Foundations of Data Science

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5.3 Areas and Volumes Computing areas and volumes is a classical problem. For many regular figures in two and three dimensions there are closed form formulae. In Chapter 2, we saw how to compute volume of a high dimensional sphere by integration. For general convex sets in d-space, there are no closed form formulae. Can we estimate volumes of d-dimensional convex sets in time that grows as a polynomial function of d? The MCMC method answes this question in the affirmative. One way to estimate the area of the region is to enclose it in a rectangle and estimate the ratio of the area of the region to the area of the rectangle by picking random points in the rectangle and seeing what proportion land in the region. Such methods fail in high dimensions. Even for a sphere in high dimension, a cube enclosing the sphere has exponentially larger area, so exponentially many samples are required to estimate the volume of the sphere. It turns out that the problem of estimating volumes of sets is reducible to the problem of drawing uniform random samples from sets. Suppose one wants to estimate the volume of a convex set R. Create a concentric series of larger and larger spheres S 1 , S 2 , . . . , S k such that S 1 is contained in R and S k contains R. Then Vol(R) = Vol(S k ∩ R) = Vol(S k ∩ R) Vol(S k−1 ∩ R) Vol(S k−1 ∩ R) Vol(S k−2 ∩ R) · · · Vol(S 2 ∩ R) Vol(S 1 ∩ R) Vol(S 1) If the radius of the sphere S i is 1 + 1 d times the radius of the sphere S i−1, then the value of Vol(S k−1 ∩ R) Vol(S k−2 ∩ R) can be estimated by rejection sampling provided one can select points at random from a d-dimensional region. Since the radii of the spheres grows as 1+ 1 , the volume of a sphere d is ( 1 + d) 1 d < e times the volume of the preceding sphere and the number of spheres is at most O(log 1+(1/d) r) = O(rd) where r is the ratio of the radius of S k to the radius of S 1 . It remains to show how to draw a uniform random sample from a d-dimensional set. It is at this point that we require the set to be convex so that the Markov chain technique will converge quickly to its stationary probability. To select a random sample from a d-dimensional convex set, impose a grid on the region and do a random walk on the grid points. At each time, pick one of the 2d coordinate neighbors of the current grid point, each with probability 1/(2d) and go to the neighbor if it is still in the set; otherwise, stay put and repeat. If the grid length in each of the d coordinate directions is at most some a, the total number of grid points in the set is at most a d . Although this is exponential in d, the Markov chain turns out to be rapidly mixing (the proof is beyond our scope here) 150
Figure 5.4: A network with a constriction. and leads to polynomial time bounded algorithm to estimate the volume of any convex set in R d . 5.4 Convergence of Random Walks on Undirected Graphs The Metropolis-Hasting algorithm and Gibbs sampling both involve a random walk. Initial states of the walk are highly dependent on the start state of the walk. Both these walks are random walks on edge-weighted undirected graphs. Such Markov chains are derived from electrical networks. Recall the following notation which we will use throughout this section. Given a network of resistors, the conductance of edge (x, y) is denoted c xy and the normalizing constant c x equals ∑ y c xy. The Markov chain has transition probabilities p xy = c xy /c x . We assume the chain is connected. Since c x p xy = c xy = c yx = c y c yx /c y = c y p xy the stationary probabilities are proportional to c x where the normalization constant is c 0 = ∑ x c x. An important question is how fast the walk starts to reflect the stationary probability of the Markov process. If the convergence time was proportional to the number of states, the algorithms would not be very useful since the number of states can be exponentially large. There are clear examples of connected chains that take a long time to converge. A chain with a constriction, see Figure 5.4, takes a long time to converge since the walk is unlikely to reach the narrow passage between the two halves, both of which are reasonably big. We will show in Theorem 5.5 that the time to converge is quantitatively related to 151
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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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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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Page 125 and 126: Destination Figure 4.17: For d < 2,
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Page 131 and 132: Exercise 4.7 Let f (n) be a functio
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Page 135 and 136: Exercise 4.38 Consider a model of a
Page 137 and 138: Exercise 4.53 Consider graph 3-colo
Page 139 and 140: 5 Random Walks and Markov Chains A
Page 141 and 142: A B C (a) A B C (b) Figure 5.1: (a)
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Page 171 and 172: Theorem 5.13 Let G be an undirected
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Page 181 and 182: Exercise 5.10 Let p be a probabilit
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Page 187 and 188: Exercise 5.40 Suppose that the cliq
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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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