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Mathematical Foundations for AI

Begin your innovative career with the Mathematics of Machine Learning course. Learn how to design & understand the next generation of AI models.

(MATHS-ML.AJ1) / ISBN : 979-8-90059-011-0
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About This Course

Tired of treating machine learning models like black boxes? The Mathematics of Machine Learning course gives you the rigorous foundation to design and troubleshoot AI. 

The power of ML lies within the mathematics of machine learning—linear algebra, calculus & probability. As quoted by Galileo: Mathematics is the language in which God has written the universe.

We turn that language into AI expertise. Mastering machine learning math is the most critical differentiator for securing high-end roles in the fields of data science & AI engineering. 

Skills You’ll Get

  • Linear Algebra & Geometry: Master vector spaces, matrices & linear algebra in practice, including eigenvalues, matrix factorizations & SVD. 
  • Calculus & Optimization: Conquer differentiation, integration & optimization techniques for both single & multivariable functions. 
  • Probability & statistics: Grasp the fundamentals of probability, random variables & expected value—the statistical backbone of all the ML models. 
  • Foundational Theory: Build a strong foundational theoretical base with mathematical logic, set theory, & complex numbers to fully understand the structure of Machine learning mathematics. 
Download Course Outline

1

Introduction

  • What is this course about?
  • How to read this course
  • Conventions used
  • What this course covers
2

Vectors and Vector Spaces

  • What is a vector space?
  • The basis
  • Vectors in practice
  • Summary
  • Problems
3

The Geometric Structure of Vector Spaces

  • Norms and distances
  • Inner products, angles, and lots of reasons to care about them
  • Summary
  • Problems
4

Linear Algebra in Practice

  • Vectors in NumPy
  • Matrices, the workhorses of linear algebra
  • Summary
  • Problems
5

Linear Transformations

  • What is a linear transformation?
  • Change of basis
  • Linear transformations in the Euclidean plane
  • Determinants, or how linear transformations affect volume
  • Summary
  • Problems
6

Matrices and Equations

  • Linear equations
  • The LU decomposition
  • Determinants in practice
  • Summary
  • Problems
7

Eigenvalues and Eigenvectors

  • Eigenvalues of matrices
  • Finding eigenvalue-eigenvector pairs
  • Eigenvectors, eigenspaces, and their bases
  • Summary
  • Problems
8

Matrix Factorizations

  • Special transformations
  • Self-adjoint transformations and the spectral decomposition theorem
  • The singular value decomposition
  • Orthogonal projections
  • Computing eigenvalues
  • The QR algorithm
  • Summary
  • Problems
9

Matrices and Graphs

  • The directed graph of a nonnegative matrix
  • Benefits of the graph representation
  • The Frobenius normal form
  • Summary
  • Problems
10

Functions

  • Functions in theory
  • Functions in practice
  • Summary
  • Problems
11

Numbers, Sequences, and Series

  • Numbers
  • Sequences
  • Series
  • Summary
  • Problems
12

Topology, Limits, and Continuity

  • Topology
  • Limits
  • Continuity
  • Summary
  • Problems
13

Differentiation

  • Differentiation in theory
  • Differentiation in practice
  • Summary
  • Problems
14

Optimization

  • Minima, maxima, and derivatives
  • The basics of gradient descent
  • Why does gradient descent work?
  • Summary
  • Problems
15

Integration

  • Integration in theory
  • Integration in practice
  • Summary
  • Problems
16

Multivariable Functions

  • What is a multivariable function?
  • Linear functions in multiple variables
  • The curse of dimensionality
  • Summary
17

Derivatives and Gradients

  • Partial and total derivatives
  • Derivatives of vector-valued functions
  • Summary
  • Problems
18

Optimization in Multiple Variables

  • Multivariable functions in code
  • Minima and maxima, revisited
  • Gradient descent in its full form
  • Summary
  • Problems
19

What is Probability?

