Essential Mathematics for Machine Learning | Important concepts of Mathematics for Machine Learning

Nowadays, machine learning is one of the most trending technologies among researchers, industries and enthusiastic learners because of making human life easier. It is being widely used in almost all areas of the real world, from Google Assistant to self-driving cars. It is about developing models that can automatically extract important information and patterns from data. But here, an important question arises: what is the magic behind ML, and the answer is mathematics. Mathematics is the core of designing ML algorithms that can automatically learn from data and make predictions. Therefore, it is very important to understand the Maths before going into the deep understanding of ML algorithms.

Mathematics has always been a good friend for some people and a phobia or anxiety for some people. Many students don't find interest in mathematics around the globe as they think that topics covered in mathematics are less or not relevant to practical or real-world problems. But with the growth of machine learning, people are getting motivated to learn mathematics as it is directly used in designing ML algorithms. It is also very helpful to learn the concepts behind this. In this topic, we will learn all the essential concepts of Mathematics that are used in Machine Learning.

Note: It is not required to go deep in learning Mathematics for working with simple machine learning models; rather, knowing essential Maths concepts is enough to understand how it is applied in ML.

Why to learn Mathematics for Machine Learning?

There is always a question in enthusiast learners that what is the need of mathematics in machine learning? As computers can solve mathematics problems faster than humans. So, the answer is, learning mathematics in machine learning is not about solving a maths problem, rather understanding the application of maths in ML algorithms and their working. Other below points explain the significance of maths in ML:

• Mathematics defines the concept behind the ML algorithms & helps in choosing the right algorithm by considering accuracy, training time, the complexity of the model, number of features.
• Computers understand data differently than humans, such as an image is seen as a 2D-3D matrix by a computer for which mathematics is required.
• With Maths, we can correctly determine the interval & uncertainty.
• It helps in selecting correct parameter values and validation methods.
• Understanding the Bias-Variance trade-off helps us identify underfitting and overfitting issues that are the main issues in ML models.

Essential Mathematics for Machine Learning

After understanding the need for Maths, the next question arises: what level of maths is required and what concepts one needs to understand. To answer this question, we have provided the basic level of mathematics required for an ML Engineer/ Scientist. Apart from the below concepts, the level of maths also depends upon the individual's interest and the type of research someone is working on.

• Linear algebra
• Multivariate Calculus
• Probability Theory
• Discrete Mathematics
• Statistics
• Algorithm & Optimization
• Others

The below graph shows the importance of different Maths concepts in Machine Learning. As shown in the graph, the most important part of Mathematics is Linear Algebra, which is widely used in ML

1. Linear Algebra for Machine Learning

Linear algebra is about the study of vectors and some rules of manipulating these vectors. The concepts of linear algebra are widely used in developing algorithms in machine learning. It enables the ML algorithms to run on a huge number of datasets. It can perform the following task:

• It is used almost everywhere in the ML world.
• Linear algebra helps in optimizing data.
• It is used in loss functions, regularisation, covariance matrices, Singular Value Decomposition (SVD), Matrix Operations, and support vector machine classification.
• It is also used in Linear Regression in Machine Learning.

Different topics of linear algebra are used in ML such as Principal Component Analysis (PCA), Singular Value Decomposition (SVD), Eigen decomposition of a matrix, LU Decomposition, QR Decomposition/Factorization, Symmetric Matrices, Orthogonalization & Orthonormalization, Matrix Operations, Projections, Eigenvalues & Eigenvectors, Vector Spaces, and Norms. These topics are needed for understanding the optimization methods.

Besides these uses, linear algebra is also widely used in neural networks and the data science field. In short, Linear Algebra provides a Platform or base for all ML algorithms to show their results.

Although linear algebra is a must-known part of mathematics for machine learning, it is not required to get in deep with this. It means it is not required to be an expert in linear algebra; instead, only good knowledge of these concepts is enough for machine learning.

2. Calculus for Machine Learning

Calculus Mathematics is an integral part of Machine learning, but it is not required to be a master of it; rather, only knowledge of basic concepts is enough. Multivariate calculus helps in solving optimization problems in machine learning. Different ML algorithms optimize an objective function with respect to a set of desired model parameters that control how well a model explains the data. The process of getting the best parameters is known as optimization, and multivariate calculus helps solve optimization problems in the ML model. It helps in optimization and getting good results from the model.

Multivariate calculus is used in algorithm training and gradient descent. We need to learn and implement some important concepts of multivariate calculus, such as Derivatives, divergence, curvature, and quadratic approximations.

Some essential topics of multivariate calculus are:

• Partial Derivatives
• Vector-Values Functions
• Directional Gradient
• Hessian, Jacobian
• Laplacian and Lagrangian Distribution.

3. Probability in Machine Learning

Probability is always an important and interesting part of Mathematics, which measures the likelihood of an event to happen. The higher the probability of an event, the more likely that event will occur. ML also helps in predicting the likelihood of future events. Probability is required to work with an ML prediction and modeling project properly. It also helps in hypothesis testing and distributions such as Gaussian distribution and Probability density function.

