Foundations of Data Science

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Exercise 2.33 If one generates points in d-dimensions with each coordinate a unit variance Gaussian, the points will approximately lie on the surface of a sphere of radius √ d. 1. What is the distribution when the points are projected onto a random line through the origin? 2. If one uses a Gaussian with variance four, where in d-space will the points lie? Exercise 2.34 Randomly generate a 100 points on the surface of a sphere in 3-dimensions and in 100-dimensions. Create a histogram of all distances between the pairs of points in both cases. Exercise 2.35 We have claimed that a randomly generated point on a ball lies near the equator of the ball, independent of the point picked to be the North Pole. Is the same claim true for a randomly generated point on a cube? To test this claim, randomly generate ten ±1 valued vectors in 128 dimensions. Think of these ten vectors as ten choices for the North Pole. Then generate some additional ±1 valued vectors. To how many of the original vectors is each of the new vectors close to being perpendicular; that is, how many of the equators is each new vector close to? Exercise 2.36 Project the vertices of a high-dimensional cube onto a line from (0, 0, . . . , 0) to (1, 1, . . . , 1). Argue that the “density” of the number of projected points (per unit distance) varies roughly as a Gaussian with variance O(1) with the mid-point of the line as center. Exercise { 2.37 Define the equator of a d-dimensional unit cube to be the hyperplane } ∑ x∣ d x i = d . 2 i=1 1. Are the vertices of a unit cube concentrated close to the equator? 2. Is the volume of a unit cube concentrated close to the equator? 3. Is the surface area of a unit cube concentrated close to the equator? Exercise 2.38 Let x be a random variable with probability density 1 4 zero elsewhere. for 0 ≤ x ≤ 4 and 1. Use Markov’s inequality to bound the probability that x > 3. 2. Make use of Prob(|x| > a) = Prob(x 2 > a 2 ) to get a tighter bound. 3. What is the bound using Prob(|x| > a) = Prob(x r > a r )? Exercise 2.39 Consider the probability distribution p(x = 0) = 1 − 1 a and p(x = a) = 1 a . Plot the probability that x is greater than or equal to a as a function of a for the bound given by Markov’s inequality and by Markov’s inequality applied to x 2 and x 4 . 34
Exercise 2.40 Consider a non orthogonal basis e 1 , e 2 , . . . , e d . The e i are a set of linearly independent unit vectors that span the space. 1. Prove that the representation of any vector in this basis is unique. 2. Calculate the squared length of z = ( √ 2 , 1) 2 e where z is expressed in the basis e 1 = (1, 0) and e 2 = (− √ 2 , √ 2 ) 2 2 3. If y = ∑ i a ie i and z = ∑ i b ie i , with 0 < a i of z is greater than the length of y? Why or why not? 4. Consider the basis e 1 = (1, 0) and e 2 = (− √ 2 2 , √ 2 2 ). (a) What is the representation of the vector (0,1) in the basis (e 1 , e 2 ). (b) What is the representation of the vector ( √ 2 2 , √ 2 2 )? (c) What is the representation of the vector (1, 2)? e 2 e 1 e 1 e 2 e 2 e 1 Exercise 2.41 Generate 20 points uniformly at random on a 900-dimensional sphere of radius 30. Calculate the distance between each pair of points. Then, select a method of projection and project the data onto subspaces of dimension k=100, 50, 10, 5, 4, 3, 2, 1 and calculate the difference between √ k times the original distances and the new pair-wise distances. For each value of k what is the maximum difference as a percent of √ k. Exercise 2.42 In d-dimensions there are exactly d-unit vectors that are pairwise orthogonal. However, if you wanted a set of vectors that were almost orthogonal you might squeeze in a few more. For example, in 2-dimensions if almost orthogonal meant at least 45 degrees apart you could fit in three almost orthogonal vectors. Suppose you wanted to find 900 almost orthogonal vectors in 100 dimensions where almost orthogonal meant an angle of between 85 and 95 degrees. How would you generate such a set? Hint: Consider projecting a 1,000 orthonormal 1,000-dimensional vectors to a random 100-dimensional space. Exercise 2.43 Exercise 2.42 finds almost orthogonal vectors using the Johnson Lindenstrauss Theorem. One could also create almost orthogonal vectors by generating random Gaussian vectors. Compare the two results to see which does a better job. 35
Page 1 and 2: Foundations of Data Science ∗ Avr
Page 3 and 4: 4.4 Branching Processes . . . . . .
Page 5 and 6: 8.2.2 Structural properties of the
Page 7 and 8: 12.4.7 Median . . . . . . . . . . .
Page 9 and 10: densities, discrete optimization, e
Page 11 and 12: 2 High-Dimensional Space 2.1 Introd
Page 13 and 14: Also, if x and y are independent, t
Page 15 and 16: 1 1 − ɛ Annulus of width 1 d Fig
Page 17 and 18: integral gives ∫ ∞ 0 e −r2 r
Page 19 and 20: The volume of the hemisphere below
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Page 23 and 24: Using the substitution 2z = y 2 , |
Page 25 and 26: For the nearest neighbor problem, i
Page 27 and 28: √ √ 2d+O(1) ≤ 2d + ∆2 −O(
Page 29 and 30: 2.10 Bibliographic Notes The word v
Page 31 and 32: Exercise 2.9 A 3-dimensional cube h
Page 33: Exercise 2.26 Explain how the volum
Page 37 and 38: 1 Exercise 2.50 Use the probability
Page 39 and 40: This decomposition of A can be view
Page 41 and 42: 3.3 Singular Vectors We now define
Page 43 and 44: orthonormal basis w 1 , w 2 , . . .
Page 45 and 46: A n × d = U n × r D r × r V T r
Page 47 and 48: 3.6 Left Singular Vectors Theorem 3
Page 49 and 50: Lemma 3.10 (Analog of eigenvalues a
Page 51 and 52: Proof: Let A = r∑ σ i u i vi T i
Page 53 and 54: a i , let dist(a i , l) denote its
Page 55 and 56: such models. Mixture models are a v
Page 57 and 58: 1. The best fit 1-dimension subspac
Page 59 and 60: This leads to the following theorem
Page 61 and 62: Thus, the maximum cut problem can b
Page 63 and 64: and this meets the claimed error bo
Page 65 and 66: (0,3) (1,1) (3,0) ⎛ M = ⎝ 1 1 0
Page 67 and 68: Exercise 3.18 Modify the power meth
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Page 75 and 76: Prob(vertex has degree k) = ( ) n
Page 77 and 78: and thus k k ≤ n. Since k! ≤ k
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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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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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