Foundations of Data Science

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3k/2 k/2 u k t Ω(k 2 ) vertices at distance k/2 from t Figure 4.19: Small worlds. distance √ n of the destination, for at least half of the starting points the path length will be at least √ n. Thus, the expected time is at least 1 2√ n and hence not in polylog time. For the general r < 2 case, we show that a local algorithm cannot find paths of length O(n (2−r)/4 ). Let δ = (2 − r)/4 and suppose the algorithm finds a path with at most n δ edges. There must be a long-distance edge on the path which terminates within distance n δ of t; otherwise, the path would end in n δ grid edges and would be too long. There are O(n 2δ ) vertices within distance n δ of t and the probability that the long distance edge from one vertex of the path ends at one of these vertices is at most n 2δ ( 1 n 2−r ) = n (r−2)/2 . To see this, recall that the lower bound on the normalizing constant is θ(n 2−r ) and hence an upper bound on the probability of a long distance edge hitting v is θ ( ) 1 n independent 2−r of where v is. Thus, the probability that the long distance edge from one of the n δ vertices on the path hits any one of the n 2δ vertices within distance n δ of t is n 2δ 1 = n r−2 n 2−r 2 . The probability that this happens for any one of the n δ vertices on the path is at most n r−2 2 n δ = n r−2 2 n 2−r 4 = n (r−2)/4 = o(1) as claimed. Short paths exist for r < 2 Finally we show for r < 2 that there are O(ln n) length paths between s and t. The proof is similar to the proof of Theorem 4.17 showing O(ln n) diameter for G(n, p) when p is Ω(ln n/n), so we do not give all the details here. We give the proof only for the case when r = 0. For a particular vertex v, let S i denote the set of vertices at distance i from v. Using only local edges, if i is O( √ ln n), then |S i | is Ω(ln n). For later i, we argue a constant factor growth in the size of S i as in Theorem 4.17. As long as |S 1 |+|S 2 |+· · ·+|S i | ≤ n 2 /2, for each of the n 2 /2 or more vertices outside, the probability that the vertex is not in 128
S i+1 is (1 − 1 ) |Si| ≤ 1 − |S i| since the long-distance edge from each vertex of S n 2 2n 2 i chooses a long-distance neighbor at random. So, the expected size of S i+1 is at least |S i |/4 and using Chernoff, we get constant factor growth up to n 2 /2. Thus, for any two vertices v and w, the number of vertices at distance O(ln n) from each is at least n 2 /2. Any two sets of cardinality at least n 2 /2 must intersect giving us a O(ln n) length path from v to w. 4.11 Bibliographic Notes The G(n, p) random graph model is from Erdös Rényi [ER60]. Among the books written on properties of random graphs a reader may wish to consult Frieze and Karonski [FK15], Jansen, Luczak and Ruciński [JLR00],or Bollobás [Bol01]. Material on phase transitions can be found in [BT87]. The work on phase transitions for CNF was started by Chao and Franco [CF86]. Further work was done in [FS96], [AP03], [Fri99], and others. The proof here that the SC algorithm produces a solution when the number of clauses is cn for c < 2 is from [Chv92]. 3 For material on the giant component consult [Kar90] or [JKLP93]. Material on branching process can be found in [AN72]. The phase transition for giant components in random graphs with given degree distributions is from Molloy and Reed [MR95a]. There are numerous papers on growth models. The material in this chapter was based primarily on [CHK + ] and [BA]. The material on small world is based on Kleinberg, [Kle00] which follows earlier work by Watts and Strogatz [WS98]. 129
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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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Page 87 and 88: Figure 4.7: A degree three vertex w
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Page 91 and 92: Size of frontier 0 ln n nθ n Numbe
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Page 97 and 98: are O(ln n) in size and the expecte
Page 99 and 100: f(x) q p 0 f(f(x)) f(x) x Figure 4.
Page 101 and 102: may be infinite. That is, ∞∑ i=
Page 103 and 104: Property cycles 1/n giant component
Page 105 and 106: Keep in mind that the leading terms
Page 107 and 108: 4.6 Phase Transitions for Increasin
Page 109 and 110: p(n) A symmetric argument shows tha
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Page 113 and 114: Proof: Say that t is a “busy time
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Page 117 and 118: problem contributes a minuscule err
Page 119 and 120: one. On the other hand, for δ = 1,
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Page 125 and 126: Destination Figure 4.17: For d < 2,
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Page 131 and 132: Exercise 4.7 Let f (n) be a functio
Page 133 and 134: Exercise 4.19 (Birthday problem) Wh
Page 135 and 136: Exercise 4.38 Consider a model of a
Page 137 and 138: Exercise 4.53 Consider graph 3-colo
Page 139 and 140: 5 Random Walks and Markov Chains A
Page 141 and 142: A B C (a) A B C (b) Figure 5.1: (a)
Page 143 and 144: in time polynomial in n. A quantity
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Page 151 and 152: Figure 5.4: A network with a constr
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Page 155 and 156: Let γ i = π 1 + π 2 + · · · +
Page 157 and 158: 5.4.1 Using Normalized Conductance
Page 159 and 160: The Gaussian distribution on the in
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Page 169 and 170: i ↑ ↓ ↑ ↓ ↑ j ↑ =⇒ i
Page 171 and 172: Theorem 5.13 Let G be an undirected
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Page 175 and 176: 1 2 4 Figure 5.11: Paths obtained f
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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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