Basic Information
Numerous Scientific Fields Have Recently Expanded Their Capabilities Thanks to Extensive Data Collection. The High-dimensional Probability Methodology Enables an Understanding of The Fundamental Limits Of Extracting Information From Such Data. The Course Will Present Its Elements, and Demonstrate Its Applicability in Statistical Inference And Learning Problems. 1. Introduction to Probability in High Dimension and Non-asymptotic Statistics. 2. Subgaussian Random Variables and Hoeffding's Inequality. Subexponential Random Variables and Bernstein's Inequality. Bernstein's Conditions. Orlicz Spaces. Applications in Estimation And Classification Problems. 3. Maximal Inequalities. Nets, Covering and Packing Numbers. Applications in Unconstrained Linear Regression and Under Sparsity Assumptions._ 4. Concentration of Matrix Norms. Concentration of Sums of Independent Matrices, and Matrix Bernstein S Inequality. Davis-kahan Inequality. Applications in Matrix Estimation, Matrix Denoising, Community Detection, and Principal Component Analysis (pca)._ 5. Minimax Lower Bounds. Basic and Tensorization Properties Of Information Divergences. Le-cam S Two Point Method And Multiple-hypotheses Fano S Method. 6. Gaussian Random Processes, and Slepain S Comparison Inequality. Sudakov-fernique Comparison and Sudakov Minorization. 7. Decoupling and Symmetrization. Dudley S Chaining Integral And Uniform Laws of Large Numbers. Connections Between Covering Numbers Numbers and Vapnik-chervonenkis Dimension The Lemmas of Sauer-shelah And Dudley. Generalization Bounds in Statistical Learning. Learning Outcomes# at The End of The Course The Students Will Be Able To# The Students Will Be Able to Formulate, Generalize and Refine Mathematical Models For Data-science Scenarions, High-dimensional Statistics, and Statistical Learning. They Will Be Able to Utilize Tools From High-dimensional Probability to Analyze Them, And Contribute to The Forefront of Research in These Topics. The Students Will Be Acquainted With The Unique Characataristics of Random Structers in High Dimension.