Foundations of Data Science

Recommendations

Info

f(x) m < 1 m > 1 p 0 m = 1 and p 1 < 1 q x Figure 4.10: Illustration of the root of equation f(x) = x in the interval [0,1). Let q be the probability that the branching process dies out. If there are i children in the first generation, then each of the i subtrees must die out and this occurs with probability q i . Thus, q equals the summation over all values of i of the product of the probability of i children times the probability that i subtrees will die out. This gives q = ∑ ∞ i=0 p iq i . Thus, q is the root of x = ∑ ∞ i=0 p ix i , that is x = f(x). This suggests focusing on roots of the equation f(x) = x in the interval [0,1]. The value ∑ x = 1 is always a root of the equation f (x) = x since f (1) = ∞ p i = 1. When is there a smaller nonnegative root? The derivative of f (x) at x = 1 is f ′ (1) = p 1 + 2p 2 + 3p 3 + · · · . Let m = f ′ (1). Thus, m is the expected number of children of a node. If m > 1, one might expect the tree to grow forever, since each node at time j is expected to have more than one child. But this does not imply that the probability of extinction is zero. In fact, if p 0 > 0, then with positive probability, the root will have no children and the process will become extinct right away. Recall that for G(n, d ), the expected number of children n is d, so the parameter m plays the role of d. If m < 1, then the slope of f (x) at x = 1 is less than one. This fact along with convexity of f (x) implies that f (x) > x for x in [0, 1) and there is no root of f(x) = x in the interval [0, 1). If m = 1 and p 1 < 1, then once again convexity implies that f(x) > x for x ∈ [0, 1) and there is no root of f(x) = x in the interval [0, 1). If m = 1 and p 1 = 1, then f(x) is the straight line f(x) = x. If m >1, then the slope of f (x) is greater than the slope of x at x = 1. This fact, along with convexity of f (x), implies f (x) = x has a unique root in [0,1). When p 0 = 0, the root is at x = 0. Let q be the smallest nonnegative root of the equation f(x) = x. For m < 1 and for i=0 98
f(x) q p 0 f(f(x)) f(x) x Figure 4.11: Illustration of convergence of the sequence of iterations f 1 (x), f 2 (x), . . . to q. m=1 and p 0 < 1, q equals one and for m >1, q is strictly less than one. We shall see that the value of q is the extinction probability of the branching process and that 1 − q is the immortality probability. That is, q is the probability that for some j, the number of children in the j th generation is zero. To see this, note that for m > 1, lim f j (x) = q for j→∞ 0 ≤ x < 1. Figure 4.11 illustrates the proof which is given in Lemma 4.10. Similarly note that when m < 1 or m = 1 with p 0 < 1, f j (x) approaches one as j approaches infinity. Lemma 4.10 Assume m > 1. Let q be the unique root of f(x)=x in [0,1). In the limit as j goes to infinity, f j (x) = q for x in [0, 1). Proof: If 0 ≤ x ≤ q, then x < f(x) ≤ f(q) and iterating this inequality x < f 1 (x) < f 2 (x) < · · · < f j (x) < f (q) = q. Clearly, the sequence converges and it must converge to a fixed point where f (x) = x. Similarly, if q ≤ x < 1, then f(q) ≤ f(x) < x and iterating this inequality x > f 1 (x) > f 2 (x) > · · · > f j (x) > f (q) = q. In the limit as j goes to infinity f j (x) = q for all x, 0 ≤ x < 1. ∑ Recall that f j (x) is the generating function ∞ Prob (z j = i) x i . The fact that in the limit the generating function equals the constant q, and is not a function of x, says that Prob (z j = 0) = q and Prob (z j = i) = 0 for all finite nonzero values of i. The remaining probability is the probability of a nonfinite component. Thus, when m >1, q is the extinction probability and 1-q is the probability that z j grows without bound, i.e., immortality. 99 i=0
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: are O(ln n) in size and the expecte
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 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?