Foundations of Data Science

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3. Prove that the stationary probability of the random walk is uniform. Is there a better word than unifrom? 4. Give an upper bound on the diameter of the network. In order to use a random walk to generate the graphs uniformly at random, the random walk must rapidly converge to the stationary probability. Proving this is beyond the material in this book. Exercise 5.18 What is the mixing time for 1. Two cliques connected by a single edge? 2. A graph consisting of an n vertex clique plus one additional vertex connected to one vertex in the clique. Exercise 5.19 What is the mixing time for 1. G(n, p) with p = log n n ? 2. A circle with n vertices where at each vertex an edge has been added to another vertex chosen at random. On average each vertex will have degree four, two circle edges, and an edge from that vertex to a vertex chosen at random, and possible some edges that are the ends of the random edges from other vertices. Exercise 5.20 Show that for the n×n×· · ·×n grid in d space, the normalized conductance is Ω(1/dn). Hint: The argument is a generalization of the argument in Exercise 5.4. Argue that for any subset S containing at most 1/2 the grid points, for at least 1/2 the grid points in S, among the d coordinate lines through the point, at least one intersects ¯S. Exercise 5.21 1. What is the set of possible harmonic functions on a connected graph if there are only interior vertices and no boundary vertices that supply the boundary condition? 2. Let q x be the stationary probability of vertex x in a random walk on an undirected graph where all edges at a vertex are equally likely and let d x be the degree of vertex x. Show that qx d x is a harmonic function. 3. If there are multiple harmonic functions when there are no boundary conditions, why is the stationary probability of a random walk on an undirected graph unique? 4. What is the stationary probability of a random walk on an undirected graph? Exercise 5.22 In Section ?? we associate a graph and edge probabilities with an electric network such that voltages and currents in the electrical network corresponded to properties of random walks on the graph. Can we go in the reverse order and construct the equivalent electrical network from a graph with edge probabilities? 182
i 1 i 2 R 1 R 2 R 3 Figure 5.13: An electrical network of resistors. Exercise 5.23 Given an undirected graph consisting of a single path of five vertices numbered 1 to 5, what is the probability of reaching vertex 1 before vertex 5 when starting at vertex 4. Exercise 5.24 Consider the electrical resistive network in Figure 5.13 consisting of vertices connected by resistors. Kirchoff’s law states that the currents at each vertex sum to zero. Ohm’s law states that the voltage across a resistor equals the product of the resistance times the current through it. Using these laws calculate the effective resistance of the network. Exercise 5.25 Consider the electrical network of Figure 5.14. 1. Set the voltage at a to one and at b to zero. What are the voltages at c and d? 2. What is the current in the edges a to c, a to d, c to d. c to b and d to b? 3. What is the effective resistance between a and b? 4. Convert the electrical network to a graph. What are the edge probabilities at each vertex? 5. What is the probability of a walk starting at c reaching a before b? a walk starting at d reaching a before b? 6. What is the net frequency that a walk from a to b goes through the edge from c to d? 7. What is the probability that a random walk starting at a will return to a before reaching b? Exercise 5.26 Consider a graph corresponding to an electrical network with vertices a and b. Prove directly that c eff c a must be less than or equal to one. We know that this is the escape probability and must be at most 1. But, for this exercise, do not use that fact. 183
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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 −
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 151 and 152: Figure 5.4: A network with a constr
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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 181: Exercise 5.10 Let p be a probabilit
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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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6.16 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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