Introduction to Data Science
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Contents 1 Introduction 9 2 High-Dimensional Space 12 2 1 Introduction 12 2 2 The Law of Large
Foundations of Data Science
Avrim Blum, John Hopcroft, and Ravindran Kannan
Thursday 4
thJanuary, 2018Copyright 2015. All rights reserved
1Contents
1 Introduction 9
2 High-Dimensional Space 12
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
122.2 The Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . . . . . .
122.3 The Geometry of High Dimensions . . . . . . . . . . . . . . . . . . . . . .
152.4 Properties of the Unit Ball . . . . . . . . . . . . . . . . . . . . . . . . . . .
172.4.1 Volume of the Unit Ball . . . . . . . . . . . . . . . . . . . . . . . .
172.4.2 Volume Near the Equator . . . . . . . . . . . . . . . . . . . . . . .
192.5 Generating Points Uniformly at Random from a Ball . . . . . . . . . . . .
222.6 Gaussians in High Dimension . . . . . . . . . . . . . . . . . . . . . . . . .
232.7 Random Projection and Johnson-Lindenstrauss Lemma . . . . . . . . . . .
252.8 Separating Gaussians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
272.9 Fitting a Spherical Gaussian to Data . . . . . . . . . . . . . . . . . . . . .
292.10 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
312.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
323 Best-Fit Subspaces and Singular Value Decomposition (SVD) 40
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
403.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
413.3 Singular Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
423.4 Singular Value Decomposition (SVD) . . . . . . . . . . . . . . . . . . . . .
453.5 Best Rank-kApproximations . . . . . . . . . . . . . . . . . . . . . . . . .47
3.6 Left Singular Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
483.7 Power Method for Singular Value Decomposition . . . . . . . . . . . . . . .
513.7.1 A Faster Method . . . . . . . . . . . . . . . . . . . . . . . . . . . .
513.8 Singular Vectors and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . .
543.9 Applications of Singular Value Decomposition . . . . . . . . . . . . . . . .
543.9.1 Centering Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
543.9.2 Principal Component Analysis . . . . . . . . . . . . . . . . . . . . .
563.9.3 Clustering a Mixture of Spherical Gaussians . . . . . . . . . . . . .
563.9.4 Ranking Documents and Web Pages . . . . . . . . . . . . . . . . .
623.9.5 An Application of SVD to a Discrete Optimization Problem . . . .
633.10 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
653.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
674 Random Walks and Markov Chains 76
4.1 Stationary Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
804.2 Markov Chain Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . .
814.2.1 Metropolis-Hasting Algorithm . . . . . . . . . . . . . . . . . . . . .
834.2.2 Gibbs Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
844.3 Areas and Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
862
4.4 Convergence of Random Walks on Undirected Graphs . . . . . . . . . . . .88
4.4.1 Using Normalized Conductance to Prove Convergence . . . . . . . .
944.5 Electrical Networks and Random Walks . . . . . . . . . . . . . . . . . . . .
974.6 Random Walks on Undirected Graphs with Unit Edge Weights . . . . . . .
1024.7 Random Walks in Euclidean Space . . . . . . . . . . . . . . . . . . . . . .
1094.8 The Web as a Markov Chain . . . . . . . . . . . . . . . . . . . . . . . . . .
1124.9 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1164.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1185 Machine Learning 129
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1295.2 The Perceptron algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . .
1305.3 Kernel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1325.4 Generalizing to New Data . . . . . . . . . . . . . . . . . . . . . . . . . . .
1345.5 Overtting and Uniform Convergence . . . . . . . . . . . . . . . . . . . . .
1355.6 Illustrative Examples and Occam's Razor . . . . . . . . . . . . . . . . . . .
1385.6.1 Learning Disjunctions . . . . . . . . . . . . . . . . . . . . . . . . .
1385.6.2 Occam's Razor . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1395.6.3 Application: Learning Decision Trees . . . . . . . . . . . . . . . . .
1 405.7 Regularization: Penalizing Complexity . . . . . . . . . . . . . . . . . . . .
1415.8 Online Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1415.8.1 An Example: Learning Disjunctions . . . . . . . . . . . . . . . . . .
14 25.8.2 The Halving Algorithm . . . . . . . . . . . . . . . . . . . . . . . . .
1435.8.3 The Perceptron Algorithm . . . . . . . . . . . . . . . . . . . . . . .
1435.8.4 Extensions: Inseparable Data and Hinge Loss . . . . . . . . . . . .
1 455.9 Online to Batch Conversion . . . . . . . . . . . . . . . . . . . . . . . . . .
1465.10 Support-Vector Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1475.11 VC-Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1485.11.1 Denitions and Key Theorems . . . . . . . . . . . . . . . . . . . . .
