Foundations of Data Science

Recommendations

Info

⎛ ⎜ ⎝ 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 adjacency matrix ⎞ ⎟ ⎠ ⎛ ⎜ ⎝ 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 singular vectors ⎞ ⎫ ⎪⎬ ⎪⎭ ⎫ ⎪⎬ ⎪⎭ ⎟ } ⎠ Figure 8.3: In spectral clustering the vertices of a graph, one finds a few top singular vectors of the adjacency matrix and forms a new matrix whose columns are these singular vectors. The rows of this matrix are points in a lower dimensional space. These points can be clustered with k-means cluster. In the above example there are three clusters that are cliques with no connecting vertices. If one removed a few edges from the cliques and added a few edges connecting vertices in different cliques, the rows of a clique would map to clusters of points as shown in the top right rather than all to a single point. Sometimes there is clearly a correct target clustering we want to find, but approximately optimal k-means or k-median clusterings can be very far from the target. We describe two important stochastic models where this situation arises. In these cases as well as many others, a clustering algorithm based on SVD, called Spectral Clustering is useful. Spectral Clustering first projects data points onto the space spanned by the top k singular vectors of the data matrix and works in the projection. It is widely used and indeed finds a clustering close to the target clustering for data arising from many stochastic models. We will prove this in a generalized setting which includes both stochastically generated data and data with no stochastic models. 8.6.1 Stochastic Block Model Stochastic Block Models are models of communities. Suppose there are k communities C 1 , C 2 , . . . , C k among a population of n people. Suppose the probability of two people in the same community knowing each other is p and if they are in different communities, the 276
probability is q (where, q < p). 32 We assume the events that person i knows person j are independent across all i and j. Specifically, we are given an n × n data matrix A, where, a ij = 1 if and only if i and j know each other. We assume the a ij are independent random variables, and use a i to denote the i th row of A. It is useful to think of A as the adjacency matrix of a graph, such as the friendship network in Facebook. We will also think of the rows a i as data points. The clustering problem is to classify the data points into the communities they belong to. In practice, the graph is fairly sparse, i.e., p and q are small, namely, O(1/n) or O(ln n/n). Consider the simple case of two communities with n/2 people in each and with p = α n ; q = β n , where, α, β ∈ O(ln n). Let u and v be the centroids of the data points in community one and community two respectively; so, u i ≈ p for i ∈ C 1 and u j ≈ q for j ∈ C 2 and v i ≈ q for i ∈ C 1 and v j ≈ p for j ∈ C 2 . We have |u − v| 2 ≈ n∑ (u j − v j ) 2 = j=1 (α − β)2 n 2 n = (α − β)2 . n Inter-centroid distance ≈ α − β √ n . (8.1) On the other hand, the distance between a data point and its cluster centroid is much greater: For i ∈ C 1 , E ( |a i − u| 2) n∑ = E((a ij − u j ) 2 ) = n [p(1 − p) + q(1 − q)] ∈ Ω(α + β). 2 j=1 We see that if α, β ∈ O(ln n), then the ratio (√ ) distance between cluster centroids ln n distance of data point to its cluster centroid ∈ O √ . n So the centroid of a cluster is much farther from data points in its own cluster than it is to the centroid of the other cluster. Now, consider the k-median objective function. Suppose we wrongly classify a point in C 1 as belonging to C 2 . The extra cost we incur is at most the distance between centroids which is only O( √ ln n/ √ n) times the k-median cost of the data point. So just by examining the cost, we cannot rule out a ε-approximate k-median clustering from misclassifying all points. A similar argument can also be made about k-means objective function. 32 More generally, for each pair of communities a and b, there could be a probability p ab that a person from community a knows a person from community b. But for the discussion here, we take p aa = p for all a and p ab = q, for all a ≠ b. 277
Page 1 and 2:
Foundations of Data Science ∗ Avr
Page 3 and 4:
4.4 Branching Processes . . . . . .
Page 5 and 6:
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
