Contents Writing proofs Tim Hsu, San Jos ́e State University Revised February 2016 I Fundamentals 5 1 Definitions and theorems 5 2 What is a proof ? 5 3 A word about definitions 6 II The structure of proofs 8 4 Assumptions and conclusions 8 5 The if-then method 8 6 Sets, elements, and the […]
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Chapter 0 Prologue Look around you. Computers and networks are everywhere, enabling an intricate web of com- plex human activities: education, commerce, entertainment, research, manufacturing, health management, human communication, even war. Of the two main technological underpinnings of this amazing proliferation, one is obvious: the breathtaking pace with which advances in microelectronics and chip design
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Design Patterns Elements of reusable Object-Oriented Software. Preface to book ………………………………………………………………………………………………… 10 Foreword…………………………………………………………………………………………………………. 12 Guide to readers……………………………………………………………………………………………….. 13 Introduction …………………………………………………………………………………………………….. 14 What is a Design Pattern?………………………………………………………………………… 15 Design Patterns in Smalltalk MVC …………………………………………………………… 17 Describing Design Patterns ……………………………………………………………………… 18 The Catalog of Design Patterns ………………………………………………………………… 20 Organizing the Catalog……………………………………………………………………………. 21 How Design Patterns Solve Design
Margins Perceptron and Online Perceptron Daniel Hsu (COMS 4771) Let S be a collection of labeled examples from Rd × {−1, +1}. We say S is linearly separable if there exists w ∈ Rd such that min y⟨w, x⟩ > 0, (x,y)∈S and we call w a linear separator for S. The (minimum) margin of
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Machine learning lecture slides COMS 4771 Fall 2020 0/32 Prediction theory Outline Statistical model for binary outcomes Plug-in principle and IID model Maximum likelihood estimation Statistical model for binary classification Analysis of nearest neighbor classifier Estimating the error rate of a classifier Beyond binary classificaiton and the IID
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Euclidean spaces Linear algebra review Daniel Hsu (COMS 4771) For each natural number n, the n-dimensional Euclidean space is denoted by Rn, and it is a vector space over the real field R (i.e., Rn is a real vector space). Vectors v1, . . . , vk ∈ Rn are linearly dependent if there exist
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Linear Algebra in Twenty Five Lectures Tom Denton and Andrew Waldron March 27, 2012 Edited by Katrina Glaeser, Rohit Thomas & Travis Scrimshaw 1 Contents 1 What is Linear Algebra? 12 2 Gaussian Elimination 19 2.1 NotationforLinearSystems ………………. 19 2.2 ReducedRowEchelonForm ………………. 21 3 Elementary Row Operations 27 4 Solution Sets for Systems of Linear
Machine learning lecture slides COMS 4771 Fall 2020 0/12 Classification II: Margins and SVMs Outline Perceptron Margins Support vector machines Soft-margin SVM 1/12 Perceptron (1) Perceptron: a variant of SGD Uses hinge loss: lhinge(s) := max{0, 1 − s} Uses conservative updates: only update when there is classification
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Machine learning lecture slides COMS 4771 Fall 2020 0/18 Classification III: Classification objectives Outline Scoring functions Cost-sensitive classification Conditional probability estimation Reducing multi-class to binary Fairness in classification 1/18 Scoring functions in general Statistical model: (X, Y ) ∼ P for distribution P over X × {−1, +1}
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Machine learning lecture slides COMS 4771 Fall 2020 0/35 Multivariate Gaussians and PCA Outline Multivariate Gaussians Eigendecompositions and covariance matrices Principal component analysis Principal component regression and spectral regularization Singular value decomposition Examples: topic modeling and matrix completion 1/35 Multivariate Gaussians: Isotropic Gaussians Start with X = (X1,…,Xd)
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