# Some Course Notes and Slides

## Notes

• Basics of Algebra and Analysis (manuscript) (html)
• Linear Algebra and Optimization With Applications to Machine Learning, Vol. I and II (html)
• Linear Algebra and Optimization With Applications to Machine Learning, Vol. I. (pdf)
• Linear Algebra and Optimization With Applications to Machine Learning, Vol. II. (pdf)
• Applications of Scientific Computation; EAS205, Some Notes (pdf)
• Solving the Elastic Net and Lasso Regression Problems (pdf)
• Spectral Theory of Unsigned and Signed Graphs
Applications to Graph Clustering: a Survey (pdf)
• Logarithms and Square Roots of Real Matrices (pdf)
• Chapters 1, 2, 3, 4 on Mathematical Reasoning and Logic, functions, relations, from "Discrete Mathematics, Second Edition:" (pdf)
• Chapter 5 from GMA (2nd edition); Basics of Projective Geometry (pdf)
• Chapter 9 from GMA (2nd edition); The Quaternions and the Spaces S^3, SU(2), SO(3), and RP^3 (pdf)
• Chapter 10 from GMA (2nd edition); Dirichlet-Voronoi Diagrams and Delaunay Triangulations (pdf)

## Slides

• Some Matlab code
• bezier-parabola (m)
• bezier-cubic (m)
• bezier function, degree 2 (m)
• bezier function, degree 3 (m)
• Lemniscate (m)
• Solving a triangular system by backsubstitution, v1 (m)
• Solving a triangular system by backsubstitution, v2 (m)
• Solving a triangular system; some examples (m)
• Computes a point on a curve using de Casteljau's algorithm (m)
• Linear (affine!) interpolation (m)
• To display the construction of a point using de Casteljau's algorithm (m)
• Running de Casteljau's algorithm; examples (m)
• The Steiner Roman surface (m)

• Example of a Learning Problem   (slides, pdf)
• Motivations: Fitting Data   (slides, pdf)
• Problems, Questions and Motivations; Vector Spaces, Bases, Linear Maps, The dual space   (slides, pdf)
• Matrices and Linear Maps   (slides, pdf)
• Haar Bases and Haar Wavelets   (slides, pdf)
• Direct Sums, Affine Maps   (slides, pdf)
• Determinants and Applications   (slides, pdf)
• Determinants "a la Michael Artin"   (slides, pdf)
• Gaussian, LU, and Choleski Decompositions   (slides, pdf)
• Normed spaces and matrix norms; condition number of a matrix   (slides, pdf)
• Iterative Methods for Solving Linear Systems   (slides, pdf)
• The Dual Space, Duality   (slides, pdf)
• Euclidean Spaces   (slides, pdf)
• QR-Decomposition for Arbitrary Matrices   (slides, pdf)
• Hermitian Spaces   (slides, pdf)
• Eigenvectors and Eigenvalues   (slides, pdf)
• Spectral Theorems in Euclidean and Hermitian Spaces   (slides, pdf)
• Introduction to the Finite Elements Method   (slides, pdf)
• Singular Value Decomposition (SVD) and Polar Form   (slides, pdf)
• Applications of SVD and Pseudo-Inverses   (slides, pdf)
• Quadratic Optimization Problems   (slides, pdf)

• Basic Notions of Topology   (slides, pdf)
• Review of Multivariate Calculus   (slides, pdf)
• Derivatives (Directional, Total), Series   (slides, pdf)
• Extrema of real-valued functions   (slides, pdf)
• Lagrange Multipliers (Equality constraints)   (slides, pdf)
• Introduction to Nonlinear Optimization   (slides, pdf)
• Convex Sets and Convex Functions   (slides, pdf)
• Active Constraints and Qualified Constaints   (slides, pdf)
• The Karush-Kuhn-Tucker Conditions   (slides, pdf)
• Lagrangian Duality   (slides, pdf)
• Weak and Strong Duality   (slides, pdf)
• Handling Equality Constraints Explicitly   (slides, pdf)
• Hard Margin Support Vector Machine: Version I   (slides, pdf)
• Hard Margin Support Vector Machine: Version II   (slides, pdf)
• Dual of the Hard Margin Support Vector Machine   (slides, pdf)
• Introduction to Soft Margin Support Vector Machines   (slides, pdf)
• Soft Margin Support Vector Machines   (slides, pdf)
• Classification of Data Points: Terminology   (slides, pdf)
• Classification of the Data Points in Terms of nu   (slides, pdf)
• Solving SVM Using ADMM   (slides, pdf)
• Ridge Regression   (slides, pdf)
• Ridge Regression: Learning an Affine Function   (slides, pdf)
• Lasso Regression   (slides, pdf)
• Lasso Regression: Learning an Affine Function   (slides, pdf)
• Elastic Net Regression   (slides, pdf)

Slides on the spectral theory of unsigned and signed graphs
with applications to graph clustering

• Graphs and graph Laplacians   (slides, pdf)
• Spectral Graph Drawing   (slides, pdf)
• Graph Clustering using Normalized Cuts; 2 clusters   (slides, pdf)
• Graph Clustering using Normalized Cuts; K clusters   (slides, pdf)
• Graph Clustering using Normalized Cuts; Finding a discrete solution   (slides, pdf)
• Signed Graphs ;   (slides, pdf)
• Graph Clustering Using Ratio Cuts   (slides, pdf)
• Appendix; Rayleigh Ratios, Rayleigh-Ritz Theorem, Courant-Fischer Theorem   (slides, pdf)

Other slides

• Rotation Logic (talk given at the Robotics Symposium, Sept. 27, 2013) (slides, pdf)
• Some Mathematical Methods in Machine Learning (Two lectures (each 1h 25mn)
given in Paris at ENS Cachan, Sept. 8, 2020) (slides, pdf)
• Dirichlet-Voronoi Diagrams and Delaunay Triangulations (pdf)
• The Quaternions and the Spaces S^3, SU(2), SO(3), and RP^3 (pdf)

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