Mathematical Foundations for Machine Learning #

Machine Learning is built on mathematical principles that allow models to:

• represent data
• learn patterns
• optimise performance

flowchart LR
    DATA[Data]
    MATH[Math Models]
    OPT[Optimisation]
    MODEL[Trained Model]

    DATA --> MATH
    MATH --> OPT
    OPT --> MODEL

ML requires core mathematical tools to understand how ML algorithms work internally. Algebra deals with relationships between variables and quantities, while Calculus focuses on change and optimization.

Core Mathematical Pillars in Machine Learning #

flowchart TB
    ML[Maths for Machine Learning]

    LA[Linear Algebra]
    PR[Probability]
    ST[Statistics]
    CA[Calculus]
    GR[Graphs]

    ML --> LA
    ML --> PR
    ML --> ST
    ML --> CA
    ML --> GR

    %% Colours (your pastel scheme)
    style ML fill:#E1F5FE,stroke:#333
    style LA fill:#C8E6C9,stroke:#333
    style PR fill:#BBDEFB,stroke:#333
    style ST fill:#90CAF9,stroke:#333
    style CA fill:#64B5F6,stroke:#333
    style GR fill:#FFCCBC,stroke:#333

• Linear Algebra
  • Linear Systems
    • Systems of Linear Equations
    • Matrices
    • Matrix Transposition
    • Solving Linear Systems
    • Inverse Matrix
    • Convex Combination
  • Vector Spaces
    • Linear Independence
    • Basis and Rank
    • Norm
    • Inner Products
    • Lengths and Distances
    • Angles and Orthogonality
    • Orthonormal Basis
    • Feature Space
  • Matrix Decompositions
    • Special Matrices
    • Determinant and Trace
    • Eigenvalues and Eigenvectors
    • Diagonalization
    • Cholesky Decomposition
    • Eigen Decomposition
    • Singular Value Decomposition (SVD)
    • Matrix Approximation
  • Dimensionality reduction and PCA
    • Principal Component Analysis (PCA)
    • PCA Theory
    • PCA in Practice
    • Latent Variable Perspective
    • Mathematical Preliminaries of SVM
    • Nonlinear SVM and Kernels
• Calculus
  • Vector Calculus
    • Differentiation of Univariate Functions
    • Partial Differentiation and Gradients
    • Gradients of Vector-Valued and Matrix Functions
    • Useful Gradient Identities
    • Backpropagation and Automatic Differentiation
  • Continuous Optimisation
    • Optimisation using Gradient Descent
    • Constrained Optimisation
    • Lagrange Multipliers
    • Convex Optimisation
  • Nonlinear Optimisation
    • Challenges in Gradient-Based Optimisation
    • Stochastic Gradient Descent (SGD)
    • Momentum-Based Learning
    • Adaptive Methods: AdaGrad, RMSProp, Adam
    • Tuning Hyperparameters and Preprocessing

Why Mathematics Matters in ML #

• Machine learning models are mathematical functions
• Training a model means optimising equations
• Understanding maths helps to:
  • Debug models
  • Improve performance
  • Choose the right algorithms

mindmap
  Mathematics for Machine Learning
    Linear Algebra
      Vectors
      Matrices
      Eigenvalues
      Matrix Multiplication

    Probability
      Random Variables
      Probability Distributions
      Conditional Probability
      Bayes Theorem

    Statistics
      Mean
      Variance
      Sampling
      Hypothesis Testing

    Calculus
      Derivatives
      Gradients
      Chain Rule
      Optimisation

    Graphs
      Line Plots
      Loss Curves
      Decision Boundaries
      Data Visualisation

Optimisation (Cross-Cutting Concept) #

Optimisation is the process of finding the best model parameters by minimising or maximising an objective function.

• a mechanism that relies on multiple mathematical areas.

Depends on:

• Linear Algebra → parameter updates
• Calculus → gradient computation
• Probability & Statistics → loss functions
• Graphs → convergence analysis

Used in:

• Gradient descent and its variants
• Training deep neural networks
• Regularisation and generalisation

Refrences #

MML-Book

Calculus & Algebra

Essence of linear algebra

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