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Data-Powered Algorithms: Advances in Linear Programming and Decision-Making Tools

Explore the innovative realm of data-driven algorithms developed by Bernard Chazelle at Princeton University. This research delves into linear programming, dimension reduction, and optimization techniques vital for processing vast datasets in applications such as face and voice recognition. It highlights concepts like the Johnson-Lindenstrauss Transform, Fast Johnson-Lindenstrauss Transform, and property testing within graph theory. This comprehensive study bridges theoretical underpinnings and practical applications, offering insights into algorithmic efficiency, randomization, and approximation in tackling complex problems.

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Data-Powered Algorithms: Advances in Linear Programming and Decision-Making Tools

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  1. Data-Powered Algorithms Bernard Chazelle Princeton University

  2. Tools

  3. Linear Programming

  4. N constraints and d variables

  5. N constraints and d variables

  6. Dimension Reduction 25 10000 Images (face recognition) Signals (voice recognition) Text (NLP) . . . Nearest neighbor searching Clustering . . .

  7. Dimension reduction All pairwise distances nearly preserved

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

  9. 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)

  10. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 + + + + + + + + + + + + + + + + - - - - - - - - - - - - - - - - Friendlier JLT d c log n 2 d log n 2 = ( )

  11. 1 1 1 1 1 1 1 + + + + + + + - - - - - - - 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

  12. Main Tool: Uncertainty Principle Heisenberg Time Frequency

  13. 1 1 1 1 + + + + - - - - c log n 2 log3 n 2 = O( + d log d + d ) Fast Johnson-Lindenstrauss Transform (FJLT) d d d Discrete Fourier Transform 0 N(0,1) d . . . Optimal ??

  14. Data-Powered Algorithms

  15. theory experimentation

  16. theory experimentation computation

  17. theory experimentation 1950... computation

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

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

  20. input output PTAS for ETSP randomization approximation

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

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

  23. Property Testing

  24. Distance is 3

  25. Distance is 4 edit distance

  26. no bipartite yes

  27. no anything bipartite yes [GR’97]

  28. Mixingcase 18 17 7 62 bipartite! non-bipartite! polylog cycles Birthday paradox

  29. Non-mixingcase Nonmixing implies small cuts [M’89]

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

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

  32. 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 • planeEMST

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