Foundations of Data Science

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and hence dist 2 (X 1 , X 2 ) = ‖X 1 ‖ 2 F − ∥ ∥ ∥X 2 X2 T X 1 2 . F Intuitively, the distance between X 1 and X 2 is the Frobenius norm of the component of X 1 not in the space spanned by the columns of X 2 . If X 1 and X 2 are 1-dimensional unit length vectors, dist 2 (X 1 , X 2 ) is the sin squared of the angle between the spaces. Example: Consider two subspaces in four dimensions ⎛ ⎞ √1 ⎛ 2 0 1 0 √3 ⎟ X 1 = ⎜ ⎝ 1 √ 2 1 √3 0 1 √3 ⎟ ⎠ Here ⎛ ⎞ 1 ⎛ √ 2 0 dist 2 0 √3 1 (X 1 , X 2 ) = ⎜ 1 ⎝ √ √3 1 ⎟ 2 ⎠ − ⎜ ⎝ ∥ 1 0 √3 ⎛ ⎞ 2 0 0 0 0 = ⎜ ⎝ √1 √3 1 ⎟ 2 ⎠ = 7 6 ∥ ∥ 0 1 √3 F 1 0 0 1 0 0 0 0 X 2 = ⎞ ⎟ ⎠ ⎜ ⎝ 1 0 0 1 0 0 0 0 ⎞ ⎟ ⎠ ( 1 0 0 0 0 1 0 0 ⎛ ) ⎜ ⎝ √1 2 0 1 0 √3 √1 √3 1 2 0 1 √3 In essence, we projected each column vector of X 1 onto X 2 and computed the Frobenius norm of X 1 minus the projection. The Frobenius norm of each column is the sin squared of the angle between the original column of X 1 and the space spanned by the columns of X 2 . 12.8 Generating Functions ∑ A sequence a 0 , a 1 , . . ., can be represented by a generating function g(x) = ∞ a i x i . The advantage of the generating function is that it captures the entire sequence in a closed form that can be manipulated as an entity. For example, if g(x) is the generating function for the sequence a 0 , a 1 , . . ., then x d g(x) is the generating function for the sequence dx 0, a 1 , 2a 2, 3a 3 , . . . and x 2 g ′′ (x) + xg ′ (x) is the generating function for the sequence for 0, a 1 , 4a 2 , 9a 3 , . . . Example: The generating function for the sequence 1, 1, . . . is ∞ ∑ x i = 1 . The gener- 1−x ating function for the sequence 0, 1, 2, 3, . . . is 418 i=0 ⎞ ⎟ ⎠ ∥ i=0 2 F
∞∑ ∑ ix i = ∞ i=0 i=0 x d dx xi = x d dx ∞∑ x i = x d i=0 1 = dx 1−x x . (1−x) 2 Example: If A can be selected 0 or 1 times and B can be selected 0, 1, or 2 times and C can be selected 0, 1, 2, or 3 times, in how many ways can five objects be selected. Consider the generating function for the number of ways to select objects. The generating function for the number of ways of selecting objects, selecting only A’s is 1+x, only B’s is 1+x+x 2 , and only C’s is 1 + x + x 2 + x 3 . The generating function when selecting A’s, B’s, and C’s is the product. (1 + x)(1 + x + x 2 )(1 + x + x 2 + x 3 ) = 1 + 3x + 5x 2 + 6x 3 + 5x 4 + 3x 5 + x 6 The coefficient of x 5 is 3 and hence we can select five objects in three ways: ABBCC, ABCCC, or BBCCC. The generating functions for the sum of random variables ∑ Let f(x) = ∞ p i x i be the generating function for an integer valued random variable i=0 ∑ where p i is the probability that the random variable takes on value i. Let g(x) = ∞ q i x i be the generating function of an independent integer valued random variable where q i is the probability that the random variable takes on the value i. The sum of these two random variables has the generating function f(x)g(x). This is because the coefficient of x i in the product f(x)g(x) is ∑ i k=0 p kq k−i and this is also the probability that the sum of the random variables is i. Repeating this, the generating function of a sum of independent nonnegative integer valued random variables is the product of their generating functions. 12.8.1 Generating Functions for Sequences Defined by Recurrence Relationships Consider the Fibonacci sequence defined by the recurrence relationship 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, . . . f 0 = 0 f 1 = 1 f i = f i−1 + f i−2 i ≥ 2 Multiply each side of the recurrence by x i and sum from i equals two to infinity. ∞∑ f i x i = i=2 ∞∑ f i−1 x i + i=2 ∞∑ f i−2 x i f 2 x 2 + f 3 x 3 + · · · = f 1 x 2 + f 2 x 3 + · · · + f 0 x 2 + f 1 x 3 + · · · i=2 = x ( f 1 x + f 2 x 2 + · · ·) + x 2 (f 0 + f 1 x + · · ·) (12.1) i=0 419
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Foundations of Data Science ∗ Avr
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4.4 Branching Processes . . . . . .
