Foundations of Data Science

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3.11 Exercises Exercise 3.1 (Least squares vertical error) In many experiments one collects the value of a parameter at various instances of time. Let y i be the value of the parameter y at time x i . Suppose we wish to construct the best linear approximation to the data in the sense that we wish to minimize the mean square error. Here error is measured vertically rather than perpendicular to the line. Develop formulas for m and b to minimize the mean square error of the points {(x i , y i ) |1 ≤ i ≤ n} to the line y = mx + b. Exercise 3.2 Given five observed variables, height, weight, age, income, and blood pressure of n people, how would one find the best least squares fit affine subspace of the form a 1 (height) + a 2 (weight) + a 3 (age) + a 4 (income) + a 5 (blood pressure) = a 6 Here a 1 , a 2 , . . . , a 6 are the unknown parameters. If there is a good best fit 4-dimensional affine subspace, then one can think of the points as lying close to a 4-dimensional sheet rather than points lying in 5-dimensions. Why might it be better to use the perpendicular distance to the affine subspace rather than vertical distance where vertical distance is measured along the coordinate axis corresponding to one of the variables? Exercise 3.3 Manually find the best fit lines (not subspaces which must contain the origin) through the points in the sets below. Subtract the center of gravity of the points in the set from each of the points in the set and find the best fit line for the resulting points. Does the best fit line for the original data go through the origin? 1. (4,4) (6,2) 2. (4,2) (4,4) (6,2) (6,4) 3. (3,2.5) (3,5) (5,1) (5,3.5) Exercise 3.4 Manually determine the best fit line through the origin for each of the following sets of points. Is the best fit line unique? Justify your answer for each of the subproblems. 1. {(0, 1) , (1, 0)} 2. {(0, 1) , (2, 0)} Exercise 3.5 Manually find the left and right-singular vectors, the singular values, and the SVD decomposition of the matrices in Figure 3.5. Exercise 3.6 Consider the matrix A = ⎛ ⎜ ⎝ 1 2 −1 2 1 −2 −1 −2 ⎞ ⎟ ⎠ 64
(0,3) (1,1) (3,0) ⎛ M = ⎝ 1 1 0 3 3 0 ⎞ ⎠ (0,2) (1,3) (3,1) M = ⎛ ⎜ ⎝ 0 2 2 0 1 3 3 1 ⎞ ⎟ ⎠ Figure 3.5 a (2,0) Figure 3.5 b Figure 3.5: SVD problem 1. Run the power method starting from x = ( 1 1) for k = 3 steps. What does this give as an estimate of v 1 ? 2. What actually are the v i ’s, σ i ’s, and u i ’s? It may be easiest to do this by computing the eigenvectors of B = A T A. 3. Suppose matrix A is a database of restaurant ratings: each row is a person, each column is a restaurant, and a ij represents how much person i likes restaurant j. What might v 1 represent? What about u 1 ? How about the gap σ 1 − σ 2 ? Exercise 3.7 Let A be a square n × n matrix whose rows are orthonormal. Prove that the columns of A are orthonormal. Exercise 3.8 Suppose A is a n × n matrix with block diagonal structure with k equal size blocks where all entries of the i th block are a i with a 1 > a 2 > · · · > a k > 0. Show that A has exactly k nonzero singular vectors v 1 , v 2 , . . . , v k where v i has the value ( k n )1/2 in the coordinates corresponding to the i th block and 0 elsewhere. In other words, the singular vectors exactly identify the blocks of the diagonal. What happens if a 1 = a 2 = · · · = a k ? In the case where the a i are equal, what is the structure of the set of all possible singular vectors? Hint: By symmetry, the top singular vector’s components must be constant in each block. Exercise 3.9 Interpret the first right and left-singular vectors for the document term matrix. Exercise 3.10 Verify that the sum of r-rank one matrices r ∑ i=1 c i x i y i T can be written as XCY T , where the x i are the columns of X, the y i are the columns of Y, and C is a diagonal matrix with the constants c i on the diagonal. Exercise 3.11 Let ∑ r i=1 σ iu i v T i be the SVD of A. Show that ∣ ∣u T 1 A ∣ ∣ = max ∣u T A ∣ equals |u|=1 σ 1 . 65
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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
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
Page 21 and 22: points onto the circle. In higher d
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 and 34: Exercise 2.26 Explain how the volum
Page 35 and 36: Exercise 2.40 Consider a non orthog
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
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Page 59 and 60: This leads to the following theorem
Page 61 and 62: Thus, the maximum cut problem can b
Page 63: and this meets the claimed error bo
Page 67 and 68: Exercise 3.18 Modify the power meth
Page 69 and 70: 2. What percent of the Forbenius no
Page 71 and 72: City Bei- Tian- Shang- Chong- Hoh-
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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
Page 79 and 80: For x to be equal to zero, it must
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Page 83 and 84: Our first task is to figure out wha
Page 85 and 86: √ Thus, the probability for all s
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
Page 93 and 94: For a small number i of steps, the
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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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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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