Foundations of Data Science

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Contents 1 Introduction 8 2 High-Dimensional Space 11 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 The Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 The Geometry of High Dimensions . . . . . . . . . . . . . . . . . . . . . . 14 2.4 Properties of the Unit Ball . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4.1 Volume of the Unit Ball . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4.2 Most of the Volume is Near the Equator . . . . . . . . . . . . . . . 17 2.5 Generating Points Uniformly at Random from a Ball . . . . . . . . . . . . 20 2.6 Gaussians in High Dimension . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.7 Random Projection and Johnson-Lindenstrauss Lemma . . . . . . . . . . . 23 2.8 Separating Gaussians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.9 Fitting a Single Spherical Gaussian to Data . . . . . . . . . . . . . . . . . 27 2.10 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3 Best-Fit Subspaces and Singular Value Decomposition (SVD) 38 3.1 Introduction and Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.3 Singular Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.4 Singular Value Decomposition (SVD) . . . . . . . . . . . . . . . . . . . . . 44 3.5 Best Rank-k Approximations . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.6 Left Singular Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.7 Power Method for Computing the Singular Value Decomposition . . . . . . 49 3.7.1 A Faster Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.8 Singular Vectors and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . 52 3.9 Applications of Singular Value Decomposition . . . . . . . . . . . . . . . . 52 3.9.1 Centering Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.9.2 Principal Component Analysis . . . . . . . . . . . . . . . . . . . . . 53 3.9.3 Clustering a Mixture of Spherical Gaussians . . . . . . . . . . . . . 54 3.9.4 Ranking Documents and Web Pages . . . . . . . . . . . . . . . . . 59 3.9.5 An Application of SVD to a Discrete Optimization Problem . . . . 60 3.10 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4 Random Graphs 71 4.1 The G(n, p) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.1.1 Degree Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.1.2 Existence of Triangles in G(n, d/n) . . . . . . . . . . . . . . . . . . 77 4.2 Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.3 The Giant Component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 2
4.4 Branching Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.5 Cycles and Full Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.5.1 Emergence of Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.5.2 Full Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.5.3 Threshold for O(ln n) Diameter . . . . . . . . . . . . . . . . . . . . 105 4.6 Phase Transitions for Increasing Properties . . . . . . . . . . . . . . . . . . 107 4.7 Phase Transitions for CNF-sat . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.8 Nonuniform and Growth Models of Random Graphs . . . . . . . . . . . . . 114 4.8.1 Nonuniform Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.8.2 Giant Component in Random Graphs with Given Degree Distribution114 4.9 Growth Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.9.1 Growth Model Without Preferential Attachment . . . . . . . . . . . 116 4.9.2 Growth Model With Preferential Attachment . . . . . . . . . . . . 122 4.10 Small World Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.11 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5 Random Walks and Markov Chains 139 5.1 Stationary Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.2 Markov Chain Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . 145 5.2.1 Metropolis-Hasting Algorithm . . . . . . . . . . . . . . . . . . . . . 146 5.2.2 Gibbs Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 5.3 Areas and Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.4 Convergence of Random Walks on Undirected Graphs . . . . . . . . . . . . 151 5.4.1 Using Normalized Conductance to Prove Convergence . . . . . . . . 157 5.5 Electrical Networks and Random Walks . . . . . . . . . . . . . . . . . . . . 160 5.6 Random Walks on Undirected Graphs with Unit Edge Weights . . . . . . . 164 5.7 Random Walks in Euclidean Space . . . . . . . . . . . . . . . . . . . . . . 171 5.8 The Web as a Markov Chain . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5.9 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 5.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 6 Machine Learning 190 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 6.2 Overfitting and Uniform Convergence . . . . . . . . . . . . . . . . . . . . . 192 6.3 Illustrative Examples and Occam’s Razor . . . . . . . . . . . . . . . . . . . 194 6.3.1 Learning disjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 194 6.3.2 Occam’s razor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 6.3.3 Application: learning decision trees . . . . . . . . . . . . . . . . . . 196 6.4 Regularization: penalizing complexity . . . . . . . . . . . . . . . . . . . . . 197 6.5 Online learning and the Perceptron algorithm . . . . . . . . . . . . . . . . 198 6.5.1 An example: learning disjunctions . . . . . . . . . . . . . . . . . . . 198 6.5.2 The Halving algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 199 3
Page 1: Foundations of Data Science ∗ Avr
Page 5 and 6: 8.2.2 Structural properties of the
Page 7 and 8: 12.4.7 Median . . . . . . . . . . .
