Linear Algebra for Machine Learning: Critical Concepts, Why Learn Before ML

Machine learning, robotics, data science, artificial intelligence, and computer vision are amongst the areas that have been instrumental in bringing our technology up to the level it is at now. As you start to acquire more knowledge about these technologies, you will come across a set of jargons or specific words that are common to these technologies.

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Some of these terms include lasso regression, KKT conditions, kernel PCA, support vector machines (SVM), Lagrange multipliers, and ridge regression, amongst others. Now, these jargons may be coined just to keep the outsiders away, but they say a lot about their association with the typical linear algebra that we know of from our days at the school. 

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So, it becomes imperative for every individual who is learning machine learning or data science to first come to terms with what linear algebra and optimization theory are. You also need to learn data science and know how to use them when solving problems using ML or when making more sense of the enormous data available using data science.

In this blog, we will focus on how machine learning and linear algebra are related and how a better understanding of the latter can help you master the former. 

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There are concepts in machine learning, such as SVM and regression that you won’t be able to properly understand if you aren’t aware of their linear algebra connection. You can go without going deep into linear algebra and how it is associated with machine learning if you are just running through these concepts to know what these actually are and have no desire of pursuing their study any further.

However, if you are planning to become a machine learning engineer who is going to be training machines going forward or do research and make significant contributions in the field, you will have to dig deep. There is no other alternative. Having a firm background in linear algebra is a must. Our main objective of writing this blog is to put before you the fundamentals of linear algebra, ensuring that we present how they are used in machine learning. Let us start by understanding what linear algebra exactly is.

What is Linear Algebra?

In simple words, it is a branch of mathematics that finds significant applications in engineering and science. Though it holds such importance and has applications that go far beyond our imaginations, we see our scientists lagging behind when it comes to having a deeper understanding of it. The main reason behind this is because it is not discrete mathematics that we find most scientists using on a frequent basis.

It belongs to the continuous part of mathematics, which makes it less interesting for scientists and people working in the technology domain. Now let us make one thing very clear. If you don’t even have a basic understanding of how linear algebra works, you will find it very tough to learn and use several machine learning algorithms, including the deep learning ones. 

When you are done with how machine learning fundamentally works and how and where you can use its algorithms, you will then be required to give a little more time to learning math. This will help you understand a lot of new things about machine learning algorithms that you previously didn’t. You will know a lot about their limitations, underlying assumptions, and whatnot. 

Now you will come across different areas in mathematics that you study at this point to learn to do more with machine learning. You can study geometry, algebra, calculus, and statistics amongst other topics; however, you need to be wise here and select the area that you think is really going to help you enrich your experience and provide you with a more firm footing as you make your way ahead in your machine learning career. You can even ask experts to help you make a decision. 

The next question you will be asking yourself now will be how you need to go about this learning process. You can’t study linear algebra from scratch. You will have to pick and choose topics that are used in machine learning in one way or the other. In the next section, we are going to discuss a few of those linear algebra topics that you can choose to study.

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Important Linear Algebra Concepts

It is very important to have sufficient knowledge of a few linear algebra concepts if you are looking to understand the underlying concepts behind machine learning. If you don’t know the math behind these advanced machine learning algorithms, you can’t wish to develop a mastery over them. Here are a few concepts of linear algebra that you need to learn about for knowing how machine learning works.

1. Vectors and Matrix

It won’t be wrong to say that these two concepts are arguably the two most important ones that you need to learn considering their close allegiance with machine learning. Vectors consist of an array of numbers while a matrix comprises 2-D vectors that are usually mentioned in uppercase.

Now let us see how they are linked to machine learning algorithms. Vectors find themselves useful in supervised machine learning algorithms where they are present in the form of target variables. On the other hand, features available in the data form the matrix. You can perform a number of operations using the matrix – conjugate, multiplication, rank, transformation, and others. Two vectors having the same number of elements and shape equality can also be used to perform subtraction and addition. 

2. Symmetric Matrix

Symmetric matrix holds importance in both linear algebra and machine learning. Linear algebra matrices are mostly used to carry functions. Most of the time, these functions are symmetrical, and so are the matrices that correspond to them. These functions and the values they hold can be used to measure feature distance. They can also be used to measure feature covariance. Listed below are a few properties of symmetric matrices:

• Symmetric matrices and their inverse are both symmetrical.
• All values in the eigenvalues are real numbers. No complex numbers are present.
• A symmetric matrix is formed when a matrix is multiplied with its transpose.
• Symmetric matrices also hold the property of factorization.
• For matrices that have linearly independent columns, the result when the matrix is multiplied with its transpose is invertible.