Page 21 and 22:
points onto the circle. In higher d
Page 23 and 24:
Using the substitution 2z = y 2 , |
Page 25 and 26:
For the nearest neighbor problem, i
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
Page 39 and 40:
This decomposition of A can be view
Page 41 and 42:
3.3 Singular Vectors We now define
Page 43 and 44:
orthonormal basis w 1 , w 2 , . . .
Page 45 and 46:
A n × d = U n × r D r × r V T r
Page 47 and 48:
3.6 Left Singular Vectors Theorem 3
Page 49 and 50:
Lemma 3.10 (Analog of eigenvalues a
Page 51 and 52:
Proof: Let A = r∑ σ i u i vi T i
Page 53 and 54:
a i , let dist(a i , l) denote its
Page 55 and 56:
such models. Mixture models are a v
Page 57 and 58:
1. The best fit 1-dimension subspac
Page 59 and 60:
This leads to the following theorem
Page 61 and 62:
Thus, the maximum cut problem can b
Page 63 and 64:
and this meets the claimed error bo
Page 65 and 66:
(0,3) (1,1) (3,0) ⎛ M = ⎝ 1 1 0
Page 67 and 68:
Exercise 3.18 Modify the power meth
Page 69 and 70:
2. What percent of the Forbenius no
Page 71 and 72:
City Bei- Tian- Shang- Chong- Hoh-
Page 73 and 74:
5 10 15 20 25 30 35 40 4 9 14 19 24
Page 75 and 76:
Prob(vertex has degree k) = ( ) n
Page 77 and 78:
and thus k k ≤ n. Since k! ≤ k
Page 79 and 80:
For x to be equal to zero, it must
Page 81 and 82:
No items E(x) ≥ 0.1 At least one
Page 83 and 84:
Our first task is to figure out wha
Page 85 and 86:
√ Thus, the probability for all s
Page 87 and 88:
Figure 4.7: A degree three vertex w
Page 89 and 90:
ftp://ftp.cs.rochester.edu/pub/u/jo
Page 91 and 92:
Size of frontier 0 ln n nθ n Numbe
Page 93 and 94:
For a small number i of steps, the
Page 95 and 96:
same or are disjoint, ⎛( n∑ )
Page 97 and 98:
are O(ln n) in size and the expecte
Page 99 and 100:
f(x) q p 0 f(f(x)) f(x) x Figure 4.
Page 101 and 102:
may be infinite. That is, ∞∑ i=
Page 103 and 104:
Property cycles 1/n giant component
Page 105 and 106:
Keep in mind that the leading terms
Page 107 and 108:
4.6 Phase Transitions for Increasin
Page 109 and 110:
p(n) A symmetric argument shows tha
Page 111 and 112:
simple. We supply some intuition be
Page 113 and 114:
Proof: Say that t is a “busy time
Page 115 and 116:
Consider a graph in which half of t
Page 117 and 118:
problem contributes a minuscule err
Page 119 and 120:
one. On the other hand, for δ = 1,
Page 121 and 122:
for all δ greater than δ critical
Page 123 and 124:
expected degree d i = 1, 2, 3 √ t
Page 125 and 126:
Destination Figure 4.17: For d < 2,
Page 127 and 128:
For at least half of the pairs of v
Page 129 and 130:
S i+1 is (1 − 1 ) |Si| ≤ 1 −
Page 131 and 132:
Exercise 4.7 Let f (n) be a functio
Page 133 and 134:
Exercise 4.19 (Birthday problem) Wh
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)
Page 143 and 144:
in time polynomial in n. A quantity
Page 145 and 146:
5.2 Markov Chain Monte Carlo The Ma
Page 147 and 148:
p(a) = 1 2 p(b) = 1 4 p(c) = 1 8 p(
Page 149 and 150:
5 8 7 12 1 3 1 6 1 8 1 3 3,1 3,2 3,
Page 151 and 152:
Figure 5.4: A network with a constr
Page 153 and 154:
There is a simple interpretation of
Page 155 and 156:
Let γ i = π 1 + π 2 + · · · +
Page 157 and 158:
5.4.1 Using Normalized Conductance
Page 159 and 160:
The Gaussian distribution on the in
Page 161 and 162:
6 6 1 8 1 5 8 4 5 5 Graph with boun
Page 163 and 164:
and then reaching a from y before r
Page 165 and 166:
every vertex? Hitting time The hitt
Page 167 and 168:
clique of size n/2 x y } {{ } n/2 F
Page 169 and 170:
i ↑ ↓ ↑ ↓ ↑ j ↑ =⇒ i
Page 171 and 172:
Theorem 5.13 Let G be an undirected
Page 173 and 174:
0 1 2 3 4 12 20 Number of resistors
Page 175 and 176:
1 2 4 Figure 5.11: Paths obtained f
Page 177 and 178:
whereas, with k self-loops, the equ
Page 179 and 180:
or p[I − (1 − α)A] = α (1, 1,
Page 181 and 182:
Exercise 5.10 Let p be a probabilit
Page 183 and 184:
i 1 i 2 R 1 R 2 R 3 Figure 5.13: An
Page 185 and 186:
(a) 1 2 3 4 (b) 1 2 3 4 (c) 1 2 3 4
Page 187 and 188:
Exercise 5.40 Suppose that the cliq
Page 189 and 190:
Exercise 5.55 Using a web browser b
Page 191 and 192:
will allow us to make mathematical
Page 193 and 194:
Not spam { }} { { Spam }} { x 1 x 2
Page 195 and 196:
1 ∨ x 8 = 1}, or more succinctly,
Page 197 and 198:
described using O(k log d) bits: lo
Page 199 and 200:
In fact, we can show this bound is
Page 201 and 202:
y definition of w ∗ . Similarly,
Page 203 and 204:
