Linear Algebra and Optimization for Machine Learning

Abstract

3.3.9 Cholesky Factorization: Symmetric LU Decomposition This textbook introduces linear algebra and optimization in the context of machine learning. Examples and exercises are provided throughout this text book together with access to a solutions manual. This textbook targets graduate level students and professors in computer science, mathematics and data science. Advanced undergraduate students can also use this textbook. The chapters for this textbook are organized as follows: 1. Linear algebra and its applications: The chapters focus on the basics of linear algebra together with their common applications to singular value decomposition, matrix factorization, similarity matrices (kernel methods), and graph analysis. Numerous machine learning applications have been used as examples, such as spectral clustering, kernel-based classification, and outlier detection. The tight integration of linear algebra methods with examples from machine learning differentiates this book from generic volumes on linear algebra. The focus is clearly on the most relevant aspects of linear algebra for machine learning and to teach readers how to apply these concepts. 2. Optimization and its applications: Much of machine learning is posed as an optimization problem in which we try to maximize the accuracy of regression and classification models. The "parent problem" of optimization-centric machine learning is least-squares regression. Interestingly, this problem arises in both linear algebra and optimization, and is one of the key connecting problems of the two fields. Least-squares regression is also the starting point for support vector machines, logistic regression, and recommender systems. Furthermore, the methods for dimensionality reduction and matrix factorization also require the development of optimization methods. A general view of optimization in computational graphs is discussed together with its applications to back propagation in neural networks. A frequent challenge faced by beginners in machine learning is the extensive background required in linear algebra and optimization. One problem is that the existing linear algebra and optimization courses are not specific to machine learning; therefore, one would typically have to complete more course material than is necessary to pick up machine learning. Furthermore, certain types of ideas and tricks from optimization and linear algebra recur more frequently in machine learning than other application-centric settings. Therefore, there is significant value in developing a view of linear algebra and optimization that is better suited to the specific perspective of machine learning Intro -- Contents -- Preface -- Acknowledgments -- Author Biography -- 1 Linear Algebra and Optimization: An Introduction -- 1.1 Introduction -- 1.2 Scalars, Vectors, and Matrices -- 1.2.1 Basic Operations with Scalars and Vectors -- 1.2.2 Basic Operations with Vectors and Matrices -- 1.2.3 Special Classes of Matrices -- 1.2.4 Matrix Powers, Polynomials, and the Inverse -- 1.2.5 The Matrix Inversion Lemma: Inverting the Sum of Matrices -- 1.2.6 Frobenius Norm, Trace, and Energy -- 1.3 Matrix Multiplication as a Decomposable Operator 1.3.1 Matrix Multiplication as Decomposable Row and ColumnOperators -- 1.3.2 Matrix Multiplication as Decomposable Geometric Operators -- 1.4 Basic Problems in Machine Learning -- 1.4.1 Matrix Factorization -- 1.4.2 Clustering -- 1.4.3 Classification and Regression Modeling -- 1.4.4 Outlier Detection -- 1.5 Optimization for Machine Learning -- 1.5.1 The Taylor Expansion for Function Simplification -- 1.5.2 Example of Optimization in Machine Learning -- 1.5.3 Optimization in Computational Graphs -- 1.6 Summary -- 1.7 Further Reading -- 1.8 Exercises -- 2 Linear Transformations and Linear Systems 2.1 Introduction -- 2.1.1 What Is a Linear Transform? -- 2.2 The Geometry of Matrix Multiplication -- 2.3 Vector Spaces and Their Geometry -- 2.3.1 Coordinates in a Basis System -- 2.3.2 Coordinate Transformations Between Basis Sets -- 2.3.3 Span of a Set of Vectors -- 2.3.4 Machine Learning Example: Discrete Wavelet Transform -- 2.3.5 Relationships Among Subspaces of a Vector Space -- 2.4 The Linear Algebra of Matrix Rows and Columns -- 2.5 The Row Echelon Form of a Matrix -- 2.5.1 LU Decomposition -- 2.5.2 Application: Finding a Basis Set -- 2.5.3 Application: Matrix Inversion 2.5.4 Application: Solving a System of Linear Equations -- 2.6 The Notion of Matrix Rank -- 2.6.1 Effect of Matrix Operations on Rank -- 2.7 Generating Orthogonal Basis Sets -- 2.7.1 Gram-Schmidt Orthogonalization and QR Decomposition -- 2.7.2 QR Decomposition -- 2.7.3 The Discrete Cosine Transform -- 2.8 An Optimization-Centric View of Linear Systems -- 2.8.1 Moore-Penrose Pseudoinverse -- 2.8.2 The Projection Matrix -- 2.9 Ill-Conditioned Matrices and Systems -- 2.10 Inner Products: A Geometric View -- 2.11 Complex Vector Spaces -- 2.11.1 The Discrete Fourier Transform -- 2.12 Summary 2.13 Further Reading -- 2.14 Exercises -- 3 Eigenvectors and Diagonalizable Matrices -- 3.1 Introduction -- 3.2 Determinants -- 3.3 Diagonalizable Transformations and Eigenvectors -- 3.3.1 Complex Eigenvalues -- 3.3.2 Left Eigenvectors and Right Eigenvectors -- 3.3.3 Existence and Uniqueness of Diagonalization -- 3.3.4 Existence and Uniqueness of Triangulization -- 3.3.5 Similar Matrix Families Sharing Eigenvalues -- 3.3.6 Diagonalizable Matrix Families Sharing Eigenvectors -- 3.3.7 Symmetric Matrices -- 3.3.8 Positive Semidefinite Matrices