📊 Math for Machine Learning

Master the mathematical foundations essential for understanding machine learning algorithms

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Math for Machine Learning Curriculum

12
Core Units
~120
Mathematical Concepts
25+
Key Algorithms
60+
Practical Applications
1

Linear Algebra Fundamentals

Build the foundation of vectors, matrices, and linear transformations essential for ML.

  • Vectors and vector spaces
  • Matrix operations
  • Linear transformations
  • Eigenvalues and eigenvectors
  • Matrix decompositions
  • Norms and inner products
  • Orthogonality
  • Computational considerations
2

Calculus and Optimization

Master derivatives, gradients, and optimization techniques for training ML models.

  • Multivariable calculus
  • Partial derivatives
  • Gradients and Jacobians
  • Chain rule
  • Optimization theory
  • Gradient descent
  • Constrained optimization
  • Lagrange multipliers
3

Probability Theory

Understand probability distributions, Bayes' theorem, and uncertainty quantification.

  • Probability axioms
  • Conditional probability
  • Bayes' theorem
  • Random variables
  • Probability distributions
  • Joint distributions
  • Independence
  • Expectation and variance
4

Statistics and Inference

Learn statistical methods for model evaluation, hypothesis testing, and parameter estimation.

  • Descriptive statistics
  • Sampling distributions
  • Central limit theorem
  • Confidence intervals
  • Hypothesis testing
  • Maximum likelihood estimation
  • Bayesian inference
  • Bootstrap methods
5

Information Theory

Explore entropy, mutual information, and information-theoretic foundations of learning.

  • Entropy and mutual information
  • Kullback-Leibler divergence
  • Cross-entropy
  • Information gain
  • Channel capacity
  • Data compression
  • Rate-distortion theory
  • Information bottleneck
6

Numerical Methods

Study computational algorithms for solving mathematical problems in machine learning.

  • Numerical stability
  • Iterative methods
  • Root finding
  • Numerical integration
  • Matrix computations
  • Solving linear systems
  • Finite differences
  • Approximation methods
7

Graph Theory

Understand graphs, networks, and their applications in machine learning and data analysis.

  • Graph representations
  • Graph algorithms
  • Shortest paths
  • Network centrality
  • Spectral graph theory
  • Graph embeddings
  • Random walks
  • Community detection
8

Functional Analysis

Explore function spaces, operators, and theoretical foundations for advanced ML methods.

  • Metric spaces
  • Normed spaces
  • Hilbert spaces
  • Linear operators
  • Reproducing kernel Hilbert spaces
  • Functional derivatives
  • Variational methods
  • Approximation theory
9

Convex Optimization

Master convex sets, functions, and optimization algorithms crucial for many ML problems.

  • Convex sets and functions
  • Convex optimization problems
  • Duality theory
  • KKT conditions
  • Interior point methods
  • Subgradient methods
  • Proximal methods
  • ADMM
10

Matrix Factorization

Study matrix decomposition techniques for dimensionality reduction and data analysis.

  • Principal Component Analysis
  • Singular Value Decomposition
  • Non-negative matrix factorization
  • Matrix completion
  • Low-rank approximations
  • Tensor decompositions
  • Sparse coding
  • Dictionary learning
11

Differential Geometry

Explore manifolds, metrics, and geometric approaches to machine learning.

  • Manifolds and charts
  • Tangent spaces
  • Riemannian metrics
  • Geodesics
  • Curvature
  • Lie groups
  • Differential forms
  • Information geometry
12

Advanced Topics

Explore cutting-edge mathematical concepts in modern machine learning research.

  • Optimal transport
  • Tropical geometry
  • Algebraic topology
  • Persistent homology
  • Quantum computing basics
  • Category theory
  • Computational complexity
  • Mathematical foundations of deep learning

Unit 1: Linear Algebra Fundamentals

Build the foundation of vectors, matrices, and linear transformations essential for ML.

Vectors and Vector Spaces

Understand the fundamental building blocks of linear algebra and their geometric interpretations.

Vector Operations Linear Combinations Basis
A vector space V over a field F is a set equipped with two operations (addition and scalar multiplication) that satisfy eight axioms including associativity, commutativity, and distributivity.
# Vector Space Properties
vector_space = {
  "definition": "Set V with addition and scalar multiplication",
  "axioms": {
    "closure": "u + v ∈ V for all u, v ∈ V",
    "associativity": "(u + v) + w = u + (v + w)",
    "commutativity": "u + v = v + u",
    "zero_element": "∃ 0 ∈ V such that v + 0 = v",
    "additive_inverse": "∃ -v such that v + (-v) = 0",
    "scalar_closure": "αv ∈ V for α ∈ F, v ∈ V",
    "distributivity": "α(u + v) = αu + αv",
    "compatibility": "α(βv) = (αβ)v"
  },
  "examples": ["R^n", "Polynomial spaces", "Function spaces", "Matrix spaces"],
  "ml_applications": ["Feature vectors", "Parameter spaces", "Hidden representations"]
}

Matrix Operations

Master matrix arithmetic, properties, and their role in linear transformations and ML algorithms.

Essential Matrix Operations:
• Addition and scalar multiplication
• Matrix multiplication (not commutative!)
• Transpose and conjugate transpose
• Inverse (when it exists)
• Determinant and trace
Matrix Multiplication Complexity:
Standard algorithm: O(n³) for n×n matrices. More efficient algorithms exist (Strassen's O(n^2.807), current best ~O(n^2.373)) but have large constants.
# Matrix Operations Framework
matrix_ops = {
  "multiplication": {
    "definition": "(AB)_ij = Σ_k A_ik * B_kj",
    "properties": ["Associative", "Distributive", "NOT commutative"],
    "complexity": "O(n³) for n×n matrices",
    "ml_use": "Forward propagation, weight updates"
  },
  "transpose": {
    "definition": "(A^T)_ij = A_ji",
    "properties": ["(A^T)^T = A", "(AB)^T = B^T A^T"],
    "ml_use": "Gradient computation, covariance matrices"
  },
  "inverse": {
    "definition": "A^(-1) such that AA^(-1) = I",
    "existence": "Only for square, full-rank matrices",
    "ml_use": "Solving linear systems, computing pseudoinverse"
  }
}

Eigenvalues and Eigenvectors

Explore the fundamental concept that underlies PCA, spectral methods, and many ML algorithms.

Eigenvalue Problem:
For matrix A, find λ (eigenvalue) and v (eigenvector) such that:
Av = λv

This is equivalent to solving: det(A - λI) = 0
Spectral Theorem:
Every symmetric real matrix can be diagonalized by an orthogonal matrix. This forms the mathematical foundation for Principal Component Analysis (PCA).
# Eigendecomposition Applications
eigendecomposition = {
  "definition": "A = QΛQ^(-1) where Λ is diagonal",
  "symmetric_case": "A = QΛQ^T (orthogonal eigenvectors)",
  "ml_applications": {
    "pca": "Find principal components via covariance matrix eigendecomposition",
    "spectral_clustering": "Use eigenvectors of graph Laplacian",
    "markov_chains": "Steady state via dominant eigenvector",
    "neural_networks": "Analyze layer dynamics and gradients"
  },
  "computational_notes": {
    "power_method": "Iterative method for dominant eigenvalue",
    "qr_algorithm": "Standard method for all eigenvalues",
    "sparse_methods": "Arnoldi/Lanczos for large sparse matrices"
  }
}