1495.11.2 Examples: VC-Dimension and Growth Function . . . . . . . . . . .
1515.11.3 Proof of Main Theorems . . . . . . . . . . . . . . . . . . . . . . . .
15 35.11.4 VC-Dimension of Combinations of Concepts . . . . . . . . . . . . .
15 65.11.5 Other Measures of Complexity . . . . . . . . . . . . . . . . . . . . .
1565.12 Strong and Weak Learning - Boosting . . . . . . . . . . . . . . . . . . . . .
1575.13 Stochastic Gradient Descent . . . . . . . . . . . . . . . . . . . . . . . . . .
1605.14 Combining (Sleeping) Expert Advice . . . . . . . . . . . . . . . . . . . . .
1625.15 Deep Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1645.15.1 Generative Adversarial Networks (GANs) . . . . . . . . . . . . . . .
1705.16 Further Current Directions . . . . . . . . . . . . . . . . . . . . . . . . . . .
1715.16.1 Semi-Supervised Learning . . . . . . . . . . . . . . . . . . . . . . .
1715.16.2 Active Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1745.16.3 Multi-Task Learning . . . . . . . . . . . . . . . . . . . . . . . . . .
17 45.17 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1753
5.18 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .176
6 Algorithms for Massive Data Problems: Streaming, Sketching, and
Sampling 181
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1816.2 Frequency Moments of Data Streams . . . . . . . . . . . . . . . . . . . . .
1826.2.1 Number of Distinct Elements in a Data Stream . . . . . . . . . . .
1836.2.2 Number of Occurrences of a Given Element. . . . . . . . . . . . . .
1866.2.3 Frequent Elements . . . . . . . . . . . . . . . . . . . . . . . . . . .
1876.2.4 The Second Moment . . . . . . . . . . . . . . . . . . . . . . . . . .
1896.3 Matrix Algorithms using Sampling . . . . . . . . . . . . . . . . . . . . . .
1926.3.1 Matrix Multiplication using Sampling . . . . . . . . . . . . . . . . .
1936.3.2 Implementing Length Squared Sampling in Two Passes . . . . . . .
1976.3.3 Sketch of a Large Matrix . . . . . . . . . . . . . . . . . . . . . . . .
1976.4 Sketches of Documents . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2016.5 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2036.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2047 Clustering 208
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2087.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2087.1.2 Two General Assumptions on the Form of Clusters . . . . . . . . .
2097.1.3 Spectral Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . .
2117.2k-Means Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .211
7.2.1 A Maximum-Likelihood Motivation . . . . . . . . . . . . . . . . . .
2117.2.2 Structural Properties of thek-Means Objective . . . . . . . . . . .212
7.2.3 Lloyd's Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2137.2.4 Ward's Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2157.2.5k-Means Clustering on the Line . . . . . . . . . . . . . . . . . . . .215
7.3k-Center Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .215
7.4 Finding Low-Error Clusterings . . . . . . . . . . . . . . . . . . . . . . . . .
2167.5 Spectral Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2167.5.1 Why Project? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2167.5.2 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2187.5.3 Means Separated by
(1) Standard Deviations . . . . . . . . . . . . 2197.5.4 Laplacians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2217.5.5 Local spectral clustering . . . . . . . . . . . . . . . . . . . . . . . .
22 17.6 Approximation Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2247.6.1 The Conceptual Idea . . . . . . . . . . . . . . . . . . . . . . . . . .
2247.6.2 Making this Formal . . . . . . . . . . . . . . . . . . . . . . . . . . .
2247.6.3 Algorithm and Analysis . . . . . . . . . . . . . . . . . . . . . . . .
2257.7 High-Density Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2277.7.1 Single Linkage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22 74
7.7.2 Robust Linkage . . . . . . . . . . . . . . . . . . . . . . . . . . . . .228
7.8 Kernel Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2287.9 Recursive Clustering based on Sparse Cuts . . . . . . . . . . . . . . . . . .
2297.10 Dense Submatrices and Communities . . . . . . . . . . . . . . . . . . . . .
2307.11 Community Finding and Graph Partitioning . . . . . . . . . . . . . . . . .
2337.12 Spectral clustering applied to social networks . . . . . . . . . . . . . . . . .
2367.13 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2397.14 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2408 Random Graphs 245
8.1 TheG(n;p) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .245
8.1.1 Degree Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . .
24 68.1.2 Existence of Triangles inG(n;d=n) . . . . . . . . . . . . . . . . . .250
8.2 Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2528.3 Giant Component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2618.3.1 Existence of a giant component . . . . . . . . . . . . . . . . . . . .
