Foundations of Data Science

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5.10 Exercises Exercise 5.1 The Fundamental Theorem of Markov chains proves that for a connected Markov chain, the long-term average distribution a t converges to a stationary distribution. Does the t step distribution p t also converge for every connected Markov Chain ? Consider the following examples: (i) A two-state chain with p 12 = p 21 = 1. (ii) A three state chain with p 12 = p 23 = p 31 = 1 and the other p ij = 0. Generalize these examples to produce Markov Chains with many states. Exercise 5.2 Let p(x), where x = (x 1 , x 2 , . . . , x d ) x i ∈ {0, 1}, be a multivariate probability distribution. For d = 100, how would you estimate the marginal distribution p(x 1 ) = ∑ p(x 1 , x 2 , . . . , x d )? x 2 ,...,x d Exercise 5.3 Prove |p − q| 1 = 2 ∑ i (p i − q i ) + for probability distributions p and q. Proposition 5.4 Exercise 5.4 Suppose S is a subset of at most n 2 /2 points in the n × n lattice. Show that for |T | ≤ |S|/2. T = {(i, j) ∈ S ∣ ∣all elements in row i and all elements in column j are in S} Exercise 5.5 Show that the stationary probabilities of the chain described in the Gibbs sampler is the correct p. Exercise 5.6 A Markov chain is said to be symmetric if for all i and j, p ij = p ji . What is the stationary distribution of a connected symmetric chain? Prove your answer. Exercise 5.7 How would you integrate a high dimensional multivariate polynomial distribution over some convex region? this exercise needs to be clarified, ARE WE IN HIGH DIMENSIONS, IS REGION CONVEX? Exercise 5.8 Given a time-reversible Markov chain, modify the chain as follows. At the current state, stay put (no move) with probability 1/2. With the other probability 1/2, move as in the old chain. Show that the new chain has the same stationary distribution. What happens to the convergence time in this modification? Exercise 5.9 Using the Metropolis-Hasting Algorithm create a Markov chain whose stationary probability is that given in the following table. x 1 x 2 00 01 02 10 11 12 20 21 22 Prob 1/16 1/8 1/16 1/8 1/4 1/8 1/16 1/8 1/16 180
Exercise 5.10 Let p be a probability vector (nonnegative components adding up to 1) on the vertices of a connected graph which is sufficiently large that it cannot be stored in a computer. Set p ij (the transition probability from i to j) to p j for all i ≠ j which are adjacent in the graph. Show that the stationary probability vector is p. Is a random walk an efficient way to sample according to a distribution close to p? Think, for example, of the graph G being the n × n × n × · · · n grid. Exercise 5.11 Construct the edge probability for a three state Markov chain where each pair of states is connected by an edge so that the stationary probability is ( 1 2 , 1 3 , 1 6) . Exercise 5.12 Consider a three state Markov chain with stationary probability ( 1 2 , 1 3 , 1 6) . Consider the Metropolis-Hastings algorithm with G the complete graph on these three vertices. What is the expected probability that we would actually make a move along a selected edge? Exercise 5.13 Try Gibbs sampling on p (x) = Metropolis Hasting Algorithm do? ( 1 2 0 0 1 2 ) . What happens? How does the Exercise 5.14 Consider p(x), where, x = (x 1 , . . . , x 100 ) and p (0) = 1 2 , p (x) = 1 (2 100 −1) x ≠ 0. How does Gibbs sampling behave? Exercise 5.15 Construct, program, and execute an algorithm to compute the volume of a unit radius sphere in 20 dimensions by carrying out a random walk on a 20 dimensional grid with 0.1 spacing. Exercise 5.16 Given a connected graph G and an integer k how would you generate connected subgraphs of G with k vertices with probability proportional to the number of edges in the subgraph induced on those vertices? The probabilities need not be exactly proportional to the number of edges and you are not expected to prove your algorithm for this problem. Exercise 5.17 Suppose one wishes to generate uniformly at random regular, degree three undirected, connected multi-graphs each with 1,000 vertices. A multi-graph may have multiple edges between a pair of vertices and self loops. One decides to do this by a Markov Chain Monte Carlo technique. They design a network where each vertex is a regular degree three, 1,000 vertex multi-graph. For edges they say that the vertices corresponding to two graphs are connected by an edge if one graph can be obtained from the other by a flip of a pair of disjoint edges. In a flip, a pair of edges (a, b) and (c, d) are replaced by (a, c) and (b, d). 1. Prove that a swap on a connected multi-graph results in a connected multi-graph. 2. Prove that the network whose vertices correspond to the desired graphs is connected. 181
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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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Page 131 and 132: Exercise 4.7 Let f (n) be a functio
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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
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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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Page 183 and 184: i 1 i 2 R 1 R 2 R 3 Figure 5.13: An
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Page 187 and 188: Exercise 5.40 Suppose that the cliq
Page 189 and 190: Exercise 5.55 Using a web browser b
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Page 205 and 206: Theorem 6.11 (Online to Batch via R
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Page 209 and 210: Corollary 6.16 (VC-dimension sample
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Page 215 and 216: work on a variety of measures. One
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Page 219 and 220: Stochastic Gradient Descent: Given:
Page 221 and 222: sleeping experts problem. Combining
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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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