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Dimension Reduction using Rademacher Series on Dual BCH Codes

This paper presents an innovative algorithmic technique for dimension reduction, leveraging Rademacher series in conjunction with dual BCH codes. The proposed method effectively removes redundancy from data by employing a sampling strategy that measures the concentration phenomenon. We address the challenge of finding effective random projections from high-dimensional spaces (Rd) to lower dimensions (Rk) with controlled distortion. The results not only improve computational efficiency but also enhance the handling of random bits in the dimension reduction process, offering significant advancements over traditional techniques.

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Dimension Reduction using Rademacher Series on Dual BCH Codes

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  1. Dimension ReductionusingRademacher SeriesonDual BCH Codes Nir Ailon Edo Liberty

  2. Dimension Reduction • Algorithmic technique: • often as “black box” • Removes redundancy from data • A sampling method • Measure concentration phenomenon • (Existential) metric embedding • Recently: computational aspects • time • randomness

  3. x x Rd  Rk Challenge Find random projection  from Rd to Rk (d big, k small) such that for every x  Rd,||x||2=1, 0< with probability exp{-k ||x||2 = 1 ± O(

  4. x1 x1 x3 x2 x2 x3 Rd  Rk Usage If you have n vectors x1..xn  Rd: set k=O(log n) by union bound: for all i,j ||xi- xj|| ≈||xi- xj|| low-distortion metric embedding “tight”

  5. Solution: Johnson-Lindenstrauss (JL) “dense random matrix”  = k d

  6. So what’s the problem? • running time (kd) • number of random bits (kd) • can we do better?

  7. Fast JL A, Chazelle 2006 • = Sparse.Hadamard.Diagonal time = O(k3 + dlog d), randomness=O(k3 + d) beats JL (kd) bound for: log d < k < d1/3 k Fourier d

  8. This Talk • O(d logk) for k < d1/2 • beats JL up to k < d1/2 • O(d) random bits

  9. Algorithm (k=d1/2) A, Liberty 2007 • = BCH .Diagonal . .... Error Correcting Code k d

  10.  binary “dual-BCH code of designed distance 4” Bx computable in time d logd given x Error Correcting Codes d B = k = d1/2 0  k-1/2 1  -k-1/2 columns closed under addition row set subset of d x d Hadamard 4-wise independence

  11. Error Correcting Codes Fact (easily from properties of dual BCH): ||Bt||24 = O(1) for yt Rk with ||y||2 = 1: ||yB||4 = O(1) x x ... Rd D B Rk=d

  12. Rademacher Serieson Error Correcting Codes look at r.v. BDx  (Rk, l2) BDx = DiixiBi Dii R{1} i=1...d = DiiMi x x ... Rd D B Rk=d

  13. Talagrand’s Concentration Boundfor Rademacher Series Z =||DiiMi||p (in our case p=2) Pr[ |Z-EZ| >  ] = O(exp{-2/4||M||2p2})

  14. Rademacher Serieson Error Correcting Codes look at r.v. BDx  (Rk, l2) Z = ||BDx||2 = ||DiiMi||2 ||M||22||x||4||Bt||24 (Cauchy-Schwartz) by ECC properties: ||M||22||x||4O(1) trivial: EZ = ||x||2 = 1 x x ... Rd D B Rk=d

  15. Rademacher Serieson Error Correcting Codes look at r.v. BDx  (Rk, l2) Z = ||BDx||2 = ||DiiMi||2 ||M||22=O(||x||4) EZ = 1 Pr[ |Z-EZ| >  ] = O(exp{-2/4||M||222}) • Pr[|Z-1| > ] = O(exp{-2/||x||42}) how to get ||x||42 = O(k-1=d-1/2) ? x x Challenge: w. prob. exp{-k deviation of more than  ... Rd D B Rk=d

  16. Controlling ||x||42 how to get ||x||42 = O(k-1=d-1/2) ? • if you think about it for a second... • “random” x has ||x||42=O(d-1/2) • but “random” x easy to reduce: just output first k dimensions • are we asking for too much? • no: truly random x has strong bound on||x||p for all p>2 x x ... Rd D B Rk=d

  17. Controlling ||x||42 how to get ||x||42 = O(k-1=d-1/2) ? • can multiply x by orthogonal matrix • try matrix HD • Z = ||HDx||4= ||DiixiHi||4 =||DiiMi||4 • by Talagrand:Pr[ |Z-EZ| > t ] = O(exp{-t2/4||M||242}) EZ = O(d-1/4) (trivial) ||M||24||H||4/34||x||4(Cauchy Schwartz) (HD used in [AC06] to control ||HDx||) x x ... Rd D B Rk=d

  18. x x ... Rd D B Rk=d Controlling ||x||42 how to get ||x||42 = O(k-1=d-1/2) ? Z =||DiiMi||4 Mi = xiHi Pr[ |Z-EZ| > t ] = O(exp{-t2/4||M||242}) (Talagrand)EZ = O(d-1/4) (trivial)||M||24||H||4/34||x||4 (Cauchy Schwartz) ||H||4/34  d-1/4 (Hausdorff-Young) • ||M||24d-1/4||x||4 • Pr[ ||HDx||4 > d-1/4+ t ] = exp{-t2/d-1/2||x||42}

  19. x x ... Rd D B Rk=d Controlling ||x||42 how to get ||x||42 = O(d-1/2) ? Pr[ ||HDx||4 > d-1/4+ t ] = exp{-t2/d-1/2||x||42} need some slack k=d-1/2- max error probability for challenge: exp{-k} k = t2/d-1/2||x||42  t = k1/2d1/4||x||4 = ||x||4d-/2

  20. x x ... Rd D B Rk=d Controlling ||x||42 how to get ||x||42 = O(d-1/2) ? first round: ||HDx||4 < d-1/4 + ||x||4 d-/2 second round: ||HD’’HD’x||4 < d-1/4 + d-1/4-/2 + ||x||4 d- r=O(1/)’th round: ||HD(r)...HD’x||4 < O(r)d-1/4 ... with probability 1-O(exp{-k})

  21. Algorithm for k=d-1/2- x x ... Rk Rd D(1) H D(r) H D B running time O(d logd) randomness O(d)

  22. Open Problems • Go beyond k=d1/2Conjecture: can do O(d log d) for k=d1-  • Approximate linear l2-regressionminimize ||Ax-b||2 given A,b (overdetermined) • State of the art for general inputs: Õ(linear time) for #{variables} < #{equations}1/3 • Conjecture: can do Õ(linear time) always

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