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Satyaki Mahalanabis Daniel Štefankovič

Density estimation in linear time (+approximating L 1 -distances). Satyaki Mahalanabis Daniel Štefankovič. University of Rochester. Density estimation. f 6. f 1. f 2. +. DATA. f 4. f 3. f 5. F = a family of densities. density. Density estimation - example. 0.418974, 0.848565,

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Satyaki Mahalanabis Daniel Štefankovič

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  1. Density estimation in linear time (+approximating L1-distances) Satyaki Mahalanabis Daniel Štefankovič University of Rochester

  2. Density estimation f6 f1 f2 + DATA f4 f3 f5 F = a family of densities density

  3. Density estimation - example 0.418974, 0.848565, 1.73705, 1.59579, -1.18767, -1.05573, -1.36625 N(,1) + F = a family of normal densities with =1 

  4. Measure of quality: g=TRUTH f=OUTPUT L1 – distance from the truth |f-g|1 =  |f(x)-g(x)| dx WhyL1? 1) small L1 all events estimated with small additive error 2) scale invariant

  5. Obstacles to “quality”: + DATA F bad data  ? weak class of densities dist1(g,F)

  6. What is bad data ? | h-g |1 g = TRUTH h = DATA (empirical density) = 2max |h(A)-g(A)| AY(F) Y(F) = Yatracos class of F Aij={ x | fi(x)>fj(x) } f2 f3 f1 A12 A13 A23

  7. Density estimation F f + with small |g-f|1 DATA (h) assuming these are small: dist1(g,F) = 2max |h(A)-g(A)| AY(F)

  8. Why would these be small ??? dist1(h,F) = 2max |h(A)-g(A)| AY(F) They will be if: 1) pick a large enough F 2) pick a small enough F so that VC-dimension of Y(F) is small 3) data are iid from h E[max|h(A)-g(A)|] Theorem (Haussler,Dudley, Vapnik, Chervonenkis): VC(Y) samples AY

  9. How to choose from 2 densities? f1 f2

  10. How to choose from 2 densities? f1 f2 +1 +1 +1 -1

  11. How to choose from 2 densities? T f1 T f2 Th  f1 f2 +1 +1 +1 -1 T

  12. How to choose from 2 densities? T f1 T f2 Th  f1 f2 Scheffé: if T h > T (f1+f2)/2  f1 else  f2 Theorem (see DL’01): |f-g|1 3dist1(g,F) + 2 +1 +1 +1 -1 T

  13. Density estimation F f + with small |g-f|1 DATA (h) assuming these are small: dist1(g,F) = 2max |h(A)-g(A)| AY(F)

  14. Test functions F={f1,f2,...,fN} Tij (x) = sgn(fi(x) – fj(x)) Tij(fi – fj) =  (fi-fj)sgn(fi-fj) = |fi– fj|1 Tijh fj wins fi wins Tijfj Tijfi

  15. Density estimation algorithms Scheffé tournament: Pick the density with the most wins. Theorem (DL’01): |f-g|1 9dist1(g,F)+8 n2 Minimum distance estimate (Y’85): Output fk F that minimizes max |(fk-h) Tij| n3 ij Theorem (DL’01): |f-g|1 3dist1(g,F)+2

  16. Density estimation algorithms Can we do better? Scheffé tournament: Pick the density with the most wins. Theorem (DL’01): |f-g|1 9dist1(g,F)+8 n2 Minimum distance estimate (Y’85): Output fk F that minimizes max |(fk-h) Tij| n3 ij Theorem (DL’01): |f-g|1 3dist1(g,F)+2

  17. Our algorithm: Efficient minimum loss-weight repeat until one distribution left 1) pick the pair of distributions in F that are furthest apart (in L1) 2) eliminate the loser Theorem [MS’08]: |f-g|1 3dist1(g,F)+2 n * Take the most “discriminative” action. * after preprocessing F

  18. Tournament revelation problem INPUT: a weighed undirected graph G (wlog all edge-weights distinct) OUTPUT: REPORT: heaviest edge {u1,v1} in G ADVERSARY eliminates u1 or v1 G1 REPORT: heaviest edge {u2,v2} in G1 ADVERSARY eliminates u2 or v2 G2 ..... OBJECTIVE: minimize total time spent generating reports

  19. Tournament revelation problem A report the heaviest edge 4 3 2 B 5 6 D C 1

  20. Tournament revelation problem A report the heaviest edge BC 4 3 2 B 5 6 D C 1

  21. Tournament revelation problem A report the heaviest edge BC 3 2 eliminate B report the heaviest edge D C 1

  22. Tournament revelation problem A report the heaviest edge BC 3 2 eliminate B report the heaviest edge D C 1 AD

  23. Tournament revelation problem report the heaviest edge BC eliminate B report the heaviest edge D C 1 AD eliminate A report the heaviest edge CD

  24. Tournament revelation problem A BC B C 4 3 2 AD BD B A D B D 5 6 D DC AC AB C AD 1 2O(F) preprocessing  O(F) run-time O(F2 log F) preprocessing  O(F2) run-time WE DO NOT KNOW: Can get O(F) run-time with polynomial preprocessing ???

  25. Efficient minimum loss-weight repeat until one distribution left 1) pick the pair of distributions that are furthest apart (in L1) 2) eliminate the loser (in practice 2) is more costly) 2O(F) preprocessing  O(F) run-time O(F2 log F) preprocessing  O(F2) run-time WE DO NOT KNOW: Can get O(F) run-time with polynomial preprocessing ???

