Data-Powered Algorithms Bernard Chazelle Princeton University Bernard Chazelle Princeton University

Linear Programming Linear Programming

N constraints and d variables

Dimension Reduction   25 Images (face recognition) Signals (voice recognition) Text (NLP) Nearest neighbor searching Clustering...

Dimension reduction All pairwise distances nearly preserved

Johnson-Lindenstrauss Transform (JLT) c log n  2 d Random Orthogonal Matrix v d

Friendly JLT c log n  2 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) 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 JLT c log n  2 d d log n  2  2 =  

Sparse JLT ? c log n  d 1 d o(1)-Fraction non-zeros

Main Tool: Uncertainty Principle Time Frequency Heisenberg

Fast Johnson-Lindenstrauss Transform (FJLT) d Discrete Fourier Transform dd c log n  N(0,1) = O  + d log d + d  log 3 n  2  2d Optimal ??

theory experimentation

computation theory experimentation

computation theory experimentation

input output Most interesting problems are too hard !! Most interesting problems are too hard !!

input output randomization approximation So, we change the model… So, we change the model…

input output randomization approximation PTAS for ETSP

input output randomization approximation Impossible to approximate chromatic chromatic number within a factor of… Impossible to approximate chromatic chromatic number within a factor of…

input output randomization approximation Property Testing [RS’96, GGR’96] Property Testing [RS’96, GGR’96] Berkeley “school” (program checking & probabilistic proofs) Berkeley “school” (program checking & probabilistic proofs)

Distance is 3

Distance is 4

nono yesyes bipartitebipartite

nono yesyes bipartitebipartite anythinganything [GR’97][GR’97]

Birthday paradox polylog cycles 1717 Mixing case

[M’89][M’89] Nonmixing implies small cuts Non-mixing case

Dense graphs [GGR98, AK99] Hofstadter. Godel, Escher, Bach. Is graph k-colorable?

Main tool Szemerédi’s Regularity Lemma Far from k-colorable Lots of witnesses

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