Machine learning (ML) isn’t just about artificial intelligence and neural networks. It’s primarily a mathematical discipline, where algorithms and models are built on a solid foundation of linear algebra, probability theory, and mathematical analysis. Without an understanding of these areas, it’s difficult to succeed in ML.
In this article, we’ll explore what mathematical knowledge is essential, how to improve it, and where it can be applied in practice.
What is machine learning?
— is a field of artificial intelligence in which algorithms learn from data, identify patterns, and make predictions without being explicitly programmed.
The main stages of ML:
- Data collection and preparation
- Model selection
- Model training
- Quality assessment
- Application of the model
Mathematics in Machine Learning: Why is it Needed?
Mathematics forms the basis for constructing and understanding algorithms, enabling effective work with data and models. Without knowledge of linear algebra, probability theory, and mathematical analysis, it is impossible to evaluate the accuracy of forecasts, identify patterns, and optimize model parameters.
Furthermore, mathematical concepts help interpret results, identify errors, and adapt algorithms to specific tasks. A deep understanding of numerical methods and statistics ensures confident decision-making and increases the likelihood of successfully applying ML in real-world projects.
“To be a good machine learning engineer or data scientist, you need to understand the mathematics behind machine learning algorithms.” – Goran Sukovic, PhD in Mathematics.
Linear Algebra for ML
Linear algebra is the foundation of most algorithms, as it allows data to be represented as vectors and matrices and operated on. It is used to process large datasets, transform features, and simplify computations when training models.
Key elements include:
- Vectors
- Matrices
- Dot product
- Vector product
- Eigenvalues
- Eigenvectors
- Singular value decomposition (SVD)
- Determinants
- Inverse matrices
- Matrix rank
Probability theory
Machine learning is used in ML for uncertainty modeling, risk assessment, and statistical model building. It helps understand data distributions, predict events, and make decisions based on probabilistic estimates. Key concepts include random variables, conditional probabilities, distributions, and the law of large numbers.
Main elements:
| Element | Description | Application in ML |
| Random variable | A variable that takes on values randomly | Modeling experimental results |
| Probability of an event | A number between 0 and 1 that represents the probability of an outcome | Assessment, forecasts |
| Conditional probability | The probability of an event under a known condition | Bayesian models, naive classifier |
| Distribution | A function that describes the probability of each outcome | Generation, modeling |
| Law of large numbers | Behavior of means as sample size increases | Stabilization of estimates, hypothesis testing |
Mathematical analysis
Mathematical analysis studies the properties of functions, their changes, and their behavior over intervals, which is critical for optimizing ML algorithms. It allows one to find extreme points, estimate convergence, and evaluate derivatives, which are used in model training methods, such as gradient descent.
Key topics include:
- Limits of functions
- Continuity of functions
- Derivatives
- Gradients
- Partial derivatives
- Integrals
- Series, their convergence
- Optimization of functions
- Extreme points
- Differential equations
Practical application
| Concept | Application | Example of an algorithm or problem | Additional benefits |
| Linear algebra | Data representation, feature transformation | Linear regression, PCA | Acceleration of calculations, reduction of dimensionality |
| Probability theory | Forecasting, risk assessment | Naive Bayes classifier | Uncertainty management, constructing confidence intervals |
| Mathematical analysis | Loss function optimization, gradients | Gradient descent, neural networks | Improving convergence, controlling function changes |
| Statistics | Quality metrics, hypothesis testing | Precision assessment, confidence intervals | Analysis of data scatter, interpretation of results |
| Numerical methods | Solution of systems of equations, approximations | Solution of SLAE, optimization of parameters | Reducing computational errors and model stability |
| Matrix decompositions | Data compression, revealing hidden patterns | SVD, PCA | Noise reduction, identification of key features |
| Logarithmic functions | Working with probabilities, scaling | Logistic regression, normalization | Overflow avoidance, linear approximation |
| Differential equations | Modeling of dynamic processes | Time series forecasting, control systems | Assessing changes over time, predicting trends |
| Normalization, standardization | Data preparation | Min-Max Scaling, Z-score | Improving convergence, equality of feature scale |
| Optimization methods | Selection of model parameters | Stochastic gradient descent, Adam | Fast local minima detection, speed control |
How to improve your math?
1. Refresher of basic knowledge:
It’s best to start with fundamental concepts: algebra, geometry, and arithmetic. A solid grasp of these fundamentals will facilitate mastering linear algebra, probability theory, mathematical analysis, and statistics.
2. Specialized materials:
Explore ML-focused courses and textbooks. Videos, lectures, exercises, and tasks help you structure your knowledge, reinforce your skills, and gradually increase the complexity of topics.
3. Practice:
Working with real data sets accelerates learning. Analysis, preparation, and problem-solving demonstrate the connection between mathematical formulas and algorithms and help reinforce theory in practice.
4. Interaction with experts:
Connecting with communities, participating in webinars, and consulting with mentors accelerates progress. Feedback, new resources, and advice help find effective approaches to learning complex topics.
Mathematics Courses for Data Science
To get started in Data Science, it is important to systematically study the mathematical foundations.
Specialized courses help students master linear algebra, probability theory, statistics, and mathematical analysis, structure the material, and offer practical assignments to reinforce knowledge.
Most programs include interactive exercises, quizzes, and case studies, allowing students to apply theory to real-world data. This approach accelerates learning, improves skills, provides feedback, and helps students confidently grasp complex mathematical concepts for effective application in data science.
Recommended books
- “Mathematics for Machine Learning” (by Mark P.J. van der Linden, Gerald L. Schafer, Joshua M. Murray)
- Linear Algebra and Its Applications by David Lay
- “Introduction to Probability” by William Feller
- “Deep Learning: Theory and Practice” – Ian Goodfellow, Yoshua Bengio, Aaron Courville
- “Linear Algebra and Analytic Geometry” by A.S. Kirkinsky.
- “Mathematics for Data Science” by Thomas Nield, “Mathematics in Machine Learning” by Mark Peter Deisenroth, and A. Aldo Faisal
Success story
Kirill, a technical university graduate, struggled with ML due to his weak mathematical background. He started with basic courses, gradually moving on to specialized data science programs. Hands-on practice with real data, problem-solving, and participation in mini-projects helped solidify his knowledge and develop his skills. A year later, Kirill completed an internship at a major IT company, and six months later, he landed a position as a data scientist, applying mathematics to model development and optimization.
Conclusion
Mathematics is an integral part of machine learning. By mastering its fundamentals, you’ll be able to better understand algorithms, build effective models, and make informed decisions. Start with the basics and gradually delve into more complex topics. Don’t be afraid of mistakes—they’re part of the process.