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Slides are available at http://grigory.us/big-data.html. Sublinear Algorihms for Big Data. Lecture 4. Grigory Yaroslavtsev http://grigory.us. Today. Dimensionality reduction AMS as dimensionality reduction Johnson- Lindenstrauss transform Sublinear time algorithms

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today
Today
  • Dimensionality reduction
    • AMS as dimensionality reduction
    • Johnson-Lindenstrauss transform
  • Sublinear time algorithms
    • Definitions: approximation, property testing
    • Basic examples: approximating diameter, testing properties of images
    • Testing sortedness
    • Testing connectedness
norm estimation
-norm Estimation
  • Stream: updates that define vector where .
  • Example: For
  • -norm:
norm estimation1
-norm Estimation
  • -norm:
  • Two lectures ago:
    • -moment
    • -moment (via AMS sketching)
  • Space:
  • Technique: linear sketches
    • for random set s
    • for random signs
ams as dimensionality reduction
AMS as dimensionality reduction
  • Maintain a “linear sketch” vector

, where

  • Estimator for :
  • “Dimensionality reduction”: “heavy” tail
normal distribution
Normal Distribution
  • Normal distribution
    • Range:
    • Density:
    • Mean = 0, Variance = 1
  • Basic facts:
    • If and are independent r.v. with normal distribution then has normal distribution
    • If are independent, then
johnson lindenstrauss transform
Johnson-Lindenstrauss Transform
  • Instead of let be i.i.d. random variables from normal distribution
  • We still have because:
    • “variance of “
  • Define and define:
  • JL Lemma: There exists s.t. for small enough
proof of jl lemma
Proof of JL Lemma
  • JL Lemma: s.t. for small enough
  • Assume .
  • We have and
  • Alternative form of JL Lemma:
proof of jl lemma1
Proof of JL Lemma
  • Alternative form of JL Lemma:
  • Let and
  • For every we have:
  • By Markov and independence of :
  • We have , hence:
proof of jl lemma2
Proof of JL Lemma
  • Alternative form of JL Lemma:
  • For every we have:
  • Let and recall that
  • A calculation finishes the proof:
johnson lindenstrauss transform1
Johnson-Lindenstrauss Transform
  • Single vector:
    • Tight: [Woodruff’10]
  • vectors simultaneously:
    • Tight: [Molinaro, Woodruff, Y. ’13]
  • Distances between vectors = vectors: