Foundations of Data Science

Recommendations

Info

Theorem 4.11 Consider a tree generated by a branching process. Let f(x) be the generating function for the number of children at each node. 1. If the expected number of children at each node is less than or equal to one, then the probability of extinction is one unless the probability of exactly one child is one. 2. If the expected number of children of each node is greater than one, then the probability of extinction is the unique solution to f(x) = x in [0, 1). Proof: Let p i be the probability of i children at each node. Then f(x) = p 0 + p 1 x + p 2 x 2 + · · · is the generating function for the number of children at each node and f ′ (1) = p 1 + 2p 2 + 3p 3 + · · · is the slope of f(x) at x = 1. Observe that f ′ (1) is the expected number of children at each node. Since the expected number of children at each node is the slope of f(x) at x = 1, if the expected number of children is less than or equal to one, the slope of f(x) at x = 1 is less than or equal to one and the unique root of f(x) = x in (0, 1] is at x = 1 and the probability of extinction is one unless f ′ (1) = 1 and p 1 = 1. If f ′ (1) = 1 and p 1 = 1, f(x) = x and the tree is an infinite degree one chain. If the slope of f(x) at x = 1 is greater than one, then the probability of extinction is the unique solution to f(x) = x in [0, 1). A branching process with m 1, then the branching process will die out with some probability less than one unless p 0 = 0 in which case it cannot die out, since a node always has at least one descendent. Note that the branching process corresponds to finding the size of a component in an infinite graph. In a finite graph, the probability distribution of descendants is not a constant as more and more vertices of the graph get discovered. The simple branching process defined here either dies out or goes to infinity. In biological systems there are other factors, since processes often go to stable populations. One possibility is that the probability distribution for the number of descendants of a child depends on the total population of the current generation. Expected size of extinct families We now show that the expected size of an extinct family is finite, provided that m ≠ 1. Note that at extinction, the size must be finite. However, the expected size at extinction could conceivably be infinite, if the probability of dying out did not decay fast enough. To see how the expected value of a random variable that is always finite could be infinite, let x be an integer valued random variable. Let p i be the probability that x = i. If ∞∑ p i = 1, then with probability one, x will be finite. However, the expected value of x i=1 100
may be infinite. That is, ∞∑ i=0 ip i = ∞. For example, if for i >0, p i = 6 π 2 1 i 2 , then ∞ ∑ i=1 p i = 1, ∑ but ∞ ip i = ∞. The value of the random variable x is always finite, but its expected i=1 value is infinite. This does not happen in a branching process, except in the special case where the slope m = f ′ (1) equals one and p 1 ≠ 1 Lemma 4.12 If the slope m = f ′ (1) does not equal one, then the expected size of an extinct family is finite. If the slope m equals one and p 1 = 1, then the tree is an infinite degree one chain and there are no extinct families. If m=1 and p 1 < 1, then the expected size of the extinct family is infinite. Proof: Let z i be the random variable denoting the size of the i th generation and let q be the probability of extinction. The probability of extinction for a tree with k children in the first generation is q k since each of the k children has an extinction probability of q. Note that the expected size of z 1 , the first generation, over extinct trees will be smaller than the expected size of z 1 over all trees since when the root node has a larger number of children than average, the tree is more likely to be infinite. By Bayes rule Prob (z 1 = k|extinction) = Prob (z 1 = k) Prob (extinction|z 1 = k) Prob (extinction) = p k q k q = p kq k−1 . Knowing the probability distribution of z 1 given extinction, allows us to calculate the expected size of z 1 given extinction. ∞∑ E (z 1 |extinction) = kp k q k−1 = f ′ (q) . k=0 We now prove, using independence, that the expected size of the i th generation given extinction is E (z i |extinction) = ( f ′ (q)) i . For i = 2, z 2 is the sum of z 1 independent random variables, each independent of the random variable z 1 . So, E(z 2 |z 1 = j and extinction) = E( sum of j copies of z 1 |extinction) = jE(z 1 |extinction). Summing over all values of j ∞∑ E(z 2 |extinction) = E(z 2 |z 1 = j and extinction)Prob(z 1 = j|extinction) = j=1 ∞∑ jE(z 1 |extinction)Prob(z 1 = j|extinction) j=1 = E(z 1 |extinction) ∞∑ jProb(z 1 = j|extinction) = E 2 (z 1 |extinction). j=1 101
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: f(x) q p 0 f(f(x)) f(x) x Figure 4.
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 and 276:
Theorem 8.5 Assume A satisfies (c,
Page 277 and 278:
probability is q (where, q < p). 32
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?