  26. Efficient minimum loss-weight repeat until one distribution left 1) pick the pair of distributions that are furthest apart (in L1) 2) eliminate the loser Theorem: |f-g|1 3dist1(g,F)+2 n Proof: “that guy lost even more badly!” For every f’ to which f loses |f-f’|1 max |f’-f’’|1 f’ loses to f’’

  27. Proof: “that guy lost even more badly!” For every f’ to which f loses |f-f’|1 max |f’-f’’|1 f’ loses to f’’ 2hT23 f2T23 + f3T23 f1 (f1-f2)T12 (f2-f3) T23 (f4-h)T23  (fi-fj)(Tij-Tkl) 0 bad loss f3 |f1-g|1 3|f2-g|1+2 BEST=f2

  28. Application: kernel density estimates (Akaike’54,Parzen’62,Rosenblatt’56) K = kernel h = density kernel used to smooth empirical g (x1,x2,...,xn i.i.d. samples from h) n 1  K(y-xi) h * K n as n i=1 = g * K

  29. What K should we choose? g * K n 1  = K(y-xi) h * K n as n i=1 Dirac  is not good Dirac  would be good Something in-between: bandwidth selection for kernel density estimates K(x/s) as s 0 Ks(x) Dirac  Ks(x)= s Theorem (see DL’01): as s 0 with sn |g*K – h|1 0

  30. Data splitting methods for kernel density estimates How to pick the smoothing factor ? n ( ) 1  y-xi K ns s i=1 n-m ( )  y-xi 1 K x1,...,xn-m fs = s (n-m)s i=1 x1,x2,...,xn choose s using density estimation xn-m+1,...,xn

  31. Kernels we will use: ( ) 1  y-xi K ns s piecewise uniform piecewise linear

  32. Bandwidth selection for uniform kernels E.g. Nn1/2 mn5/4 N distributions each is piecewise uniform with n pieces m datapoints Goal: run the density estimation algorithm efficiently TIME MD EMLW (fi+fj)Tij gTij n+m log n N 2 (fk-h) Tkj N2 n+m log n |fi-fj|1 n N2

  33. Bandwidth selection for uniform kernels Can speed this up? E.g. Nn1/2 mn5/4 N distributions each is piecewise uniform with n pieces m datapoints Goal: run the density estimation algorithm efficiently TIME MD EMLW (fi+fj)Tij gTij n+m log n N 2 (fk-h) Tkj N2 n+m log n |fi-fj|1 n N2

  34. Bandwidth selection for uniform kernels Can speed this up? E.g. Nn1/2 mn5/4 N distributions each is piecewise uniform with n pieces m datapoints absolute error bad relative error good Goal: run the density estimation algorithm efficiently TIME MD EMLW (fi+fj)Tij gTij n+m log n N 2 (fk-h) Tkj N2 n+m log n |fi-fj|1 n N2

  35. Approximating L1-distances between distributions N piecewise uniform densities (each n pieces) (N2+Nn) (log N) WE WILL DO: 2 TRIVIAL (exact): N2n

  36. Dimension reduction for L2 |S|=n Johnson-Lindenstrauss Lemma (’82) : L2 Lt2t = O(-2 ln n) ( x,y  S) d(x,y)  d((x),(y))  (1+)d(x,y) N(0,t-1/2)

  37. Dimension reduction for L1 |S|=n Cauchy Random Projection (Indyk’00) : L1 Lt1t = O(-2 ln n) ( x,y  S) d(x,y) est((x),(y))  (1+)d(x,y) N(0,t-1/2) C(0,1/t) (Charikar, Brinkman’03 : cannot replace est by d)

  38. Cauchy distribution C(0,1) density function: 1 (1+x2) FACTS: XC(0,1) aXC(0,|a|) XC(0,a), YC(0,b) X+YC(0,a+b)

  39. Cauchy random projection for L1 (Indyk’00) A B D X1 X2 X3 X4 X5 X6 X7 X8 X9 X1C(0,z) A(X2+X3) + B(X5+X6+X7+X8) z

  40. Cauchy random projection for L1 (Indyk’00) A B D X1 X2 X3 X4 X5 X6 X7 X8 X9 X1C(0,z) A(X2+X3) + B(X5+X6+X7+X8) D(X1+X2+...+X8+X9) z Cauchy(0,|-|1)

  41. All pairs L1-distances piece-wise linear densities

  42. All pairs L1-distances piece-wise linear densities R=(3/4)X1 + (1/4)X2 B=(3/4)X2 + (1/4)X1 R-BC(0,1/2) X1 X2  C(0,1/2)

  43. All pairs L1-distances piece-wise linear densities Problem: too many intersections! Solution: cut into even smaller pieces! Stochastic measures are useful.

  44. Brownian motion 1 exp(-x^2/2) (2)1/2 Cauchy motion 1 (1+x)2

  45. Brownian motion 1 exp(-x^2/2) (2)1/2 computing integrals is easy f:RRd f dL = Y  N(0,S)

  46. Cauchy motion 1 (1+x)2 computing integrals is easy f:RRd f dL = Y  C(0,s) for d=1 computing integrals is hard d>1 * * obtaining explicit expression for the density

  47. X1 X2 X3 X4 X5 X6 X7 X8 X9 What were we doing? (f1,f2,f3) dL = (w1)1,(w2)1,(w3)1

  48. X1 X2 X3 X4 X5 X6 X7 X8 X9 What were we doing? (f1,f2,f3) dL = (w1)1,(w2)1,(w3)1 Can we efficiently compute integrals dL for piecewise linear?

  49. Can we efficiently compute integrals dL for piecewise linear? : R R2 (z)=(1,z) (X,Y)= dL

  50. : R R2 (z)=(1,z) (X,Y)= dL u+v,u-v (2(X-Y),2Y) has density at 2

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