Data-Powered%20Algorithms

1
Data-Powered Algorithms

• Bernard Chazelle
• Princeton University

2

• Tools

3
Linear Programming
4
(No Transcript)
5
(No Transcript)
6
(No Transcript)
7
(No Transcript)
8
(No Transcript)
9
(No Transcript)
10
(No Transcript)
11
(No Transcript)
12
N constraints and d variables
13
N constraints and d variables
14
Dimension Reduction
?25
?10000
Images (face recognition) Signals (voice
recognition) Text (NLP) . . . Nearest neighbor
searching Clustering . . .
15
Dimension reduction
All pairwise distances nearly preserved
16
Johnson-Lindenstrauss Transform (JLT)
d
v
Random Orthogonal Matrix
c log n ?2
d
17
Friendly JLT
d
N(0,1)
N(0,1)
N(0,1)
N(0,1)
N(0,1)
N(0,1)
N(0,1)
N(0,1)
c log n ?2
N(0,1)
N(0,1)
N(0,1)
N(0,1)
N(0,1)
N(0,1)
N(0,1)
N(0,1)
18
Friendlier JLT
d
c log n ?2
d log n ?2
?( )
19
Sparse JLT ?
d
0
. . .
0
0
0
0
0
0
c log n ?2
1
d
0
0
0
0
0
. . .
o(1)-Fraction non-zeros
0
20
Main Tool Uncertainty Principle
Heisenberg
Time
Frequency
21
Fast Johnson-Lindenstrauss Transform (FJLT)
d
d
d
Discrete Fourier Transform
0 N(0,1)
d
. . .
Optimal ??
22

• Data-Powered

• Algorithms

23
theory
experimentation
24
theory
experimentation
computation
25
theory
experimentation

• 1950...

computation
26
input
output
Most interesting problems are too hard !!
27
input
output
So, we change the model
randomization
approximation
28
input
output
PTAS for ETSP
randomization
approximation
29
input
output
Impossible to approximate chromatic number
within a factor of
randomization
approximation
30
input
output
Berkeley school (program checking
probabilistic proofs)
randomization
Property Testing RS96, GGR96
approximation
31

• Property Testing

32
Distance is 3
33
Distance is 4

• edit distance

34
no
bipartite
yes
35
no
anything
bipartite
yes
GR97
36
(No Transcript)
37
(No Transcript)
38
Mixing case
18
17
7
62

• bipartite!

• non-bipartite!

polylog cycles
Birthday paradox
39
Non-mixing case
Nonmixing implies small cuts
M89
40
Dense graphs
Is graph k-colorable?
GGR98, AK99
41
Main tool
Szemerédis Regularity Lemma
Far from k-colorable
Lots of witnesses
42
Property Testing
http//www.cs.princeton.edu/chazelle/

• Graph algorithms
• connectivity
• acyclicity
• k-way cuts
• clique
• Distributions
• independence
• entropy
• monotonicity
• distances
• Geometry
• convexity
• disjointness
• delaunay
• plane EMST