Foundations of Data Science

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Definition 3.1 If p is a probability density in d space, the best fit line for p is the line l = {cv 1 : c ∈ R} where [ v 1 = arg max E (v T x) 2] . |v|=1 x∼p For a spherical Gaussian centered at the origin, it is easy to see that any line passing through the origin is a best fit line. Our next lemma shows that the best fit line for a spherical Gaussian centered at µ ≠ 0 is the line passing through µ and the origin. Lemma 3.16 Let the probability density p be a spherical Gaussian with center µ ≠ 0. The unique best fit 1-dimensional subspace is the line passing through µ and the origin. If µ = 0, then any line through the origin is a best-fit line. Proof: For a randomly chosen x (according to p) and a fixed unit length vector v, [ E (v T x) 2] [ (v = E T (x − µ) + v T µ ) ] 2 x∼p x∼p [ (v = E T (x − µ) ) 2 ( + 2 v T µ ) ( v T (x − µ) ) + ( v T µ ) ] 2 x∼p = E x∼p [ (v T (x − µ) ) 2 ] + 2 ( v T µ ) E [ v T (x − µ) ] + ( v T µ ) 2 = E x∼p [ (v T (x − µ) ) 2 ] + ( v T µ ) 2 = σ 2 + ( v T µ ) 2 where the fourth line follows from the fact that E[v T (x − µ)] = 0, and the fifth line follows from the fact that E[(v T (x − µ)) 2 ] is the variance in the direction v. The best fit line v maximizes E x∼p [(v T x) 2 ] and therefore maximizes ( v T µ ) 2 . This is maximized when v is aligned with the center µ. To see uniqueness, just note that if µ ≠ 0, then v T µ is strictly less when v is not aligned with the center. We now extend Definition 3.1 to k-dimensional subspaces. Definition 3.2 If p is a probability density in d-space then the best-fit k-dimensional subspace V k is [ V k = argmax E |proj(x, V )| 2 ] , V :dim(V )=k x∼p where proj(x, V ) is the orthogonal projection of x onto V . Lemma 3.17 For a spherical Gaussian with center µ, a k-dimensional subspace is a best fit subspace if and only if it contains µ. Proof: If µ = 0, then by symmetry any k−dimensional subspace is a best-fit subspace. If µ ≠ 0, then, the best-fit line must pass through µ by Lemma (3.16). Now, as in the greedy algorithm for finding subsequent singular vectors, we would project perpendicular to the first singular vector. But after the projection, the mean of the Gaussian becomes 0 and any vectors will do as subsequent best-fit directions. 58
This leads to the following theorem. Theorem 3.18 If p is a mixture of k spherical Gaussians, then the best fit k-dimensional subspace contains the centers. In particular, if the means of the Gaussians are linearly independent, the space spanned by them is the unique best-fit k dimensional subspace. Proof: Let p be the mixture w 1 p 1 +w 2 p 2 +· · ·+w k p k . Let V be any subspace of dimension k or less. Then, [ E ] |proj(x, V )| 2 = x∼p k∑ i=1 w i E x∼pi [ |proj(x, V )| 2 ] If V contains the centers of the densities p i , by Lemma 3.17, each term in the summation is individually maximized, which implies the entire summation is maximized, proving the theorem. For an infinite set of points drawn according to the mixture, the k-dimensional SVD subspace gives exactly the space of the centers. In reality, we have only a large number of samples drawn according to the mixture. However, it is intuitively clear that as the number of samples increases, the set of sample points will approximate the probability density and so the SVD subspace of the sample will be close to the space spanned by the centers. The details of how close it gets as a function of the number of samples are technical and we do not carry this out here. 3.9.4 Ranking Documents and Web Pages An important task for a document collection is to rank the documents according to their intrinsic relevance to the collection. A good candidate definition of “intrinsic relevance” is a document’s projection onto the best-fit direction for that collection, namely the top left-singular vector of the term-document matrix. An intuitive reason for this is that this direction has the maximum sum of squared projections of the collection and so can be thought of as a synthetic term-document vector best representing the document collection. Ranking in order of the projection of each document’s term vector along the best fit direction has a nice interpretation in terms of the power method. For this, we consider a different example, that of the web with hypertext links. The World Wide Web can be represented by a directed graph whose nodes correspond to web pages and directed edges to hypertext links between pages. Some web pages, called authorities, are the most prominent sources for information on a given topic. Other pages called hubs, are ones that identify the authorities on a topic. Authority pages are pointed to by many hub pages and hub pages point to many authorities. One is led to what seems like a circular definition: a hub is a page that points to many authorities and an authority is a page that is pointed to by many hubs. 59
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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
Page 7 and 8: 12.4.7 Median . . . . . . . . . . .
Page 9 and 10: densities, discrete optimization, e
Page 11 and 12: 2 High-Dimensional Space 2.1 Introd
Page 13 and 14: Also, if x and y are independent, t
Page 15 and 16: 1 1 − ɛ Annulus of width 1 d Fig
Page 17 and 18: integral gives ∫ ∞ 0 e −r2 r
Page 19 and 20: The volume of the hemisphere below
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Page 23 and 24: Using the substitution 2z = y 2 , |
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Page 27 and 28: √ √ 2d+O(1) ≤ 2d + ∆2 −O(
Page 29 and 30: 2.10 Bibliographic Notes The word v
Page 31 and 32: Exercise 2.9 A 3-dimensional cube h
Page 33 and 34: Exercise 2.26 Explain how the volum
Page 35 and 36: Exercise 2.40 Consider a non orthog
Page 37 and 38: 1 Exercise 2.50 Use the probability
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Page 41 and 42: 3.3 Singular Vectors We now define
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Page 51 and 52: Proof: Let A = r∑ σ i u i vi T i
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Page 55 and 56: such models. Mixture models are a v
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Page 67 and 68: Exercise 3.18 Modify the power meth
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Page 75 and 76: Prob(vertex has degree k) = ( ) n
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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
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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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