Page 9 and 10: densities, discrete optimization, e
Page 11 and 12: 2 High-Dimensional Space 2.1 Introd
Page 13 and 14: Also, if x and y are independent, t
Page 15 and 16: 1 1 − ɛ Annulus of width 1 d Fig
Page 17 and 18: integral gives ∫ ∞ 0 e −r2 r
Page 19 and 20: The volume of the hemisphere below
Page 21 and 22: points onto the circle. In higher d
Page 23 and 24: Using the substitution 2z = y 2 , |
Page 25 and 26: For the nearest neighbor problem, i
Page 27 and 28: √ √ 2d+O(1) ≤ 2d + ∆2 −O(
Page 29 and 30: 2.10 Bibliographic Notes The word v
Page 31 and 32: Exercise 2.9 A 3-dimensional cube h
Page 33 and 34: Exercise 2.26 Explain how the volum
Page 35 and 36: Exercise 2.40 Consider a non orthog
Page 37 and 38: 1 Exercise 2.50 Use the probability
Page 39 and 40: This decomposition of A can be view
Page 41 and 42: 3.3 Singular Vectors We now define
Page 43 and 44: orthonormal basis w 1 , w 2 , . . .
Page 45 and 46: A n × d = U n × r D r × r V T r
Page 47 and 48: 3.6 Left Singular Vectors Theorem 3
Page 49 and 50: Lemma 3.10 (Analog of eigenvalues a
Page 51 and 52: Proof: Let A = r∑ σ i u i vi T i
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a i , let dist(a i , l) denote its
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such models. Mixture models are a v
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1. The best fit 1-dimension subspac
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This leads to the following theorem
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Thus, the maximum cut problem can b
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and this meets the claimed error bo
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(0,3) (1,1) (3,0) ⎛ M = ⎝ 1 1 0
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Exercise 3.18 Modify the power meth
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2. What percent of the Forbenius no
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City Bei- Tian- Shang- Chong- Hoh-
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5 10 15 20 25 30 35 40 4 9 14 19 24
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Prob(vertex has degree k) = ( ) n
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and thus k k ≤ n. Since k! ≤ k
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For x to be equal to zero, it must
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No items E(x) ≥ 0.1 At least one
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Our first task is to figure out wha
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√ Thus, the probability for all s
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Figure 4.7: A degree three vertex w
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ftp://ftp.cs.rochester.edu/pub/u/jo
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Size of frontier 0 ln n nθ n Numbe
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For a small number i of steps, the
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same or are disjoint, ⎛( n∑ )
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are O(ln n) in size and the expecte
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f(x) q p 0 f(f(x)) f(x) x Figure 4.
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may be infinite. That is, ∞∑ i=
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Property cycles 1/n giant component
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Keep in mind that the leading terms
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4.6 Phase Transitions for Increasin
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p(n) A symmetric argument shows tha
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simple. We supply some intuition be
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Proof: Say that t is a “busy time
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Consider a graph in which half of t
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problem contributes a minuscule err
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one. On the other hand, for δ = 1,
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for all δ greater than δ critical
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expected degree d i = 1, 2, 3 √ t
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Destination Figure 4.17: For d < 2,
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For at least half of the pairs of v
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S i+1 is (1 − 1 ) |Si| ≤ 1 −
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Exercise 4.7 Let f (n) be a functio
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Exercise 4.19 (Birthday problem) Wh
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Exercise 4.38 Consider a model of a
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Exercise 4.53 Consider graph 3-colo
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5 Random Walks and Markov Chains A
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A B C (a) A B C (b) Figure 5.1: (a)
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in time polynomial in n. A quantity
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5.2 Markov Chain Monte Carlo The Ma
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p(a) = 1 2 p(b) = 1 4 p(c) = 1 8 p(
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5 8 7 12 1 3 1 6 1 8 1 3 3,1 3,2 3,
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Figure 5.4: A network with a constr
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There is a simple interpretation of
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Let γ i = π 1 + π 2 + · · · +
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5.4.1 Using Normalized Conductance
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The Gaussian distribution on the in
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6 6 1 8 1 5 8 4 5 5 Graph with boun
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and then reaching a from y before r
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every vertex? Hitting time The hitt
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clique of size n/2 x y } {{ } n/2 F
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i ↑ ↓ ↑ ↓ ↑ j ↑ =⇒ i
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Theorem 5.13 Let G be an undirected
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0 1 2 3 4 12 20 Number of resistors
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1 2 4 Figure 5.11: Paths obtained f
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whereas, with k self-loops, the equ
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or p[I − (1 − α)A] = α (1, 1,
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Exercise 5.10 Let p be a probabilit
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i 1 i 2 R 1 R 2 R 3 Figure 5.13: An
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(a) 1 2 3 4 (b) 1 2 3 4 (c) 1 2 3 4
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Exercise 5.40 Suppose that the cliq
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Exercise 5.55 Using a web browser b
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will allow us to make mathematical
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Not spam { }} { { Spam }} { x 1 x 2
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1 ∨ x 8 = 1}, or more succinctly,
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described using O(k log d) bits: lo
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In fact, we can show this bound is
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y definition of w ∗ . Similarly,
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y x φ Figure 6.4: Data that is not
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Theorem 6.11 (Online to Batch via R
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function of concept class H, will g
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Corollary 6.16 (VC-dimension sample
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A sphere in d-dimensions is a set o
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then with probability ≥ 1 − δ,
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work on a variety of measures. One
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Since weight(0) = n, the total weig
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Stochastic Gradient Descent: Given:
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sleeping experts problem. Combining
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⎫ ⎪⎬ ⎪⎭ Each gate is conn
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W 1 W 2 W 1 W 2 W 3 (a) (b) Figure
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convolution pooling Image Convoluti
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discard separators in advance that
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6.14.2 Active learning Active learn
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6.16 Exercises Exercise 6.1 (Sectio
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√ L log(L(T )) |S|) will be at mo
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7 Algorithms for Massive Data Probl
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important to know if some popular s
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We now give an example of a 2-unive
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estimate of m. To obtain a coin tha
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expected value of the sum will be z
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δ have values in ±1. There are ex
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mean. (ii) There is “no free lunc
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⎡ ⎢ ⎣ A m × n ⎤ ⎡ ⎥
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One uses the primitive in Section ?