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8.2.2 Structural properties of the
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12.4.7 Median . . . . . . . . . . .
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densities, discrete optimization, e
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2 High-Dimensional Space 2.1 Introd
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Also, if x and y are independent, t
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1 1 − ɛ Annulus of width 1 d Fig
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integral gives ∫ ∞ 0 e −r2 r
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The volume of the hemisphere below
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points onto the circle. In higher d
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Using the substitution 2z = y 2 , |
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For the nearest neighbor problem, i
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√ √ 2d+O(1) ≤ 2d + ∆2 −O(
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2.10 Bibliographic Notes The word v
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Exercise 2.9 A 3-dimensional cube h
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Exercise 2.26 Explain how the volum
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Exercise 2.40 Consider a non orthog
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1 Exercise 2.50 Use the probability
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This decomposition of A can be view
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3.3 Singular Vectors We now define
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orthonormal basis w 1 , w 2 , . . .
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A n × d = U n × r D r × r V T r
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3.6 Left Singular Vectors Theorem 3
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Lemma 3.10 (Analog of eigenvalues a
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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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Page 373 and 374: 3. f(x) = 1 2 f(2x) + 1 2 f(2x −
Page 375 and 376: 12 Appendix 12.1 Asymptotic Notatio
Page 377 and 378: these equalities by derivatives.
Page 379 and 380: Gaussian and related integrals To v
Page 381 and 382: 1 + x ≤ e x for all real x (1 −
Page 383 and 384: Thus, n! ≤ n n e −n√ ne. For
Page 385 and 386: from which the current inequality f
Page 387 and 388: Example: Let f (x) = x k for k an e
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Page 391 and 392: 12.4.7 Median One often calculates
Page 393 and 394: The density of y is the unit varian
Page 395 and 396: Generation of random numbers accord
Page 397 and 398: Example: Consider flipping a coin 1
Page 399 and 400: Setting λ = ln(1 + δ) Prob ( s >
Page 401 and 402: 12.5 Bounds on Tail Probability Aft
Page 403 and 404: Collect terms of the summation with
Page 405 and 406: The first integral is just the stan
Page 407 and 408: of a symmetric matrix, counting mul
Page 409 and 410: capture this situation. Now AA T =
Page 411 and 412: The above theorem tells us that the
Page 413 and 414: Important special cases are |x| 0 t
Page 415 and 416: Lemma 12.23 Let A be a symmetric ma
Page 417: etween vertices in different blocks
Page 421 and 422: Generating functions are useful for
Page 423 and 424: Thus, u n = (n − 1) u n−2 . The
Page 425 and 426: f(x) a c b Figure 12.3: Illustratio
Page 427 and 428: need to argue that no other sequenc
Page 429 and 430: 1. How large must δ be if we wish
Page 431 and 432: Exercise 12.40 We are given the pro
Page 433 and 434: Index 2-universal, 240 4-way indepe
Page 435 and 436: Lagrange, 423 Law of large numbers,
Page 437 and 438: References [AK] Sanjeev Arora and R
Page 439: [MR95b] [MU05] [MV10] Rajeev Motwan
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