Foundations of Data Science

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a b c i j Figure 9.9: warning propagation k and the factors by a, b, and c. Factor a sends a message m ai to each variable i that appears in the factor a called a warning. The warning is 0 or 1 depending on whether or not factor a believes that the value assigned to i is required for a to be satisfied. A factor a determines the warning to send to variable i by examining all warnings received by other variables in factor a from factors containing them. For each variable j, sum the warnings from factors containing j that warn j to take value T and subtract the warnings that warn j to take value F. If the difference says that j should take value T or F and this value for variable j does not satisfy a, and this is true for all j, then a sends a warning to i that the value of variable i is critical for factor a. Start the warning propagation algorithm by assigning 1 to a warning with probability 1/2. Iteratively update the warnings. If the warning propagation algorithm converges, then compute for each variable i the local field h i and the contradiction number c i . The local field h i is the number of clauses containing the variable i that sent messages that i should take value T minus the number that sent messages that i should take value F. The contradiction number c i is 1 if variable i gets conflicting warnings and 0 otherwise. If the factor graph is a tree, the warning propagation algorithm converges. If one of the warning messages is one, the problem is unsatisfiable; otherwise it is satisfiable. 9.13 Correlation Between Variables In many situations one is interested in how the correlation between variables drops off with some measure of distance. Consider a factor graph for a 3-CNF formula. Measure the distance between two variables by the shortest path in the factor graph. One might ask if one variable is assigned the value true, what is the percentage of satisfying assignments of the 3-CNF formula in which the second variable also is true. If the percentage is the same as when the first variable is assigned false, then we say that the two variables are uncorrelated. How difficult it is to solve a problem is likely to be related to how fast the correlation decreases with distance. Another illustration of this concept is in counting the number of perfect matchings in a graph. One might ask what is the percentage of matching in which some edge is 322
present and ask how correlated this percentage is with the presences or absence of edges at some distance d. One is interested in whether the correlation drops off with distance. To explore this concept we consider the Ising model studied in physics. The Ising or ferromagnetic model is a pairwise random Markov field. The underlying graph, usually a lattice, assigns a value of ±1, called spin, to the variable at each vertex. The probability (Gibbs measure) of a given configuration of spins is proportional to exp(β ∑ x i x j ) = ∏ e βx ix j where x i = ±1 is the value associated with vertex i. Thus (i,j)∈E (i,j)∈E p (x 1 , x 2 , . . . , x n ) = 1 Z where Z is a normalization constant. ∏ (i,j)∈E exp(βx i x j ) = 1 Z e β ∑ x i x j (i,j)∈E The value of the summation is simply the difference in the number of edges whose vertices have the same spin minus the number of edges whose vertices have opposite spin. The constant β is viewed as inverse temperature. High temperature corresponds to a low value of β and low temperature corresponds to a high value of β. At high temperature, low β, the spins of adjacent vertices are uncorrelated whereas at low temperature adjacent vertices have identical spins. The reason for this is that the probability of a configuration is proportional to e β ∑ x i x j. i∼j As β is increased, for configurations with a large number of edges whose vertices have identical spins, e β ∑ x i x j i∼j increases more than for configurations whose edges have vertices with non identical spins. When the normalization constant 1 Z is adjusted for the new value of β, the highest probability configurations are those where adjacent vertices have identical spins. Given the above probability distribution, what is the correlation between two variables x i and x j . To answer this question, consider the probability that x i equals plus one as a function of the probability that x j equals plus one. If the probability that x i equals plus one is 1 2 independent of the value of the probability that x j equals plus one, we say the values are uncorrelated. Consider the special case where the graph G is a tree. In this case a phase transition occurs at β 0 = 1 d+1 ln where d is the degree of the tree. For a sufficiently tall tree and for 2 d−1 β > β 0 , the probability that the root has value +1 is bounded away from 1 / 2 and depends on whether the majority of leaves have value +1 or -1. For β < β 0 the probability that the root has value +1 is 1 / 2 independent of the values at the leaves of the tree. Consider a height one tree of degree d. If i of the leaves have spin +1 and d − i have spin -1, then the probability of the root having spin +1 is proportional to e iβ−(d−i)β = e (2i−d)β . 323
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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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Page 325 and 326: Now consider a very tall tree. If t
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Page 333 and 334: In the discrete case, x = [x 0 , x
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Page 353 and 354: Exercise 10.18 Repeat the above exe
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Page 359 and 360: 11.3 Wavelet Systems So far we have
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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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