Foundations of Data Science

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Probability q(p) of the root being 1 1/2 as a function of p 1 0 high temperature low temperature 0 1/2 1 Probability p of a leaf being 1 at phase transition slope of q(p) equals 1 at p = 1/2 Figure 9.10: Shape of q as a function of p for the height one tree and three values of β corresponding to low temperature, the phase transition temperature, and high temperature. . decreasing. From this it follows that q is monotonically increasing. In the iteration going from p to q, we do not get the true marginal probabilities at each level since we ignored the effect of the portion of the tree above. However, when we get to the root, we do get the true marginal for the root. To get the true marginal’s for the interior nodes we need to send messages down from the root. Note: The joint probability distribution for the tree is of the form e β ∑ (ij)∈E) x i x j ∏ = e βx ix j . (i,j)∈E Suppose x 1 has value 1 with probability p. Then define a function ϕ, called evidence, such that { p for x1 = 1 ϕ (x 1 ) = 1 − p for x 1 = −1 = ( p − 1 2) x1 + 1 2 and multiply the joint probability function by ϕ. Note, however, that the marginal probability of x 1 is not p. In fact, it may be further from p after multiplying the conditional probability function by the function ϕ. 326
9.14 Exercises Exercise 9.1 Find a nonnegative factorization of the matrix ⎛ ⎞ 4 6 5 1 2 3 A = ⎜7 10 7 ⎟ ⎝6 8 4 ⎠ 6 10 11 Indicate the steps in your method and show the intermediate results. Exercise 9.2 Find a nonnegative factorization of each of the following matrices. (1) (3) ⎛ ⎞ 10 9 15 14 13 2 1 3 3 1 8 7 13 11 11 7 5 11 10 7 ⎜ 5 5 11 6 11 ⎟ ⎝ 1 1 3 1 3 ⎠ 2 2 2 2 ⎛ ⎞ 4 4 3 3 1 3 4 3 13 16 13 10 5 13 14 10 15 24 21 12 9 21 18 12 7 16 15 6 7 15 10 6 ⎜ 1 4 4 1 2 4 2 1 ⎟ ⎝ 5 8 7 4 3 7 6 4 ⎠ 3 12 12 3 6 12 6 3 (2) (4) ⎛ ⎞ 5 5 10 14 17 2 2 4 4 6 1 1 2 4 4 1 1 2 2 3 ⎜3 3 6 8 10 ⎟ ⎝5 5 10 16 18⎠ 2 2 4 6 7 ⎛ ⎞ 1 1 3 4 4 4 1 ⎜9 9 9 12 9 9 3 ⎟ ⎝6 6 12 16 15 15 4⎠ 3 3 3 4 3 3 1 Exercise 9.3 Consider the matrix A that is the product of nonnegative matrices B and C. ⎛ ⎞ ⎛ ⎞ 12 22 41 35 10 1 ( ) ⎜19 20 13 48 ⎟ ⎝11 14 16 29⎠ = ⎜ 1 9 ⎟ 1 2 4 3 ⎝ 3 4⎠ 2 2 1 5 14 16 14 36 2 6 Which rows of A are approximate positive linear combinations of other rows of A? Find an approxiamte nonnegative factorization of A Exercise 9.4 What is the probability of heads occurring after a sufficiently long sequence of transitions in Viterbi algorithm example of the most likely sequence of states? Exercise 9.5 Find optimum parameters for a three state HMM and given output sequence. Note the HMM must have a strong signature in the output sequence or we probably will not be able to find it. The following example may not be good for that reason. 327
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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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Page 325: Now consider a very tall tree. If t
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Page 333 and 334: In the discrete case, x = [x 0 , x
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Page 353 and 354: Exercise 10.18 Repeat the above exe
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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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