Foundations of Data Science

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Disappearance of isolated vertices, 103 Discovery time, 165 Distance total variation, 146 Distribution vertex degree, 72 Document ranking, 59 Effective resistance, 167 Eigenvalue, 405 Eigenvector, 52, 405 Electrical network, 160 Erdös Rényi, 71 Error correcting codes, 246 Escape probability, 164 Euler’s constant, 171 Event, 388 Expected degree vertex, 71 Expected value, 389 Exponential generating function, 421 Extinct families size, 100 Extinction probability, 96, 99 Finite fields, 423 First moment method, 80 Fourier transform, 339, 422 Frequency domain, 340 G(n,p), 71 Gamma function, 17 Gamma function , 383 Gaussian, 21, 391, 423 fitting to data, 27 tail, 387 Gaussians sparating, 25 Generating function, 97 component size, 117 for sum of two variables, 97 Generating functions, 418 Generating points in the unit ball, 20 Geometric mean, 385 Giant component, 72, 80, 85, 87, 103, 104 Gibbs sampling, 147 Graph connecntivity, 102 resistance, 170 Graphical model, 308 Greedy k-clustering, 271 Growth models, 114 nonuniform, 114 with preferential attachment, 122 without preferential attachment, 116 Haar wavelet, 355 Harmonic function, 160 Hash function, 424 universal, 240 Heavy tail, 75 Hidden Markov model, 303 Hitting time, 165, 177 Immortality probability, 99 Incoherent, 339 Increasing property, 80, 107 unsatisfiability, 109 Independence limited way, 246 Independent, 388 Indicator random variable, 83 of triangle, 78 Indicator variable, 389 Ising model, 323 Isolated vertices, 85, 103 number of, 85 Isometry restricted isometry property, 337 Jensen’s inequality, 385 Johnson-Lindenstrauss lemma, 23, 24 k-clustering, 271 k-means clustering algorithm, 267 Kernel methods, 283 Kirchhoff’s law, 162 Kleinberg, 124 434
Lagrange, 423 Law of large numbers, 11, 13 Learning, 190 Linearity of expectation, 77, 389 Lloyd’s algorithm, 267 Local algorithm, 124 Long-term probabilities, 143 m-fold, 107 Markov chain, 140 state, 145 Markov Chain Monte Carlo, 141 Markov random field, 311 Markov’s inequality, 12 Matrix multiplication by sampling, 249 diagonalizable, 405 similar, 405 Maximum cut problem, 60 Maximum likelihood estimation, 396 Maximum likelihood estimator, 27 Maximum principle, 161 MCMC, 141 Mean value theorem, 424 Median, 391 Metropolis-Hastings algorithm, 146 Mixing time, 143 Model random graph, 71 Molloy Reed, 114 Moment generating function, 421 Mutually independent, 388 Nearest neighbor problem, 25 NMF, 301 Nonnegative matrix factorization, 301 Normal distribution standard deviation, 74 Normalized conductance, 143, 152 Number of triangles in G(n, p), 78 Ohm’s law, 162 Orthonormal, 412 Page rank, 175 personalized , 178 Persistent, 140 Phase transition, 80 CNF-sat, 109 nonfinite components, 120 Poisson distribution, 394 Polar coordinates, 15 Polynomial interpolation, 246 Power iteration, 59 Power law distribution, 75 Power method, 49 Power-law distribution, 114 Prüfer, 426 Principle component analysis, 53 Probability density function, 388 Probability distribution function, 388 Psuedo random, 246 Pure-literal heuristic, 110 Queue, 111 arrival rate, 111 Radon, 210 Random graph, 71 Random projection, 23 theorem, 24 Random variable, 387 Random walk Eucleadean space, 171 in three dimensions, 173 in two dimensions, 172 on lattice, 172 undirected graph, 164 web, 175 Rapid Mixing, 145 Real spectral theorem, 407 Recommendation system, 253 Replication, 107 Resistance, 160, 170 efffective, 164 Restart, 175 value, 176 Return time, 176 435
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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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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 −
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 and 390: Random variables x 1 , x 2 , . . .
Page 391 and 392: 12.4.7 Median One often calculates
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: Index 2-universal, 240 4-way indepe
Page 437 and 438: References [AK] Sanjeev Arora and R
Page 439: [MR95b] [MU05] [MV10] Rajeev Motwan
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