3. Eigenvalues and Eigenvector

Eigenvectors are vectors that only change by a scalar factor, and there is no change in their direction at all. The eigenvalue corresponding to eigenvectors is the magnitude by which they are scaled. Eigenvalues and eigenvectors are found in the fundamentals of mathematics and computing. When we plot a vector on an XY graph, it follows a specific direction. When we apply the linear transformation on a few vectors, we see that they don’t change their direction. These vectors are very important in machine learning.

Eigenvalues and eigenvectors are used to minimize data noise. We can also use the two to improve the efficiency of the tasks that are known to be computationally intensive. They can also be used to do away with overfitting. There are several other scenarios as well in which eigenvalues and eigenvectors prove useful.

It is quite difficult to visualize the features of sound, textual, or image data. This data is usually represented in 3-D. This is where eigenvalues and eigenvectors come into the picture. They can be used to capture all the huge amount of that is stored in a matrix. Eigenvalues and eigenvectors are used in facial recognition too. 

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4. Principal Component Analysis (PCA)

There are many times when dimensionality makes things difficult when it comes to solving certain machine learning problems. In these problems, we are dealing with data whose features have a very high correlation amongst themselves and are in a dimension that is higher than usual.

The problem that comes out with this dimensionality issue is that it becomes very difficult to understand the influence that every feature has on the target variable. This is so because features with higher correlation than normal tend to influence the target in the same manner. It is also very difficult to visualize data that is in a higher dimension. 

The principal component analysis is the solution to these problems. It helps you bring down your data dimension to 2-D or 3-D. This is done ensuring that no information is lost due to changes in the maximum variance. Maths behind PCA relates to orthogonality. PCA is the best method available to make the model less complex by bringing down the number of features in the data set.

However, you should avoid using it as the initial step to eliminate overfitting. You should begin with limiting the number of features in the data or increasing data quantity. You should then try using L1 or L2 regularization. If nothing works, only then you should turn to PCA.

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Why should you learn linear algebra before machine learning?

1. Linear algebra is the key to excel in machine learning

There is no denying the fact that calculus trumps linear algebra when it comes to advanced mathematics. Integral and differential calculus help you a lot more than just with integration, differentiation, and limits, they also serve as fundamental knowledge required for applications, such as tensors and vectors.

Learning these things will help you have a better understanding of linear equations and linear functions amongst other areas. You will also know about advanced concepts, such as the Simplex method and spatial vectors. If you need help with linear programming, you can use the Simplex method. To get better in these concepts, start by giving more time to linear algebra.

2. Machine learning prediction

When you learn linear algebra, you improve the awareness or instinct that plays such an important role in machine learning. You will now be able to provide more perspectives. The matrices and vectors that you studied will help you widen your thinking and make it more unwavering. The possibilities are endless. You could start doing things that others around you will find very hard to understand. You could begin visualizing and setting up different graphs. You could start using more parameters for different machine learning components. 

3. Linear algebra helps in creating better machine learning algorithms

You can use your learning of linear algebra to build better supervised as well as unsupervised machine learning algorithms. Logistic regression, linear regression, decision trees, and support vector machines (SVM) are a few supervised learning algorithms that you can create from scratch with the help of linear algebra.

On the other hand, you can also use it for unsupervised algorithms, including single value decomposition (SVD), clustering, and components analysis. Linear algebra will help you develop a more in-depth understanding of the machine learning project you are working on, and thus will give you the flexibility to customize different parameters. You can learn more about Linear regression in machine learning.

4. Linear algebra for better graphic processing in machine learning

Machine learning projects provide you with different graphical interpretations to work on – images, audio, video, and edge detection. Machine learning algorithms have classifiers that train a part of the given data set based on their categories. Another job of classifiers is to do away with errors from the data that has already been trained.

It is at this stage that linear algebra comes in to help compute this complex and large data set. It uses matrix decomposition techniques to process and handles large data for different projects. The most popular matrix decomposition methods are Q-R and L-U decomposition. 

5. Linear algebra to improve your take on statistics 

Statistics are very important to organize and integrate data in machine learning. If you want to understand statistical concepts in a better way, you need to first know how linear algebra works. Linear algebra has methods, operations, and notations that can help integrate advanced statistical topics like multivariate analysis into your project.

Suppose you are working on patient data that includes weight, height, blood pressure, and heart rate. These are the multiple variables of the data set you are working on. Let us make an assumption here that an increase in weight will lead to an increase in blood pressure. It’s not too difficult to understand that this is a linear relationship. So to better understand how an increase in one variable affects the other, you will need to have a good understanding of linear algebra.

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Conclusion

Machine learning in itself is quite a vast topic; however, there are other concepts, like linear algebra, that are as important to learn as ML itself. Learning linear algebra and other such topics will help understand the concepts of machine learning better.

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