y x φ Figure 6.4: Data that is not
Page 205 and 206:
Theorem 6.11 (Online to Batch via R
Page 207 and 208:
function of concept class H, will g
Page 209 and 210:
Corollary 6.16 (VC-dimension sample
Page 211 and 212:
A sphere in d-dimensions is a set o
Page 213 and 214:
then with probability ≥ 1 − δ,
Page 215 and 216:
work on a variety of measures. One
Page 217 and 218:
Since weight(0) = n, the total weig
Page 219 and 220:
Stochastic Gradient Descent: Given:
Page 221 and 222:
sleeping experts problem. Combining
Page 223 and 224:
⎫ ⎪⎬ ⎪⎭ Each gate is conn
Page 225 and 226: W 1 W 2 W 1 W 2 W 3 (a) (b) Figure
Page 227 and 228: convolution pooling Image Convoluti
Page 229 and 230: discard separators in advance that
Page 231 and 232: 6.14.2 Active learning Active learn
Page 233 and 234: 6.16 Exercises Exercise 6.1 (Sectio
Page 235 and 236: √ L log(L(T )) |S|) will be at mo
Page 237 and 238: 7 Algorithms for Massive Data Probl
Page 239 and 240: important to know if some popular s
Page 241 and 242: We now give an example of a 2-unive
Page 243 and 244: estimate of m. To obtain a coin tha
Page 245 and 246: expected value of the sum will be z
Page 247 and 248: δ have values in ±1. There are ex
Page 249 and 250: mean. (ii) There is “no free lunc
Page 251 and 252: ⎡ ⎢ ⎣ A m × n ⎤ ⎡ ⎥
Page 253 and 254: One uses the primitive in Section ?
Page 255 and 256: would require s ≥ n, which clearl
Page 257 and 258: its that capture sufficient informa
Page 259 and 260: 7.6 Exercises Algorithms for Massiv
Page 261 and 262: Counting Frequent Elements The Majo
Page 263 and 264: Exercise 7.30 Blast: Given a long s
Page 265 and 266: Preliminaries: We will follow the s
Page 267 and 268: B A Figure 8.1: Example where the n
Page 269 and 270: Lloyd’s algorithm: Start with k c
Page 271 and 272: 8.2.5 k-means clustering on the lin
Page 273 and 274: a clustering that is ɛ-close to C
Page 275: Theorem 8.5 Assume A satisfies (c,
Page 279 and 280: 8.6.4 Spectral Clustering Algorithm
Page 281 and 282: But for l ≠ l ′ , by hypothesis
Page 283 and 284: 4. the σ-interior of the clusters
Page 285 and 286: Figure 8.4: Example of a bipartite
Page 287 and 288: are likely to be balanced given tha
Page 289 and 290: s 1 ∞ ∞ λ t ∞ edges and vert
Page 291 and 292: A clustering algorithm is consisten
Page 293 and 294: less than n, then richness is not r
Page 295 and 296: 8.13 Exercises Exercise 8.1 Constru
Page 297 and 298: A B Figure 8.8: insert caption Exer
Page 299 and 300: 9 Topic Models, Hidden Markov Proce
Page 301 and 302: Nonnegative matrix factorization (N
Page 303 and 304: This is a linear program. As we rem
Page 305 and 306: for t = 1 to T Prob(O 0 O 1 · ·
Page 307 and 308: a ij transition probability from st
Page 309 and 310: C 1 S 2 C 2 causes D 1 D 2 diseases
Page 311 and 312: x 1 + x 2 + x 3 x 1 + x 2 x 1 + x 3
Page 313 and 314: x 1 x 1 + x 2 + x 3 x 3 + x 4 + x 5
Page 315 and 316: the messages coming to a variable n
Page 317 and 318: y 2 y 3 x 2 x 3 y 1 x 1 x 4 y 4 y n
Page 319 and 320: two pieces of information, the valu
Page 321 and 322: graph. The vertices of Π are denot
Page 323 and 324: present and ask how correlated this
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
Page 331 and 332:
b . a . a b b b b . . b b a a b b .
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.
Page 339 and 340:
A T S A S is invertible. We will pr
Page 341 and 342:
was included, we would just multipl
Page 343 and 344:
position on genome trees = Phenotyp
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
Page 363 and 364:
the support of both sides of the eq
Page 365 and 366:
and the wavelet functions are ortho
Page 367 and 368:
11.7 Sufficient Conditions for the
Page 369 and 370:
Example of orthogonality when wavel
Page 371 and 372:
11.9 Designing a Wavelet System In
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
Page 377 and 378:
these equalities by derivatives.
Page 379 and 380:
Gaussian and related integrals To v
Page 381 and 382:
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 and 434:
Index 2-universal, 240 4-way indepe
Page 435 and 436:
Lagrange, 423 Law of large numbers,
Page 437 and 438:
References [AK] Sanjeev Arora and R
Page 439:
[MR95b] [MU05] [MV10] Rajeev Motwan
show all

Foundations of Data Science

Create successful ePaper yourself

Delete template?

Save as template?