Book Review: Basic Mathematics for Artificial Intelligence
Hala Nelson – An Author with an Unusual Destiny
Lebanese by birth, Hala Nelson has had a difficult life to become an applied mathematics teacher in the field of artificial intelligence (AI).
IN on your blog She recalls a childhood full of hardships and dangers, with Israeli bombings and checkpoints. At the age of 20, the author was able to receive an invitation to study in the United States, which opened the way to studying mathematics at a professional level.
Here is how Hala Nelson describes her beginnings in mathematics: “When I was growing up, my father would bring me a big orange book with very small print, full of math problems written in French. I would sit next to him by our iron stove, and he would read the French problems to me, and then I would translate them into English and solve them. I loved his accent and his outrageous mispronunciation as he tried to translate the math into English himself. I would do all the problems in that book by candlelight, since we almost never had electricity. Those are my first memories of serious mathematics: with my father, the big orange French book, the warmth of the old iron stove, and the candles.”
Today, Hala Nelson is a professor in the Department of Mathematics at James Madison University in Virginia, USA.
Why studying the mathematical apparatus of AI is worthy of attention of data scientists
Modern developers are spoiled by integrated environments and visual tools, so the pure mathematics from this book will cause many a “toothache” and a desire to postpone acquiring such knowledge until the moment “when it's too late.” However, do not deceive yourself – at that very moment you will not have enough time to study the complex mathematical apparatus underlying AI. It is much easier to create a schedule for yourself and slowly move through the chapters of the book. Fortunately, the basics of the mathematical apparatus of AI do not change as quickly as ML/DL tools based on software.
While it’s true that having a working knowledge of a number of tools like PyTorch and Scikit-Learn is important for data scientists to practice, this knowledge alone is not enough to fully understand and effectively apply machine learning and artificial intelligence models. These libraries provide powerful abstractions that allow complex algorithms to be implemented quickly, but without understanding the underlying mathematics, data scientists risk using them as “black boxes.”
It is the knowledge of mathematical foundations that allows data scientists to go beyond the ready-made solutions offered by libraries. They can modify existing algorithms or create new ones optimized for specific tasks. For example, understanding the principles of optimization helps to correctly select and configure loss functions and optimizers in neural networks, which can significantly improve the performance of the model.
A mathematical foundation is also critical to understanding the limitations and assumptions underlying different algorithms. This helps to select the most appropriate methods for specific problems and data, as well as to assess the potential risks and limitations of the chosen approaches.
Finally, the rapid development of the field of artificial intelligence and data science requires professionals to constantly learn and adapt to new methods and algorithms. A solid mathematical foundation makes it easier to understand new concepts and allows you to master new tools and techniques faster, which is critical to maintaining personal competitiveness in the job market in this dynamic field.
What's useful in the chapters
In this section you will find annotations to the chapters of the book “Basic Mathematics for Artificial Intelligence” by Hala Nelson. In order not to take up too much time, we made the annotations very short. The table of contents of the book for the Russian edition can be found here, and for the English one – here.
Chapter 1: Why is it Important to Learn AI Mathematics? This chapter introduces you to the world of AI, looking at its foundations and the importance of mathematics in this field. It is important for data scientists to understand how mathematical concepts underlie AI and help solve complex problems, from data processing to developing intelligent agents.
Chapter 2. Data, Data, and More DataThis chapter covers the fundamentals of working with data, including data collection, analysis, and modeling. Understanding data distributions and probabilistic models is important for data scientists, as they often work with large data sets that require complex analytical approaches to extract meaningful relationships.
Chapter 3: Fitting Features to DataThis section covers regression and classification methods, including models such as logistic regression and support vector machines. These models are the foundation of machine learning and allow data scientists to build predictive models from data.
Chapter 4. Neural Network Optimization The chapter covers various aspects of neural network optimization, including gradient descent, regularization, and normalization. For data scientists, these techniques are important for tuning and improving the performance of neural networks, which is critical for developing highly effective AI systems.
