Linear Algebra for Machine Learning

Part I – Foundations

The first part laid the groundwork by introducing the basic building blocks of linear algebra and their direct relevance to machine learning data structures and operations.

• Part 1: Vectors, Scalars, and Spaces
  ML/AI Relevance: Features, weights, data representation
  Focus: NumPy arrays, PyTorch tensors, 2D/3D plots
  We started with the fundamentals, exploring how vectors and scalars represent data points and parameters in ML, and how vector spaces provide the framework for operations.

• Part 2: Matrices as Data & Transformations
  ML/AI Relevance: Images, datasets, linear layers
  Focus: Image as matrix, reshaping
  Matrices were introduced as representations of data (like images) and as transformations (like neural network layers), showing their dual role in ML.

• Part 3: Matrix Arithmetic: Add, Scale, Multiply
  ML/AI Relevance: Linear combinations, weighted sums
  Focus: Broadcasting, matmul, matrix properties
  We covered essential operations like addition, scaling, and multiplication, critical for combining features and computing outputs in models.

• Part 4: Dot Product and Cosine Similarity
  ML/AI Relevance: Similarity, projections, word vectors
  Focus: np.dot, torch.cosine_similarity
  The dot product and cosine similarity were explored as measures of similarity, vital for tasks like recommendation systems and NLP embeddings.

• Part 5: Linear Independence & Span
  ML/AI Relevance: Feature redundancy, expressiveness
  Focus: Gram matrix, visualization
  We discussed how linear independence and span help identify redundant features and understand the expressive power of data representations.

• Part 6: Norms and Distances
  ML/AI Relevance: Losses, regularization, gradient scaling
  Focus: L1, L2 norms, distance measures
  Norms and distances were introduced as tools for measuring magnitudes and differences, underpinning loss functions and regularization techniques.

Part II – Core Theorems and Algorithms

The second part dove deeper into the theoretical underpinnings and algorithmic machinery of linear algebra, connecting them to pivotal ML techniques.

• Part 7: Orthogonality and Projections
  ML/AI Relevance: Error decomposition, PCA, embeddings
  Focus: Gram-Schmidt, projections, orthonormal basis
  Orthogonality and projections were shown to be key for decomposing data and reducing dimensions, setting the stage for PCA.

• Part 8: Matrix Inverses and Systems of Equations
  ML/AI Relevance: Solving for parameters, backpropagation
  Focus: np.linalg.solve, invertibility
  We explored how matrix inverses solve systems of equations, a concept central to finding optimal parameters in models.

• Part 9: Rank, Nullspace, and the Fundamental Theorem
  ML/AI Relevance: Data compression, under/over-determined systems
  Focus: np.linalg.matrix_rank, SVD intuition
  Rank and nullspace illuminated the structure of data and solutions, linking to compression and system solvability.

• Part 10: Eigenvalues and Eigenvectors
  ML/AI Relevance: Covariance, PCA, stability, spectral clustering
  Focus: np.linalg.eig, geometric intuition
  Eigenvalues and eigenvectors were introduced as tools for understanding data variance and stability, crucial for PCA and clustering.

• Part 11: Singular Value Decomposition (SVD)
  ML/AI Relevance: Dimensionality reduction, noise filtering, LSA
  Focus: np.linalg.svd, visual demo
  SVD was presented as a powerful decomposition method for reducing dimensions and filtering noise in data.

• Part 12: Positive Definite Matrices
  ML/AI Relevance: Covariance, kernels, optimization
  Focus: Checking PD, Cholesky, quadratic forms
  We examined positive definite matrices, essential for ensuring well-behaved optimization and valid covariance structures.

Part III – Applications in ML & Advanced Topics

The final part focused on direct applications and advanced concepts, showcasing how linear algebra drives cutting-edge ML techniques and large-scale systems.

• Part 13: Principal Component Analysis (PCA)
  ML/AI Relevance: Dimensionality reduction, visualization
  Focus: Step-by-step PCA in code
  PCA was implemented as a practical method for reducing data dimensions while retaining key information, with hands-on coding.

• Part 14: Least Squares and Linear Regression
  ML/AI Relevance: Linear models, fitting lines/planes
  Focus: Normal equations, SGD, scikit-learn
  We connected linear algebra to regression, using least squares to fit models via the normal equations and stochastic gradient descent.

• Part 15: Gradient Descent in Linear Models
  ML/AI Relevance: Optimization, parameter updates
  Focus: Matrix calculus, vectorized code
  Gradient descent was explored through matrix operations, showing how linear algebra enables efficient optimization.

• Part 16: Neural Networks as Matrix Functions
  ML/AI Relevance: Layers, forward/backward pass, vectorization
  Focus: PyTorch modules, parameter shapes
  Neural networks were framed as sequences of matrix operations, highlighting vectorization in forward and backward passes.

• Part 17: Tensors and Higher-Order Generalizations
  ML/AI Relevance: Deep learning, NLP, computer vision
  Focus: torch.Tensor, broadcasting, shape tricks
  Tensors extended matrix concepts to higher dimensions, critical for deep learning tasks in NLP and vision.

• Part 18: Spectral Methods in ML (Graph Laplacians, etc.)
  ML/AI Relevance: Clustering, graph ML, signal processing
  Focus: Laplacian matrices, spectral clustering
  Spectral methods using graph Laplacians were introduced for clustering and graph-based learning.

• Part 19: Kernel Methods and Feature Spaces
  ML/AI Relevance: SVM, kernel trick, non-linear features
  Focus: Gram matrix, RBF kernels, Mercer’s theorem
  Kernel methods enabled non-linear learning via the kernel trick, transforming data implicitly into higher-dimensional spaces.

• Part 20: Random Projections and Fast Transforms
  ML/AI Relevance: Large-scale ML, efficient computation
  Focus: Johnson-Lindenstrauss, random matrix code
  Finally, random projections and fast transforms addressed scalability, reducing dimensionality and speeding up computations for massive datasets.