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Linear Algebra for Machine Learning: Definition and Core Concepts

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Linear Algebra is a sub-field of mathematics concerned with vectors, matrices, and linear transforms. It is a key foundation to the field of Machine Learning, from notations used to describe the operation of algorithms to the implementation of algorithms in code.

In this post, we will look at Linear Algebra definition, its different examples and how it relates to vectors and matrices. Then we look through what vectors and matrices are and how to work with them, including the knotty problem of eigenvalues and eigenvectors, and how to use these to solve problems.

We would learn about the interrelation between Linear Algebra and Machine Learning. Here we will explore some common examples of Machine Learning that you may be familiar with that are used and best understood using Linear Algebra.

In this post, I would like to delve deeper into Linear Algebra for Machine Learning & also find out ways how you can improve your skills and knowledge in Linear Algebra to get more out of Machine Learning. We would also learn about linear regression python code and multiple linear regression python.

Our discussion would also include linear regression Machine Learning python. Linear regression using python has become extremely among developers and this calls for an in-depth analysis of Linear Algebra for Machine Learning.

Linear Algebra for Machine Learning

Linear Algebra

What is Linear Algebra?

Linear Algebra, which is a sub-field of mathematics, is primarily concerned with vectors, matrices, and linear transforms. Linear Algebra lays the basic premises for Machine Learning, from notations used to describe the operation of algorithms to the implementation of algorithms in code.

Linear Algebra for Machine Learning

Machine Learning addresses the question of how to build computers that improve automatically through experience. It is one of today’s most rapidly growing technical fields, lying at the intersection of computer science and statistics, and at the core of artificial intelligence and data science.

The recent development in Machine Learning has been driven both by the development of new learning algorithms and theory and by the ongoing explosion in the availability of online data and low-cost computation.

The adoption of data-intensive machine-learning methods can be found throughout science, technology, and commerce, leading to more evidence-based decision-making across many walks of life, including healthcare, manufacturing, education, financial modelling, policing, and marketing.

Linear Algebra for Machine Learning Examples

Linear Algebra for Machine Learning

Linear Algebra for Machine Learning Examples

Some of the best examples of Linear Algebra for Machine Learning include Dataset and Data Files, Images and Photographs, Linear Regression, Regularization, Deep Learning, Principal Component Analysis, and Singular-Value Decomposition.

1. Dataset and Data Files

This represents a table-like set of numbers where each row represents an observation and each column represents a feature of the observation. Each row is of the same length.

Therefore, it may be inferred that the data is vectorized where rows can be provided to a model one at a time or in a batch and the model can be pre-configured to expect rows of a fixed width.

2. Images and Photographs

Images or photos are the common instances of Linear Algebra for Machine Learning usage. Every image that you may need to work upon is itself a table structure with a width and height and one-pixel value in each cell for black and white images or 3-pixel values in each cell for a colour image.

A photo is another classic example of a matrix from Linear Algebra. Photo editing activities such as cropping, scaling, shearing, and so on are all described using the notation and operations of Linear Algebra.

3. Regularization

Regularization is another example of using Linear Algebra for Machine Learning.  In applied Machine Learning, we prefer simpler models, since these are better at generalizing from specific examples to unseen data.

In many methods that involve coefficients, such as regression methods and artificial neural networks, simpler models are often characterized by models that have smaller coefficient values.

Regularization is used to encourage a model to minimize the size of coefficients while it is being fit on data. Some of the common implementations include the L2 and L1 forms of regularization. Both forms of regularization are 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.

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4. Deep Learning

Deep learning has seen a recent spurt in the use of artificial neural networks with newer methods and faster hardware that allow for the development and training of larger and deeper (more layers) networks on very large datasets.

Deep learning methods are routinely achieving state-of-the-art results on a range of challenging problems such as machine translation, photo captioning, speech recognition, and much more.

The execution of neural networks involves Linear Algebra data structures multiplied and added together. Scaled up to multiple dimensions, deep learning methods work with vectors, matrices, and even tensors of inputs and coefficients, where a tensor is a matrix with more than two dimensions.

Linear Algebra is pivotal to the description of Deep Learning methods via matrix notation to the implementation of deep learning methods such as Google’s TensorFlow Python library.

Linear Algebra for Machine Learning


Linear Regression

What is Linear Regression?

Linear regression is a statistical model that examines the linear relationship between two (Simple Linear Regression) or more (Multiple Linear Regression) variables — a dependent variable and independent variable(s). In Linear relationship with one (or more) independent variables increase (or decrease), the dependent variable increases (or decreases) as well.

The linear relationship can be positive (independent variable goes up, the dependent variable goes up) or negative (independent variable goes up, the dependent variable goes down).  The overall idea of regression is to examine two things:

(1) does a set of predictor variables do a good job in predicting an outcome (dependent) variable? (2) Which variables, in particular, are significant predictors of the outcome variable, and in what way do they–indicated by the magnitude and sign of the beta estimates–impact the outcome variable?  

These regression estimates are used to explain the relationship between one dependent variable and one or more independent variables.

The simplest form of the regression equation with one dependent and one independent variable is defined by the formula y = c + b*x, where y = estimated dependent variable score, c = constant, b = regression coefficient, and x = score on the independent variable.

Simple Linear Regression

Linear Algebra for Machine Learning

Simple Linear Regression

Simple linear regression is required for finding the relationship between two continuous variables. One is a predictor or independent variable and other is the response or dependent variable. It looks for a statistical relationship but a not deterministic relationship.

The relationship between the two variables is said to be deterministic if one variable can be accurately expressed by the other.

For example, using temperature in degree Celsius it is possible to accurately predict Fahrenheit. Statistical relationship is not accurate in determining the relationship between two variables.

For example the relationship between height and weight.

Multiple Linear Regression

Multiple linear regression aims at modelling the relationship between two or more features and a response by fitting a linear equation to observed data. It is an extension of Simple Linear Regression.

Linear Algebra for Machine Learning


Linear Regression using Python

Linear Regression in Python can be performed in two ways: Statsmodels and sci-kit-learn. Linear regression python code is becoming increasingly popular.

(a) Linear Regression using Python: Statsmodels

Statsmodels may be defined as a Python module that provides classes and functions for the estimation of many different statistical models, and for conducting statistical tests, and statistical data exploration.

The simplest way to get or install Statsmodels is through the Anaconda package. After installing it, you will need to import it every time you want to use it.

You can perform both single and multiple Linear Regression in Statsmodels. You have the option of using as little or as many variables you want in your regression model.

(b) Linear Regression using Python: SKLearn

SKLearn has many learning algorithms, for regression, classification, and clustering and dimensionality reduction. You may use the entire dataset, even though a long and tedious process, or break up your data into a training data to train your model on, and test data.

Have a look at the Linear Regression examples in Python.

You may learn more about linear regression python code by reading blogs, discussions, and watching video tutorials.

If you want to excel as a coder or an expert in Linear Algebra for Machine Learning, you should have a strong background in mathematics. You will need to have a thorough knowledge of vectors, matrices and how to apply these to solve linear systems of equations, and how to apply these to computational problems

In this post, we have taken a look at Linear Algebra and the vital role it plays in Machine Learning. We looked at different techniques of learning linear regression using python. You may go for a refresher, crash course or a deeper video course for a better understanding.

Closing Thoughts

I hope this has sparked your interest in Linear Algebra for Machine Learning. You may look up for courseware or high-quality resources on Linear Algebra for Machine Learning.

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