# Linear Algebra for Machine learning

Machine learning has a strong connection with mathematics. Each machine learning algorithm is based on the concepts of mathematics & also with the help of mathematics, one can choose the correct algorithm by considering training time, complexity, number of features, etc. Linear Algebra is an essential field of mathematics, which defines the study of vectors, matrices, planes, mapping, and lines required for linear transformation.

The term Linear Algebra was initially introduced in the early 18th century to find out the unknowns in Linear equations and solve the equation easily; hence it is an important branch of mathematics that helps study data. Also, no one can deny that Linear Algebra is undoubtedly the important and primary thing to process the applications of Machine Learning. It is also a prerequisite to start learning Machine Learning and data science. Linear algebra plays a vital role and key foundation in machine learning, and it enables ML algorithms to run on a huge number of datasets.

The concepts of linear algebra are widely used in developing algorithms in machine learning. Although it is used almost in each concept of Machine learning, specifically, it can perform the following task:

• Optimization of data.
• Applicable in loss functions, regularisation, covariance matrices, Singular Value Decomposition (SVD), Matrix Operations, and support vector machine classification.
• Implementation of Linear Regression in Machine Learning.

Besides the above uses, linear algebra is also used in neural networks and the data science field.

Basic mathematics principles and concepts like Linear algebra are the foundation of Machine Learning and Deep Learning systems. To learn and understand Machine Learning or Data Science, one needs to be familiar with linear algebra and optimization theory. In this topic, we will explain all the Linear algebra concepts required for machine learning.

## Why learn Linear Algebra before learning Machine Learning?

Linear Algebra is just similar to the flour of bakery in Machine Learning. As the cake is based on flour similarly, every Machine Learning Model is also based on Linear Algebra. Further, the cake also needs more ingredients like egg, sugar, cream, soda. Similarly, Machine Learning also requires more concepts as vector calculus, probability, and optimization theory. So, we can say that Machine Learning creates a useful model with the help of the above-mentioned mathematical concepts.

Below are some benefits of learning Linear Algebra before Machine learning:

• Better Graphic experience
• Improved Statistics
• Creating better Machine Learning algorithms
• Estimating the forecast of Machine Learning
• Easy to Learn

### Better Graphics Experience:

Linear Algebra helps to provide better graphical processing in Machine Learning like Image, audio, video, and edge detection. These are the various graphical representations supported by Machine Learning projects that you can work on. Further, parts of the given data set are trained based on their categories by classifiers provided by machine learning algorithms. These classifiers also remove the errors from the trained data.

Moreover, Linear Algebra helps solve and compute large and complex data set through a specific terminology named Matrix Decomposition Techniques. There are two most popular matrix decomposition techniques, which are as follows:

• Q-R
• L-U

### Improved Statistics:

Statistics is an important concept to organize and integrate data in Machine Learning. Also, linear Algebra helps to understand the concept of statistics in a better manner. Advanced statistical topics can be integrated using methods, operations, and notations of linear algebra.

### Creating better Machine Learning algorithms:

Linear Algebra also helps to create better supervised as well as unsupervised Machine Learning algorithms.

Few supervised learning algorithms can be created using Linear Algebra, which is as follows:

• Logistic Regression
• Linear Regression
• Decision Trees
• Support Vector Machines (SVM)

Further, below are some unsupervised learning algorithms listed that can also be created with the help of linear algebra as follows:

• Single Value Decomposition (SVD)
• Clustering
• Components Analysis

With the help of Linear Algebra concepts, you can also self-customize the various parameters in the live project and understand in-depth knowledge to deliver the same with more accuracy and precision.

### Estimating the forecast of Machine Learning:

If you are working on a Machine Learning project, then you must be a broad-minded person and also, you will be able to impart more perspectives. Hence, in this regard, you must increase the awareness and affinity of Machine Learning concepts. You can begin with setting up different graphs, visualization, using various parameters for diverse machine learning algorithms or taking up things that others around you might find difficult to understand.

### Easy to Learn:

Linear Algebra is an important department of Mathematics that is easy to understand. It is taken into consideration whenever there is a requirement of advanced mathematics and its applications.

## Minimum Linear Algebra for Machine Learning

### Notation:

Notation in linear algebra enables you to read algorithm descriptions in papers, books, and websites to understand the algorithm's working. Even if you use for-loops rather than matrix operations, you will be able to piece things together.

### Operations:

Working with an advanced level of abstractions in vectors and matrices can make concepts clearer, and it can also help in the description, coding, and even thinking capability. In linear algebra, it is required to learn the basic operations such as addition, multiplication, inversion, transposing of matrices, vectors, etc.

### Matrix Factorization:

One of the most recommended areas of linear algebra is matrix factorization, specifically matrix deposition methods such as SVD and QR.

