Foundations of Data Science

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11.8 Expressing a Function in Terms of Wavelets Given a wavelet system with scale function φ and mother wavelet ψ we wish to express a function f(x) in terms of an orthonormal basis of the wavelet system. First we will express f(x) in terms of scale functions φ jk (x) = φ(2 j x − k). To do this we will build a tree similar to that in Figure 11.2 for the Haar system only computing the coefficients will be much more complex. Recall that the coefficients at a level in the tree are the coefficients to represent f(x) using scale functions with the precision of the level. Let f(x) = ∑ ∞ k=0 a jkφ j (x − k) where the a jk are the coefficients in the expansion of f(x) using level j scale functions. Since the φ j (x − k) are orthogonal a jk = Expanding φ j in terms of φ j+1 yields a jk = = ∫ ∞ x=−∞ ∑d−1 m=0 ∫ ∞ x=−∞ f(x) c m ∫ ∞ x=−∞ ∑d−1 m=0 ∑d−1 = c m a j+1,2k+m m=0 Let n = 2k + m. Now m = n − 2k. Then a jk = f(x)φ j (x − k)dx. c m φ j+1 (2x − 2k − m)dx f(x)φ j+1 (2x − 2k − m)dx ∑d−1 n=2k c n−2k a j+1,n (11.4) In construction the tree similar to that in Figure 11.2, the values at the leaves are the values of the function sampled in the intervals of size 2 −j . Equation 11.4 is used to compute values as one moves up the tree. The coefficients in the tree could be used if we wanted to represent f(x) using scale functions. However, we want to represent f(x) using one scale function whose scale is the support of f(x) along with wavelets which gives us an orthogonal set of basis functions. To do this we need to calculate the coefficients for the wavelets.. The value at the root of the tree is the coefficient for the scale function. We then move down the tree calculating the coefficients for the wavelets. Finish by calculating wavelet coefficients maybe add material on jpeg Example: Add example using D 4 . Maybe example using sinc 370
11.9 Designing a Wavelet System In designing a wavelet system there are a number of parameters in the dilation equation. If one uses d terms in the dilation equation, one degree of freedom can be used to satisfy ∑d−1 c i = 2 i=0 which insures the existence of a solution with a nonzero mean. Another d degrees of 2 freedom are used to satisfy ∑d−1 c i c i−2k = δ(k) i=0 which insures the orthogonal properties. The remaining d − 1 degrees of freedom can be 2 used to obtain some desirable properties such as smoothness. Smoothness appears to be related to vanishing moments of the scaling function. Material on the design of systems is beyond the scope of this book and can be found in the literature. 371
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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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Page 325 and 326: Now consider a very tall tree. If t
Page 327 and 328: 9.14 Exercises Exercise 9.1 Find a
Page 329 and 330: 10 Other Topics 10.1 Rankings Ranki
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Page 333 and 334: In the discrete case, x = [x 0 , x
Page 335 and 336: Figure 10.3: Some subgradients for
Page 337 and 338: 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: Example of orthogonality when wavel
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
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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
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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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