Mathematical Foundations of Data Science¶

This course builds the shared mathematical language for data: matrices and images, distance and clustering, high-dimensional geometry, SVD/PCA, random graphs, and learning foundations. Applications come first; the math is introduced when it explains what you already saw.

Audience: first-semester MSc Math (Data Science). Linear algebra and basic probability assumed; Python is learned in parallel through short notebooks.

See the repository README for the full course aim, topic outline, references, prerequisites, and build instructions.

• Mathematical Foundations of Data Science
• Opening, Data, Matrices, and Images
  • Course Opening: Students, Goals, and Applications
  • Foundation from the Python Data Science Handbook
  • The Basics of NumPy Arrays
  • Images as high dimensional data
  • Scalars, Vectors, Matrices and Tensors
  • Multiplying Matrices and Vectors
  • Identity and Inverse Matrices
  • Norms
• Classification and Clustering
  • k-Nearest Neighbors and Classification Evaluation Metrics
  • k-means Clustering
  • IRIS Clustering
  • Image Segmentaion
  • Discrete Optimization, Truth Tables, and Clustering
  • SAT-Table
  • N-Queen Problem
  • Clustering validation: Silhouette
  • Brute force search clustering
  • Add coordinates as spatial features to clustering
  • Vector Quantization
  • Clustering Validation Metrics
• High-Dimensional Spaces
  • High Dimensional Spaces
  • Rejection Sampling and Points in the Unit Ball
  • kNN in high dimensional spaces
  • k-means in high dimensional spaces
  • High Dimensional Data & The Curse of Dimensionality
• Linear Algebra and SVD
  • Linear Algebra
  • Linear Dependence and Span
  • Subspaces and Maps to Other Spaces
  • Special Kinds of Matrices and Vectors
  • Projection with dot product
  • Eigendecomposition
  • Singular Value Decomposition
  • The Moore-Penrose Pseudoinverse
  • The Trace Operator
  • The Determinant
• Low-Rank Approximation, PCA, and Dimensionality Reduction
  • Low-Rank Matrix Approximation
  • Principal Component Analysis, Part 1
  • Principal Component Analysis, Part 2
  • SVD for Image Compression
  • Clustering images
  • Clustering images by hierarchical clustering
• Random Graphs and Random Walks
  • Introduction to Graph Theory and Complex Networks Analysis
  • Introduction to Graph Theory and Complex Network Analysis
  • Fundamentals
  • Erdős–Rényi Random Graphs
  • Giant Component and Phase Transition
  • Light Random Walks on Graphs
• Learning Foundations
  • Bias-Variance Tradeoff
  • Learning Foundations: Classification and Generalization
  • Perceptron and Online Learning
  • Boosting Sketch, Ranking, and Voting
• Random Projections and Further Topics
  • Random Projection: Theory and Implementation
  • Gaussian Random Projection
  • Johnson–Lindenstrauss Lemma
  • Compressed Sensing (Survey)
  • Bias-Variance Tradeoff
  • Course Synthesis and Bridge to SML
• Appendix
  • Computation on Arrays: Broadcasting
  • Introduction to Network Analysis with NetworkX
  • Example - Principal Components Analysis

Instructors¶

Mahmood Amintoosi
Email: m.amintoosi AT um.ac.ir

Teaching Assistants (TA-Head)¶

Hoda MehrBagherpour
Email: mehrbagherpour AT mail.um.ac.ir