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would require s ≥ n, which clearl
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its that capture sufficient informa
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7.6 Exercises Algorithms for Massiv
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Counting Frequent Elements The Majo
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Exercise 7.30 Blast: Given a long s
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Preliminaries: We will follow the s
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B A Figure 8.1: Example where the n
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Lloyd’s algorithm: Start with k c
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8.2.5 k-means clustering on the lin
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a clustering that is ɛ-close to C
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Theorem 8.5 Assume A satisfies (c,
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probability is q (where, q < p). 32
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8.6.4 Spectral Clustering Algorithm
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But for l ≠ l ′ , by hypothesis
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4. the σ-interior of the clusters
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Figure 8.4: Example of a bipartite
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are likely to be balanced given tha
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s 1 ∞ ∞ λ t ∞ edges and vert
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A clustering algorithm is consisten
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less than n, then richness is not r
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8.13 Exercises Exercise 8.1 Constru
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A B Figure 8.8: insert caption Exer
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9 Topic Models, Hidden Markov Proce
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Nonnegative matrix factorization (N
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This is a linear program. As we rem
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for t = 1 to T Prob(O 0 O 1 · ·
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a ij transition probability from st
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C 1 S 2 C 2 causes D 1 D 2 diseases
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x 1 + x 2 + x 3 x 1 + x 2 x 1 + x 3
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x 1 x 1 + x 2 + x 3 x 3 + x 4 + x 5
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the messages coming to a variable n
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y 2 y 3 x 2 x 3 y 1 x 1 x 4 y 4 y n
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two pieces of information, the valu
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graph. The vertices of Π are denot
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present and ask how correlated this
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Now consider a very tall tree. If t
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9.14 Exercises Exercise 9.1 Find a
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10 Other Topics 10.1 Rankings Ranki
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b . a . a b b b b . . b b a a b b .
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In the discrete case, x = [x 0 , x
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Figure 10.3: Some subgradients for
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Suppose ˜x 0 were another minimum.
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A T S A S is invertible. We will pr
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was included, we would just multipl
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position on genome trees = Phenotyp
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form a matrix, called the Hessian,
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The simplex algorithm is a classica
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This problem is NP-hard. One way to
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10.9 Exercises Exercise 10.1 Select
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Exercise 10.18 Repeat the above exe
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Lemma 11.1 If a dilation equation i
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The Haar Wavelet ⎧ ⎨ 1 0 ≤ x
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11.3 Wavelet Systems So far we have
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11.5 Conditions on the Dilation Equ
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the support of both sides of the eq
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and the wavelet functions are ortho
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11.7 Sufficient Conditions for the
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Example of orthogonality when wavel
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11.9 Designing a Wavelet System In
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3. f(x) = 1 2 f(2x) + 1 2 f(2x −
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12 Appendix 12.1 Asymptotic Notatio
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these equalities by derivatives.
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Gaussian and related integrals To v
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1 + x ≤ e x for all real x (1 −
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Thus, n! ≤ n n e −n√ ne. For
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from which the current inequality f
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Example: Let f (x) = x k for k an e
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Random variables x 1 , x 2 , . . .
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12.4.7 Median One often calculates
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The density of y is the unit varian
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Generation of random numbers accord
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Example: Consider flipping a coin 1
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Setting λ = ln(1 + δ) Prob ( s >
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12.5 Bounds on Tail Probability Aft
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Collect terms of the summation with
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The first integral is just the stan
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of a symmetric matrix, counting mul
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capture this situation. Now AA T =
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The above theorem tells us that the
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Important special cases are |x| 0 t
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Lemma 12.23 Let A be a symmetric ma
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etween vertices in different blocks
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∞∑ ∑ ix i = ∞ i=0 i=0 x d d
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Generating functions are useful for
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Thus, u n = (n − 1) u n−2 . The
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f(x) a c b Figure 12.3: Illustratio
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need to argue that no other sequenc
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1. How large must δ be if we wish
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Exercise 12.40 We are given the pro
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Index 2-universal, 240 4-way indepe
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Lagrange, 423 Law of large numbers,
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References [AK] Sanjeev Arora and R
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[MR95b] [MU05] [MV10] Rajeev Motwan
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