2618.3.2 No other large components . . . . . . . . . . . . . . . . . . . . . . .
26 38.3.3 The case ofp <1=n. . . . . . . . . . . . . . . . . . . . . . . . . . .264
8.4 Cycles and Full Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . .
2658.4.1 Emergence of Cycles . . . . . . . . . . . . . . . . . . . . . . . . . .
26 58.4.2 Full Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2668.4.3 Threshold forO(lnn) Diameter . . . . . . . . . . . . . . . . . . . .268
8.5 Phase Transitions for Increasing Properties . . . . . . . . . . . . . . . . . .
2708.6 Branching Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2728.7 CNF-SAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2778.7.1 SAT-solvers in practice . . . . . . . . . . . . . . . . . . . . . . . . .
2788.7.2 Phase Transitions for CNF-SAT . . . . . . . . . . . . . . . . . . . .
2798.8 Nonuniform Models of Random Graphs . . . . . . . . . . . . . . . . . . . .
2848.8.1 Giant Component in Graphs with Given Degree Distribution . . . .
2858.9 Growth Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2868.9.1 Growth Model Without Preferential Attachment . . . . . . . . . . .
2878.9.2 Growth Model With Preferential Attachment . . . . . . . . . . . .
2938.10 Small World Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 948.11 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2998.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3019 Topic Models, Nonnegative Matrix Factorization, Hidden Markov Mod-
els, and Graphical Models 3109.1 Topic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3109.2 An Idealized Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3139.3 Nonnegative Matrix Factorization - NMF . . . . . . . . . . . . . . . . . . .
3159.4 NMF with Anchor Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3179.5 Hard and Soft Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3185
9.6 The Latent Dirichlet Allocation Model for Topic Modeling . . . . . . . . .320
9.7 The Dominant Admixture Model . . . . . . . . . . . . . . . . . . . . . . .
3229.8 Formal Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3249.9 Finding the Term-Topic Matrix . . . . . . . . . . . . . . . . . . . . . . . .
3279.10 Hidden Markov Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3329.11 Graphical Models and Belief Propagation . . . . . . . . . . . . . . . . . . .
3379.12 Bayesian or Belief Networks . . . . . . . . . . . . . . . . . . . . . . . . . .
3389.13 Markov Random Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3399.14 Factor Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3409.15 Tree Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3419.16 Message Passing in General Graphs . . . . . . . . . . . . . . . . . . . . . .
3429.17 Graphs with a Single Cycle . . . . . . . . . . . . . . . . . . . . . . . . . .
3449.18 Belief Update in Networks with a Single Loop . . . . . . . . . . . . . . . .
3469.19 Maximum Weight Matching . . . . . . . . . . . . . . . . . . . . . . . . . .
3479.20 Warning Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3519.21 Correlation Between Variables . . . . . . . . . . . . . . . . . . . . . . . . .
3519.22 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3559.23 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35710 Other Topics 360
10.1 Ranking and Social Choice . . . . . . . . . . . . . . . . . . . . . . . . . . .
36010.1.1 Randomization . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36210.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36310.2 Compressed Sensing and Sparse Vectors . . . . . . . . . . . . . . . . . . .
36410.2.1 Unique Reconstruction of a Sparse Vector . . . . . . . . . . . . . .
36510.2.2 Eciently Finding the Unique Sparse Solution . . . . . . . . . . . .
36610.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36810.3.1 Biological . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36810.3.2 Low Rank Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . .
36910.4 An Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37010.4.1 Sparse Vector in Some Coordinate Basis . . . . . . . . . . . . . . .
37010.4.2 A Representation Cannot be Sparse in Both Time and Frequency
Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37110.5 Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37310.6 Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37510.6.1 The Ellipsoid Algorithm . . . . . . . . . . . . . . . . . . . . . . . .
37 510.7 Integer Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37710.8 Semi-Denite Programming . . . . . . . . . . . . . . . . . . . . . . . . . .
37810.9 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38010.10Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3816
11 Wavelets 385
11.1 Dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38511.2 The Haar Wavelet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38611.3 Wavelet Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39011.4 Solving the Dilation Equation . . . . . . . . . . . . . . . . . . . . . . . . .
39011.5 Conditions on the Dilation Equation . . . . . . . . . . . . . . . . . . . . .
39211.6 Derivation of the Wavelets from the Scaling Function . . . . . . . . . . . .
39411.7 Sucient Conditions for the Wavelets to be Orthogonal . . . . . . . . . . .
39811.8 Expressing a Function in Terms of Wavelets . . . . . . . . . . . . . . . . .
40111.9 Designing a Wavelet System . . . . . . . . . . . . . . . . . . . . . . . . . .
40211.10Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40211.11 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40211.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40312 Appendix 406
12.1 Denitions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40612.2 Asymptotic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40612.3 Useful Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40812.4 Useful Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41312.5 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42012.5.1 Sample Space, Events, and Independence . . . . . . . . . . . . . . .
42012.5.2 Linearity of Expectation . . . . . . . . . . . . . . . . . . . . . . . .
42112.5.3 Union Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42212.5.4 Indicator Variables . . . . . . . . . . . . . . . . . . . . . . . . . . .
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