Chapter 5. Convolutional Neural Networks and Computer VisionThis chapter covers convolutional neural networks (CNNs) and their applications in computer vision. Data scientists use CNNs for pattern recognition and image classification tasks, which is one of the hottest areas of AI.
Chapter 6. Singular Value Decomposition: Image Processing, Natural Language Processing, and Social MediaThe chapter describes singular value decomposition (SVD) and its applications in image processing, text mining, and social media. SVD is important for data scientists because it helps in data compression, dimensionality reduction, and feature extraction.
Chapter 7. AI for Natural Language and Finance: Vectorization and Time SeriesThis chapter explores text vectorization techniques and time series analysis applied to natural language processing and financial analysis. Data scientists working with text or time series data will find these techniques useful for modeling and prediction.
Chapter 8. Probabilistic Generative ModelsThis chapter discusses probabilistic generative models, including GANs and variational autoencoders. These models are important for generating new data, simulations, and other creative tasks, making them valuable for data scientists working in the field of generative AI.
Chapter 9. Graph ModelsThis chapter is devoted to graph models and their application in various fields such as social networks, biology, and recommendations. For data scientists, this is an important topic, as graphs are used as a tool for modeling complex relationships in data.
Chapter 10. Operations ResearchThis chapter covers operational research methods such as linear programming and game theory. These methods help optimize complex systems and processes, which is useful for data scientists working to improve the efficiency of decisions.
Chapter 11. ProbabilityThe chapter delves into probability theory, including stochastic processes and Markov chains. These concepts are important for data scientists, as they underlie many ML/DL algorithms.
Chapter 12. Mathematical LogicDiscusses various logical structures and their application in ML. For data scientists, understanding mathematical logic helps in building models that can reason and make decisions based on complex rules.
Chapter 13. Artificial Intelligence and Differential EquationsThis chapter explores the use of partial differential equations (PDEs) in the context of AI. These equations are used to model physical processes, and knowledge of their solutions is important for data scientists working on integrating physical models into AI.
Chapter 14. Artificial Intelligence, Ethics, Mathematics, Law, and PolicyThe final chapter examines issues of ethics, law, and policy in the field of AI. It is important for data scientists to understand these aspects in order to create AI systems that comply with regulatory requirements and ethical standards.
About the terms AI, ML, DL in the book
It seemed to us that the book contains a somewhat free mix of the terms AI, ML and DL. Perhaps the frequent mention of the term AI is done for better marketing. Therefore, it is advisable to clarify the difference between the terms.
Terminology: AI is a general term for all methods of creating AI systems (this even includes chatbots, voice recognition, etc.), while ML and DL are narrower and more rigorous areas within AI related to data analysis models.
Models: ML relies on training systems using data to perform specific tasks, while DL uses complex neural networks to handle very complex tasks.
How the terms are used: AI can be used to describe or identify any AI systems, from simple to complex. ML is used when systems need to learn from data, and DL is used for the most complex tasks that require analyzing large amounts of data.
So AI is this general umbrella term that ML and DL fall under, where ML and DL are specific methods for building artificial intelligence systems, with DL being a more advanced form of ML focused on neural networks and big data.
Conclusion
If you decide to buy the Russian edition or download the English original, try working on this book for at least a few hours a week, even if it takes a year. This is a proven practice when you need to study a complex topic without burning out along the way and without limiting yourself in time.
Understanding the basics of mathematics is critical for data scientists, and Hala Nelson's book, Basic Mathematics for Artificial Intelligence, helps you master this knowledge. Mathematics is the foundation upon which all AI and machine learning algorithms and models are built. Without a deep understanding of mathematical concepts, data scientists risk using tools and techniques without fully understanding their capabilities, limitations, and potential pitfalls.
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Good luck in learning and applying mathematics in AI, ML and DL!
