Foundations of Data Science

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Exercise 6.17 VC-dimension Prove that the VC-dimension of triangles in the plane is seven. Exercise 6.18 VC-dimension Prove that the VC dimension of convex polygons in the plane is infinite. 236
7 Algorithms for Massive Data Problems: Streaming, Sketching, and Sampling 7.1 Introduction This chapter deals with massive data problems where the input data is too large to be stored in random access memory. One model for such problems is the streaming model, where n data items a 1 , a 2 , . . . , a n arrive one at a time. For example, the a i might be IP addresses being observed by a router on the internet. The goal is for our algorithm to compute some statistics, property, or summary of these data items without using too much memory, much less than n. More specifically, we assume each a i itself is a b-bit quantity where b is not too large; for example, each a i might be an integer in {1, . . . , m} where m = 2 b . Our goal will be to produce some desired output using space polynomial in b and log n; see Figure 7.1. For example, a very easy problem to solve in the streaming model is to compute the sum of all the a i . If each a i is an integer between 1 and m = 2 b , then the sum of all the a i is an integer between 1 and mn and so the number of bits of memory needed to maintain the sum is O(b + log n). A harder problem, which we discuss shortly, is computing the number of distinct numbers in the input sequence. One natural approach for tackling a number of problems in the streaming model is to perform random sampling of the input “on the fly”. To introduce the basic flavor of sampling on the fly, consider a stream a 1 , a 2 , . . . , a n from which we are to select an index i with probability proportional to the value of a i . When we see an element, we do not know the probability with which to select it since the normalizing constant depends on all of the elements including those we have not yet seen. However, the following method works. Let S be the sum of the a i ’s seen so far. Maintain S and an index i selected with probability a i . Initially i = 1 and S = a S 1. Having seen symbols a 1 , a 2 , . . . , a j , S will equal a 1 + a 2 + · · · + a j and for each i ∈ {1, . . . , j}, the selected index will be i with probability a i . On seeing a S j+1, change the selected index to j + 1 with probability a j+1 S+a j+1 and otherwise keep the same index as before with probability 1 − a j+1 S+a j+1 . If we change the index to j + 1, clearly it was selected with the correct probability. If we keep i as our selection, then by induction it will have been selected with probability ( 1 − a j+1 S + a j+1 ) ai S = S S + a j+1 a i S = a i S + a j+1 which is the correct probability for selecting index i. Finally s is updated by adding a j+1 to S. This problem comes up in many areas such as (sleeping) experts where there is a sequence of weights and we want to pick an expert with probability proportional to its weight. The a i ’s are the weights and the subscript i denotes the expert. 237
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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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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 215 and 216: work on a variety of measures. One
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Page 219 and 220: Stochastic Gradient Descent: Given:
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Page 223 and 224: ⎫ ⎪⎬ ⎪⎭ Each gate is conn
Page 225 and 226: W 1 W 2 W 1 W 2 W 3 (a) (b) Figure
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Page 231 and 232: 6.14.2 Active learning Active learn
Page 233 and 234: 6.16 Exercises Exercise 6.1 (Sectio
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Page 241 and 242: We now give an example of a 2-unive
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Page 259 and 260: 7.6 Exercises Algorithms for Massiv
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Page 263 and 264: Exercise 7.30 Blast: Given a long s
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Page 267 and 268: B A Figure 8.1: Example where the n
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Page 271 and 272: 8.2.5 k-means clustering on the lin
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Page 275 and 276: Theorem 8.5 Assume A satisfies (c,
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Page 279 and 280: 8.6.4 Spectral Clustering Algorithm
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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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