  • The language of thinking
  • The axioms of probability
  • Conditional probability
  • Summary
  • Problems
20

Random Variables and Distributions

  • Random variables
  • Discrete distributions
  • Real-valued distributions
  • Density functions
  • Summary
  • Problems
21

The Expected Value

  • Discrete random variables
  • Continuous random variables
  • Properties of the expected value
  • Variance
  • The law of large numbers
  • Information theory
  • The Maximum Likelihood Estimation
  • Summary
  • Problems
A

Appendix A: It’s Just Logic

  • Mathematical logic 101
  • Logical connectives
  • The propositional calculus
  • Variables and predicates
  • Existential and universal quantification
  • Problems
B

Appendix B: The Structure of Mathematics

  • What is a definition?
  • What is a theorem?
  • What is a proof?
  • Equivalences
  • Proof techniques
C

Appendix C: Basics of Set Theory

  • What is a set?
  • Operations on sets
  • The Cartesian product
  • The cardinality of sets
  • The Russell paradox (optional)
D

Appendix D: Complex Numbers

  • The definition of complex numbers
  • The geometric representation
  • The fundamental theorem of algebra
  • Why are complex numbers important?

1

Vectors and Vector Spaces

  • Implementing Tuple and List Operations
  • Performing NumPy Array and Vector Operations
2

The Geometric Structure of Vector Spaces

  • Analyzing Vectors and Distances
3

Linear Algebra in Practice

  • Evaluating Vector Norms and Operations
  • Applying Matrix Computations Using NumPy
  • Representing Images and Text Using Vectors and Matrices
4

Matrices and Equations

  • Solving Linear Equations Using Gaussian Elimination
  • Solving Linear Equations Using Determinants and Inverses
  • Solving Linear Models in Machine Learning
  • Performing LU Decomposition
  • Computing the Determinant Using LU Decomposition
5

Eigenvalues and Eigenvectors

  • Finding Eigenvalues and Eigenvectors of Matrices
  • Analyzing Matrices Using Characteristic Polynomials
  • Visualizing Eigenvectors and Eigenspaces in Linear Algebra
6

Matrix Factorizations

  • Implementing Spectral Decomposition and PCA
  • Performing Feature Extraction and Dimensionality Reduction
  • Performing Singular Value Decomposition
  • Implementing QR Decomposition
7

Matrices and Graphs

  • Reordering Adjacency Matrices Using SCCs
8

Functions

  • Implementing Callable Functions
9

Numbers, Sequences, and Series

  • Visualizing Mathematical Sequences and Approximations
  • Visualizing the Harmonic Series
10

Topology, Limits, and Continuity

  • Analyzing Openness, Closedness, and Compactness of Sets
11

Differentiation

  • Applying the Chain Rule
12

Optimization

  • Implementing Gradient Descent for Model Training
13

Integration

  • Approximating Integrals Using the Trapezoidal Rule
14

Multivariable Functions

  • Plotting Multivariable Function Landscapes
  • Plotting Linear Mappings and Hyperplanes for ML Models
15

Derivatives and Gradients

  • Evaluating Composite Functions
  • Analyzing Gradients, Jacobians, and Hessians in ML Optimization
  • Implementing Backpropagation in Neural Networks
16

Optimization in Multiple Variables

  • Training Machine Learning Models with Gradient Descent
17

What is Probability?

  • Simulating Random Outcomes for Probability Learning
18

Random Variables and Distributions

  • Visualizing Discrete Probability Distributions in Machine Learning
  • Visualizing Continuous Statistical Distributions
19

The Expected Value

  • Simulating Random Processes for AI Models

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  Want to Learn More?

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Data scientists, ML engineers & anyone interested in understanding the machine learning math that underpins the algorithm, like neural networks. 

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Basic calculus & linear algebra knowledge are helpful, but the course begins with foundational concepts to ensure mastery of mathematics for machine learning.

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It covers the full breadth of the mathematics of machine learning, including rigorous topics including topology, eigenvectors & matrix factorizations, which are essential for truly understanding modern AI.

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Both; it actually covers the theory while teaching how to apply concepts like gradient descent & linear algebra in practice using tools like NumPy.

Ready to Master the Math of AI?

  Translate complex theories into real-world AI solutions with a machine learning mathematics program.

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