Foundations of Data Science

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Sample space, 387 Sampling length squared, 250 Satisfying assignments expected number of, 109 Scale function, 355 Scale invariant, 290 Scale vector, 355 Second moment method, 77, 81 Sharp threshold, 80 Similar matrices, 405 Singular value decomposition, 38 Singular vector, 41 first, 41 left, 43 right, 43 second, 41 Six-degrees separation, 124 Sketch matrix, 253 Sketches documents, 256 Small world, 124 Smallest-clause heuristic, 110 Spam, 177 Spectral clustering, 275 Sperner’s lemma, 426 Standard deviation normal distribution, 74 Stanley Milgram, 124 State, 145 Stationary distribution, 143 Stirling approximation, 382 Streaming model, 237 Subgradient, 335 Symmetric matrices, 406 disappearance of isolated vertices, 85 emergence of cycles, 102 emergence of diameter two, 82 giant component plus isolated vertices, 104 Time domain, 340 Total variation distance, 146 Trace, 415 Triangle inequality, 381 Triangles, 77 Union bound, 389 Unit-clause heuristic, 110 Unitary matrix, 412 Unsatisfiability, 109 Variance, 390 variational method, 382 VC-dimension, 206 convex polygons, 209 finite sets, 211 half spaces, 209 intervals, 209 pairs of intervals, 209 rectangles, 209 spheres, 210 Viterbi algorithm, 305 Voltage probabilistic interpretation, 162 Wavelet, 354 World Wide Web, 175 Young’s inequality, 381, 384 Tail bounds, 397 Tail of Gaussian, 387 Taylor series, 377 Threshold, 79 CNF-sat, 109 diameter O(ln n), 106 436
References [AK] Sanjeev Arora and Ravindran Kannan. Learning mixtures of separated nonspherical gaussians. Annals of Applied Probability, 15(1A):6992. [Alo86] Noga Alon. Eigenvalues and expanders. Combinatorica, 6:83–96, 1986. [AM05] [AN72] [AP03] [BA] Dimitris Achlioptas and Frank McSherry. On spectral learning of mixtures of distributions. In COLT, pages 458–469, 2005. Krishna Athreya and P. E. Ney. Branching Processes, volume 107. Springer, Berlin, 1972. Dimitris Achlioptas and Yuval Peres. The threshold for random k-sat is 2 k (ln 2 - o(k)). In STOC, pages 223–231, 2003. Albert-Lszl Barabsi and Rka Albert. Emergence of scaling in random networks. Science, 286(5439). [BMPW98] Sergey Brin, Rajeev Motwani, Lawrence Page, and Terry Winograd. What can you do with a web in your pocket? Data Engineering Bulletin, 21:37–47, 1998. [Bol01] Béla Bollobás. Random Graphs. Cambridge University Press, 2001. [BT87] [CF86] [CHK + ] [Chv92] [DFK91] [DG99] [DS84] Béla Bollobás and Andrew Thomason. Threshold functions. Combinatorica, 7(1):35–38, 1987. Ming-Te Chao and John V. Franco. Probabilistic analysis of two heuristics for the 3-satisfiability problem. SIAM J. Comput., 15(4):1106–1118, 1986. Duncan S. Callaway, John E. Hopcroft, Jon M. Kleinberg, M. E. J. Newman, and Steven H. Strogatz. Are randomly grown graphs really random? 33rd Annual Symposium on Foundations of Computer Science, 24-27 October 1992, Pittsburgh, Pennsylvania, USA. IEEE, 1992. Martin Dyer, Alan Frieze, and Ravindran Kannan. A random polynomial time algorithm for approximating the volume of convex bodies. Journal of the Association for Computing Machinary, 1991. Sanjoy Dasgupta and Anupam Gupta. An elementary proof of the johnsonlindenstrauss lemma. 99(006), 1999. Peter G. Doyle and J. Laurie Snell. Random walks and electric networks, volume 22 of Carus Mathematical Monographs. Mathematical Association of America, Washington, DC, 1984. 437
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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 −
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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
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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
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Page 405 and 406: The first integral is just the stan
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Page 413 and 414: Important special cases are |x| 0 t
Page 415 and 416: Lemma 12.23 Let A be a symmetric ma
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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: Lagrange, 423 Law of large numbers,
Page 439: [MR95b] [MU05] [MV10] Rajeev Motwan
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