## Examples of Linear Algebra in Machine Learning

Below are some popular examples of linear algebra in Machine learning:

• Datasets and Data Files
• Linear Regression
• Recommender Systems
• One-hot encoding
• Regularization
• Principal Component Analysis
• Images and Photographs
• Singular-Value Decomposition
• Deep Learning
• Latent Semantic Analysis

### 1. Datasets and Data Files

Each machine learning project works on the dataset, and we fit the machine learning model using this dataset.

Each dataset resembles a table-like structure consisting of rows and columns. Where each row represents observations, and each column represents features/Variables. This dataset is handled as a Matrix, which is a key data structure in Linear Algebra.

Further, when this dataset is divided into input and output for the supervised learning model, it represents a Matrix(X) and Vector(y), where the vector is also an important concept of linear algebra.

### 2. Images and Photographs

In machine learning, images/photographs are used for computer vision applications. Each Image is an example of the matrix from linear algebra because an image is a table structure consisting of height and width for each pixel.

Moreover, different operations on images, such as cropping, scaling, resizing, etc., are performed using notations and operations of Linear Algebra.

### 3. One Hot Encoding

In machine learning, sometimes, we need to work with categorical data. These categorical variables are encoded to make them simpler and easier to work with, and the popular encoding technique to encode these variables is known as one-hot encoding.

In the one-hot encoding technique, a table is created that shows a variable with one column for each category and one row for each example in the dataset. Further, each row is encoded as a binary vector, which contains either zero or one value. This is an example of sparse representation, which is a subfield of Linear Algebra.

### 4. Linear Regression

Linear regression is a popular technique of machine learning borrowed from statistics. It describes the relationship between input and output variables and is used in machine learning to predict numerical values. The most common way to solve linear regression problems using Least Square Optimization is solved with the help of Matrix factorization methods. Some commonly used matrix factorization methods are LU decomposition, or Singular-value decomposition, which are the concept of linear algebra.

### 5. Regularization

In machine learning, we usually look for the simplest possible model to achieve the best outcome for the specific problem. Simpler models generalize well, ranging from specific examples to unknown datasets. These simpler models are often considered models with smaller coefficient values.

A technique used to minimize the size of coefficients of a model while it is being fit on data is known as regularization. Common regularization techniques are L1 and L2 regularization. Both of these forms of regularization are, in fact, a measure of the magnitude or length of the coefficients as a vector and are methods lifted directly from linear algebra called the vector norm.

### 6. Principal Component Analysis

Generally, each dataset contains thousands of features, and fitting the model with such a large dataset is one of the most challenging tasks of machine learning. Moreover, a model built with irrelevant features is less accurate than a model built with relevant features. There are several methods in machine learning that automatically reduce the number of columns of a dataset, and these methods are known as Dimensionality reduction. The most commonly used dimensionality reductions method in machine learning is Principal Component Analysis or PCA. This technique makes projections of high-dimensional data for both visualizations and training models. PCA uses the matrix factorization method from linear algebra.

### 7. Singular-Value Decomposition

Singular-Value decomposition is also one of the popular dimensionality reduction techniques and is also written as SVD in short form.

It is the matrix-factorization method of linear algebra, and it is widely used in different applications such as feature selection, visualization, noise reduction, and many more.

### 8. Latent Semantic Analysis

Natural Language Processing or NLP is a subfield of machine learning that works with text and spoken words.

NLP represents a text document as large matrices with the occurrence of words. For example, the matrix column may contain the known vocabulary words, and rows may contain sentences, paragraphs, pages, etc., with cells in the matrix marked as the count or frequency of the number of times the word occurred. It is a sparse matrix representation of text. Documents processed in this way are much easier to compare, query, and use as the basis for a supervised machine learning model.

This form of data preparation is called Latent Semantic Analysis, or LSA for short, and is also known by the name Latent Semantic Indexing or LSI.

### 9. Recommender System

A recommender system is a sub-field of machine learning, a predictive modelling problem that provides recommendations of products. For example, online recommendation of books based on the customer's previous purchase history, recommendation of movies and TV series, as we see in Amazon & Netflix.

The development of recommender systems is mainly based on linear algebra methods. We can understand it as an example of calculating the similarity between sparse customer behaviour vectors using distance measures such as Euclidean distance or dot products.

Different matrix factorization methods such as singular-value decomposition are used in recommender systems to query, search, and compare user data.

### 10. Deep Learning

Artificial Neural Networks or ANN are the non-linear ML algorithms that work to process the brain and transfer information from one layer to another in a similar way.

Deep learning studies these neural networks, which implement newer and faster hardware for the training and development of larger networks with a huge dataset. All deep learning methods achieve great results for different challenging tasks such as machine translation, speech recognition, etc. The core of processing neural networks is based on linear algebra data structures, which are multiplied and added together. Deep learning algorithms also work with vectors, matrices, tensors (matrix with more than two dimensions) of inputs and coefficients for multiple dimensions.

## Conclusion

In this topic, we have discussed Linear algebra, its role and its importance in machine learning. For each machine learning enthusiast, it is very important to learn the basic concepts of linear algebra to understand the working of ML algorithms and choose the best algorithm for a specific problem.

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