Date post: | 20-Dec-2015 |
Category: |
Documents |
View: | 213 times |
Download: | 0 times |
Data-Powered AlgorithmsData-Powered AlgorithmsData-Powered AlgorithmsData-Powered Algorithms
Bernard ChazelleBernard Chazelle
Princeton UniversityPrinceton University
Bernard ChazelleBernard Chazelle
Princeton UniversityPrinceton University
Linear ProgrammingLinear Programming Linear ProgrammingLinear Programming
N constraints and d variablesN constraints and d variables
N constraints and d variablesN constraints and d variables
Dimension ReductionDimension Reduction
1000010000 2525Images (face recognition)Images (face recognition) Signals (voice recognition)Signals (voice recognition)Text (NLP)Text (NLP). . . . . .
Nearest neighbor searchingNearest neighbor searchingClusteringClustering. . .. . .
Dimension reductionDimension reduction
All pairwise distances nearly preserved
Johnson-Lindenstrauss Transform (JLT)
c log nc log n22
dd
Random OrthogonalMatrix
vv dd
Friendly JLTFriendly JLT
c log nc log n22
dd
N(0,1)N(0,1) N(0,1)N(0,1) N(0,1)N(0,1)N(0,1)N(0,1)N(0,1)N(0,1)N(0,1)N(0,1) N(0,1)N(0,1)
N(0,1)N(0,1)
N(0,1)N(0,1) N(0,1)N(0,1) N(0,1)N(0,1)N(0,1)N(0,1)N(0,1)N(0,1)N(0,1)N(0,1) N(0,1)N(0,1)
N(0,1)N(0,1)
Friendlier JLTFriendlier JLT
c log nc log n22
dd
11++-- 11++-- 11++-- 11++--11++--11++--11++--11++--
11++--11++-- 11++--
11++-- 11++--11++-- 11++--
11++--
d log nd log n 22 = =
Sparse JLTSparse JLT? ?
c log nc log n22
11++--11++--11++--
11++-- 11++--11++--
11++--
00
00
00
00
00
00
0000
00
dd
11 dd
00
00
00
00
. .
..
. .
. .
..
. .
o(1)-Fraction non-o(1)-Fraction non-zeroszeros
Main Tool: Uncertainty Main Tool: Uncertainty PrinciplePrinciple
TimeTime
FrequencyFrequency
HeisenbergHeisenberg
Fast Johnson-Lindenstrauss Transform (FJLT)Fast Johnson-Lindenstrauss Transform (FJLT)
1+- 1+- 1+-
1+-
dd
DiscreteFourier
Transform
dddd
c log nc log n22
. . .
0N(0,1)
= =OO+ d log d + d + d log d + d loglog33 n n 22
dd
OptimalOptimal?? ??
theory experimentation
computation
theory experimentation
computation
theory experimentation
inputinput outputoutput
Most interestingMost interestingproblems areproblems are
too hard !!too hard !!
Most interestingMost interestingproblems areproblems are
too hard !!too hard !!
inputinput outputoutput
randomizationrandomization
approximationapproximation
So, we change So, we change the model…the model…
So, we change So, we change the model…the model…
inputinput outputoutput
randomizationrandomization
approximationapproximationPTAS for ETSPPTAS for ETSPPTAS for ETSPPTAS for ETSP
inputinput outputoutput
randomizationrandomization
approximationapproximation
Impossible toImpossible toapproximateapproximate chromatic chromatic
number withinnumber withina factor of… a factor of…
Impossible toImpossible toapproximateapproximate chromatic chromatic
number withinnumber withina factor of… a factor of…
inputinput outputoutput
randomizationrandomization
approximationapproximationProperty Property TestingTesting
[RS’96, [RS’96, GGR’96]GGR’96]
Property Property TestingTesting
[RS’96, [RS’96, GGR’96]GGR’96]
Berkeley “school”Berkeley “school”(program checking &(program checking &probabilistic proofs)probabilistic proofs)
Berkeley “school”Berkeley “school”(program checking &(program checking &probabilistic proofs)probabilistic proofs)
Distance is 3Distance is 3Distance is 3Distance is 3
Distance is 4Distance is 4Distance is 4Distance is 4
nononono
yesyesyesyes
bipartitebipartitebipartitebipartite
nononono
yesyesyesyesbipartitebipartitebipartitebipartite
anythinganythinganythinganything
[GR’97][GR’97][GR’97][GR’97]
Birthday paradox Birthday paradox Birthday paradox Birthday paradox
62626262
181818187777
polylog cyclespolylog cyclespolylog cyclespolylog cycles
17171717
MixingMixing casecaseMixingMixing casecase
[M’89[M’89
]][M’89[M’89
]]Nonmixing implies small cutsNonmixing implies small cutsNonmixing implies small cutsNonmixing implies small cuts
Non-mixingNon-mixing casecaseNon-mixingNon-mixing casecase
Dense graphsDense graphsDense graphsDense graphs
[GGR98, AK99][GGR98, AK99][GGR98, AK99][GGR98, AK99]
Hofstadter. Godel, Escher, Bach.
Is graph k-colorable?Is graph k-colorable?Is graph k-colorable?Is graph k-colorable?
1010001
0101011
1101100
1010011
1101101
0010110
1011001
Main Main tooltoolMain Main tooltool
Szemerédi’s Regularity Lemma Szemerédi’s Regularity Lemma Szemerédi’s Regularity Lemma Szemerédi’s Regularity Lemma
Far from k-colorableFar from k-colorableFar from k-colorableFar from k-colorable
Lots of Lots of witnesseswitnesses
Lots of Lots of witnesseswitnesses
Property Testing
Graph algorithms connectivity acyclicity k-way cuts clique
Distributions independence entropy monotonicity distances
Geometry convexity disjointness delaunay plane EMST
http://www.cs.princeton.edu/http://www.cs.princeton.edu/~chazelle/~chazelle/
http://www.cs.princeton.edu/http://www.cs.princeton.edu/~chazelle/~chazelle/