Foundations of Data Science

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( n∑ ) n∑ E x i = E(x i ) = n 1 n = 1. i=1 i=1 Note that the x i are not independent. But, linearity of expectation still applies. Example: Consider the expected number of vertices of degree d in a random graph G(n, p). The number of vertices of degree d is the sum of n indicator random variables, one for each vertex, with value one if the vertex has degree d. The expectation is the sum of the expectations of the n indicator random variables and this is just n times the expectation of one of them. Thus, the expected number of degree d vertices is n ( n d) p d (1 − p) n−d . 12.4.5 Variance In addition to the expected value of a random variable, another important parameter is the variance. The variance of a random variable x, denoted var(x) or often σ 2 (x) is E (x − E (x)) 2 and measures how close to the expected value the random variable is likely to be. The standard deviation σ is the square root of the variance. The units of σ are the same as those of x. By linearity of expectation σ 2 = E (x − E (x)) 2 = E(x 2 ) − 2E(x)E(x) + E 2 (x) = E ( x 2) − E 2 (x) . 12.4.6 Variance of the Sum of Independent Random Variables In general, the variance of the sum is not equal to the sum of the variances. However, if x and y are independent, then E (xy) = E (x) E (y) and To see this var(x + y) = var (x) + var (y) . var(x + y) = E ( (x + y) 2) − E 2 (x + y) = E(x 2 ) + 2E(xy) + E(y 2 ) − E 2 (x) − 2E(x)E(y) − E 2 (y). From independence, 2E(xy) − 2E(x)E(y) = 0 and var(x + y) = E(x 2 ) − E 2 (x) + E(y 2 ) − E 2 (y) = var(x) + var(y). More generally, if x 1 , x 2 , . . . , x n are pairwise independent random variables, then var(x 1 + x 2 + · · · + x n ) = var(x 1 ) + var(x 2 ) + · · · + var(x n ). For the variance of the sum to be the sum of the variances only requires pairwise independence not full independence. 390
12.4.7 Median One often calculates the average value of a random variable to get a feeling for the magnitude of the variable. This is reasonable when the probability distribution of the variable is Gaussian, or has a small variance. However, if there are outliers, then the average may be distorted by outliers. An alternative to calculating the expected value is to calculate the median, the value for which half of the probability is above and half is below. 12.4.8 The Central Limit Theorem Let s = x 1 + x 2 + · · · + x n be a sum of n independent random variables where each x i has probability distribution { 0 1 x i = 2 1 . 1 2 The expected value of each x i is 1 / 2 with variance σ 2 i = ( 1 2 − 0 ) 2 1 2 + ( 1 2 − 1 ) 2 1 2 = 1 4 . The expected value of s is n/2 and since the variables are independent, the variance of the sum is the sum of the variances and hence is n/4. How concentrated s is around its mean depends on the standard deviation of s which is √ n . For n equal 100 the expected 2 value of s is 50 with a standard deviation of 5 which is 10% of the mean. For n = 10, 000 the expected value of s is 5,000 with a standard deviation of 50 which is 1% of the mean. Note that as n increases, the standard deviation increases, but the ratio of the standard deviation to the mean goes to zero. More generally, if x i are independent and identically distributed, each with standard deviation σ, then the standard deviation of x 1 + x 2 + · · · + x n is √ nσ. So, x 1+x 2 √n +···+x n has standard deviation σ. The central limit theorem makes a stronger assertion that in fact x 1+x 2 √n +···+x n has Gaussian distribution with standard deviation σ. Theorem 12.2 Suppose x 1 , x 2 , . . . , x n is a sequence of identically distributed independent random variables, each with mean µ and variance σ 2 . The distribution of the random variable 1 √ (x 1 + x 2 + · · · + x n − nµ) n converges to the distribution of the Gaussian with mean 0 and variance σ 2 . 12.4.9 Probability Distributions The Gaussian or normal distribution 391
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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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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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Page 345 and 346: form a matrix, called the Hessian,
Page 347 and 348: The simplex algorithm is a classica
Page 349 and 350: This problem is NP-hard. One way to
Page 351 and 352: 10.9 Exercises Exercise 10.1 Select
Page 353 and 354: Exercise 10.18 Repeat the above exe
Page 355 and 356: Lemma 11.1 If a dilation equation i
Page 357 and 358: The Haar Wavelet ⎧ ⎨ 1 0 ≤ x
Page 359 and 360: 11.3 Wavelet Systems So far we have
Page 361 and 362: 11.5 Conditions on the Dilation Equ
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Page 367 and 368: 11.7 Sufficient Conditions for the
Page 369 and 370: Example of orthogonality when wavel
Page 371 and 372: 11.9 Designing a Wavelet System In
Page 373 and 374: 3. f(x) = 1 2 f(2x) + 1 2 f(2x −
Page 375 and 376: 12 Appendix 12.1 Asymptotic Notatio
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Page 379 and 380: Gaussian and related integrals To v
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Page 383 and 384: Thus, n! ≤ n n e −n√ ne. For
Page 385 and 386: from which the current inequality f
Page 387 and 388: Example: Let f (x) = x k for k an e
Page 389: Random variables x 1 , x 2 , . . .
Page 393 and 394: The density of y is the unit varian
Page 395 and 396: Generation of random numbers accord
Page 397 and 398: Example: Consider flipping a coin 1
Page 399 and 400: Setting λ = ln(1 + δ) Prob ( s >
Page 401 and 402: 12.5 Bounds on Tail Probability Aft
Page 403 and 404: Collect terms of the summation with
Page 405 and 406: The first integral is just the stan
Page 407 and 408: of a symmetric matrix, counting mul
Page 409 and 410: capture this situation. Now AA T =
Page 411 and 412: The above theorem tells us that the
Page 413 and 414: Important special cases are |x| 0 t
Page 415 and 416: Lemma 12.23 Let A be a symmetric ma
Page 417 and 418: etween vertices in different blocks
Page 419 and 420: ∞∑ ∑ ix i = ∞ i=0 i=0 x d d
Page 421 and 422: Generating functions are useful for
Page 423 and 424: Thus, u n = (n − 1) u n−2 . The
Page 425 and 426: f(x) a c b Figure 12.3: Illustratio
Page 427 and 428: need to argue that no other sequenc
Page 429 and 430: 1. How large must δ be if we wish
Page 431 and 432: Exercise 12.40 We are given the pro
Page 433 and 434: Index 2-universal, 240 4-way indepe
Page 435 and 436: Lagrange, 423 Law of large numbers,
Page 437 and 438: References [AK] Sanjeev Arora and R
Page 439: [MR95b] [MU05] [MV10] Rajeev Motwan
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