ML Fundamentals

Mathematical Foundations for Machine Learning

• What you Need to Know

  • Linear Algebra in Machine Learning

    • Vector spaces and linear transformations in ML contexts
    • Matrix operations for data representation and model parameters
    • Eigendecomposition and Singular Value Decomposition (SVD)
    • Resources:
      • Linear Algebra for ML - Fast.ai computational linear algebra course
      • Matrix Calculus for Deep Learning - Matrix derivatives and gradients
      • Linear Algebra Review - CS229 - Stanford ML linear algebra notes

  • Calculus and Optimization Theory

    • Gradient computation and chain rule for backpropagation
    • Convex optimization and global vs local minima
    • Lagrange multipliers and constrained optimization
    • Resources:
      • Convex Optimization - Boyd - Free convex optimization textbook
      • Optimization for Machine Learning - Modern optimization techniques
      • Calculus on Computational Graphs - Backpropagation explained

  • Probability Theory and Statistical Learning

    • Bayesian inference and maximum likelihood estimation
    • Probability distributions and their ML applications
    • Information theory and entropy in machine learning
    • Resources:
      • Probabilistic Machine Learning - Kevin Murphy's comprehensive ML book
      • Information Theory - MacKay - Information theory and inference
      • Bayesian Methods for Machine Learning - HSE University course

Supervised Learning Algorithms

• What you Need to Know

  • Linear Models and Regularization

    • Linear regression with mathematical derivation
    • Ridge, Lasso, and Elastic Net regularization
    • Logistic regression and maximum likelihood
    • Resources:
      • Linear Models - ESL - Elements of Statistical Learning chapters
      • Regularization Tutorial - Ridge and Lasso implementation
      • Logistic Regression Math - Mathematical foundations

  • Tree-Based Methods and Ensemble Learning

    • Decision trees and information gain criteria
    • Random Forest and bagging techniques
    • Gradient boosting and XGBoost implementation
    • Resources:
      • Decision Trees - Scikit-learn - Decision tree algorithms and implementation
      • Random Forest Paper - Breiman's original random forest paper
      • XGBoost Documentation - Gradient boosting framework

  • Support Vector Machines and Kernel Methods

    • SVM optimization problem and dual formulation
    • Kernel trick and non-linear transformations
    • Soft margin and regularization in SVMs
    • Resources:
      • SVM Tutorial - MIT SVM mathematical foundations
      • Kernel Methods - Learning with Kernels textbook
      • SVM Implementation - Scikit-learn SVM guide

Unsupervised Learning and Dimensionality Reduction

• What you Need to Know

  • Clustering Algorithms

    • K-means clustering and expectation-maximization
    • Hierarchical clustering and linkage criteria
    • DBSCAN and density-based clustering
    • Resources:
      • Clustering Algorithms - Comprehensive clustering guide
      • K-means Mathematical Analysis - CMU clustering lecture
      • DBSCAN Paper - Original DBSCAN algorithm

  • Principal Component Analysis and Matrix Factorization

    • PCA mathematical derivation and implementation
    • Singular Value Decomposition applications
    • Non-negative Matrix Factorization (NMF)
    • Resources:
      • PCA Tutorial - Principal Component Analysis explained
      • SVD and PCA - MIT SVD tutorial
      • Matrix Factorization - Netflix recommendation system

  • Manifold Learning and Non-linear Dimensionality Reduction

    • t-SNE for visualization and clustering
    • UMAP for dimensionality reduction
    • Autoencoders for non-linear feature learning
    • Resources:
      • t-SNE Paper - Original t-SNE algorithm
      • UMAP Documentation - Uniform Manifold Approximation
      • Autoencoder Tutorial - Keras autoencoder implementation

Deep Learning Fundamentals

• What you Need to Know

  • Neural Network Architecture and Training

    • Multi-layer perceptrons and universal approximation theorem
    • Backpropagation algorithm mathematical derivation
    • Gradient descent variants and optimization techniques
    • Resources:
      • Deep Learning Book - Goodfellow, Bengio, and Courville comprehensive text
      • Neural Networks and Deep Learning - Nielsen's online book
      • Backpropagation Calculus - 3Blue1Brown backprop explanation