Some important Probability concepts that one needs to know are given below:

• Joint, Marginal, and Conditional Probability,
• Probability Distributions (Discrete, Continuous),
• Density Estimation
• Maximum Likelihood Estimation
• Regression with Maximum Likelihood
• Bayes Theorem, etc.

4. Statistics in Machine Learning

Statistics helps in drawing logical conclusions from the given data. It is a crucial concept that every machine learning engineer/scientist must learn to understand the working of classifications algorithms like logistic regression, distributions, discrimination analysis, and hypothesis testing in Machine learning. It helps in performing the following task:

• It is a collection of tools that helps to identify the goal from the available data and information.
• Statistics helps to understand the data and transform the sample observations into meaningful information.
• No system in the world has perfect data stored and readily available as needed. Every system has data anomalies like incomplete, corrupted data, etc. Statistical concepts will be your best friend to help in such complex situations.
• It helps in finding answers to the questions such as, "Who scored the maximum & minimum in a cricket tournament?" "Which technology is on-trend in 2021?", and many more.

Some fundamental concepts of Statistics needed for ML are given below:

• Combinatorics
• Axioms
• Bayes'Bayes' Theorem
• Variance and Expectation
• Random Variables
• Conditional and Joint Distributions.

5. Discrete Mathematics in Machine Learning

Discrete mathematics is a study of mathematical concepts based on discrete (non-continuous numbers) numbers, and more often, Integers. Discrete mathematics has wide applications in different fields such as Algorithms, programming languages, cryptography, software development, etc.

There are many cases in machine learning & AI where discrete mathematics is required to use. For example, a neural network contains the integer number of nodes and interconnections, and it can have .56 nodes. For such cases, a discrete element is needed and hence required discrete mathematics. Graph structure and graph algorithms are some important topics of discrete mathematics for machine learning.

For normal ML projects, only the fundamentals of discrete mathematics are enough. At the same time, if we want to work with graphical models, relational domains, structured prediction, etc., you need to refer to a discrete mathematics book. However, for the science graduates, most of the concepts are covered during College.

6. Algorithms and Complex Optimization

The optimization algorithms are important to understand better the computational efficiency and scalability of machine learning algorithms. The conceptual knowledge of data structures (Binary Trees, Hashing, Heap, Stack, etc.), Dynamic Programming, Randomized & Sublinear Algorithm, Graphs, Gradient/Stochastic Descents and Primal-Dual methods are needed.

Best Way/Resources to learn Mathematics for Machine learning

To learn maths for machine learning is not much typical thing because there are multiple resources available, including books, online courses, and different blogs. All these resources provide plenty of knowledge of different maths topics. However, each resource is good for some concepts, so here we are providing a list of all important resources that will help you to learn Maths in a better & simple Way.

1. Text-Books to learn Maths for Machine Learning

• Mathematics for Machine Learning by Marc Peter Deisenroth is one of the best books to begin your mathematical journey for machine learning. In this book, the practical applications of the algorithms and the maths behind them are explained in detail. The concept of essential mathematics for machine learning has been explained in the best Way. The book can be downloaded from here.
• Hands-on Mathematics for deep-learning by Jay Dawani is another book for advanced maths concepts that help understand advanced ML algorithms and deep learning models. This book also provides a brief introduction to linear algebra, calculus, probability, and statistics. In the second edition of the book, you will get a detailed explanation of the mathematics of multilayer perceptron, convolutional neural networks (CNN), and recurrent neural networks (RNN). It also explains some crucial concepts such as regularization (L1 and L2 norm), dropout layers, and many more.

1. Online Videos to Learn Maths for Machine Learning

• Khan Academy

Khan Academy is popular online resource that provides best-explained maths and science courses, and that that's also for free. From these videos, you can easily learn different concepts of Mathematics on Linear Algebra, Probability & Statistics, Multivariable Calculus, and Optimization.

• Udacity

Introduction to Statistics by Udacity is another free video resource by which you can understand the fundamental concepts of statistics that are needed for Machine Learning & Data Science.

• Multivariate Calculus by Imperial College London:

Imperial College London has provided a YouTube series on some concepts of multivariate calculus and its application in various ml algorithms. If you want the entire mathematics course for Machine Learning, you need to enroll with Coursera; however, Imperial College London has made the Multivariate calculus available for free for all enthusiastic learners.

Conclusion

Mathematics is one of the most important parts of Machine Learning. However, how much maths you need to learn is completely depends on what you want to learn and how deep you are going in that topic. It means, for developing simple ML models, you don't need to go into deep with Mathematics, just with a basic knowledge of Maths concept(As studied in College) are enough, but if you want to develop complex models and go into advanced concepts then you also need to understand maths behind this. Learning Maths & applying it practically with ML algorithms will require approximately 3-4 months.