  • Convolutional Neural Networks

    • Convolution operation and feature maps
    • CNN architectures (LeNet, AlexNet, VGG, ResNet)
    • Pooling layers and spatial hierarchies
    • Resources:
      • CNN for Visual Recognition - Stanford CS231n CNN guide
      • CNN Architectures - Survey of CNN architectures
      • ResNet Paper - Deep residual learning

  • Recurrent Neural Networks and Sequence Modeling

    • RNN architecture and vanishing gradient problem
    • LSTM and GRU for long-term dependencies
    • Attention mechanisms and Transformer architecture
    • Resources:
      • Understanding LSTMs - LSTM architecture explained
      • Attention Is All You Need - Original Transformer paper
      • RNN Tutorial - Karpathy's RNN guide

Model Selection and Validation

• What you Need to Know

  • Cross-Validation Techniques

    • K-fold cross-validation and stratified sampling
    • Leave-one-out and bootstrap validation
    • Time series cross-validation for temporal data
    • Resources:
      • Cross-Validation - Scikit-learn CV documentation
      • Model Selection - ESL model assessment chapter
      • Time Series CV - Time series validation techniques

  • Bias-Variance Tradeoff and Regularization

    • Mathematical analysis of bias-variance decomposition
    • Regularization techniques and their effects
    • Early stopping and dropout as regularization
    • Resources:
      • Bias-Variance Tradeoff - Visual bias-variance explanation
      • Regularization in Deep Learning - Comprehensive regularization survey
      • Dropout Paper - Original dropout technique

  • Hyperparameter Optimization

    • Grid search and random search strategies
    • Bayesian optimization and Gaussian processes
    • Automated hyperparameter tuning techniques
    • Resources:
      • Hyperparameter Optimization - Bergstra and Bengio survey
      • Bayesian Optimization - Gaussian process optimization
      • Optuna Tutorial - Automated hyperparameter optimization

Feature Engineering and Selection

• What you Need to Know

  • Feature Extraction and Transformation

    • Polynomial features and interaction terms
    • Feature scaling and normalization techniques
    • Handling categorical variables and encoding
    • Resources:
      • Feature Engineering - O'Reilly feature engineering book
      • Preprocessing Data - Scikit-learn preprocessing guide
      • Categorical Encoding - Advanced categorical encoding techniques

  • Feature Selection Methods

    • Filter methods (correlation, mutual information)
    • Wrapper methods (recursive feature elimination)
    • Embedded methods (Lasso, tree-based importance)
    • Resources:
      • Feature Selection - Comprehensive feature selection guide
      • Information Theory Feature Selection - Mutual information methods
      • Recursive Feature Elimination - RFE implementation

Evaluation Metrics and Performance Analysis

• What you Need to Know

  • Classification Metrics

    • Accuracy, precision, recall, and F1-score analysis
    • ROC curves and Area Under Curve (AUC)
    • Multi-class and multi-label evaluation metrics
    • Resources:
      • Classification Metrics - Comprehensive metrics guide
      • ROC and AUC Explained - Google's ROC tutorial
      • Multi-class Metrics - Multi-class evaluation survey

  • Regression Metrics

    • Mean Squared Error (MSE) and Root Mean Squared Error (RMSE)
    • Mean Absolute Error (MAE) and robust metrics
    • R-squared and adjusted R-squared interpretation
    • Resources:
      • Regression Metrics - Regression evaluation guide
      • Understanding R-squared - R-squared interpretation
      • Robust Regression Metrics - Alternative regression metrics

Algorithm Implementation from Scratch

• What you Need to Know

  • Linear Algebra Implementation

    • Matrix operations without NumPy dependencies
    • Gradient computation and optimization loops
    • Numerical stability and computational efficiency
    • Resources:
      • ML Algorithms from Scratch - Pure Python implementations
      • Numerical Linear Algebra - Computational linear algebra course
      • Matrix Cookbook - Matrix identities and derivatives

  • Optimization Algorithm Implementation

    • Gradient descent variants (SGD, Adam, RMSprop)
    • Newton's method and quasi-Newton methods
    • Coordinate descent and proximal methods
    • Resources:
      • Optimization Algorithms - Overview of optimization for deep learning
      • SGD Variants - Gradient descent optimization overview
      • Proximal Algorithms - Proximal optimization methods

Ready to Engineer Data? Continue to Module 2: Data Engineering to master data pipelines, preprocessing, and feature engineering for machine learning systems.