1. Numerical Linear Algebra in the Streaming Model: Upper Bounds


    Ken Clarkson
    IBM Almaden

    joint with David Woodruff

  2. The Problems

    Given `n times d` matrix `A`, `n times d'` matrix `B`, integer `k`, estimators for:
  3. General Properties of Our Algorithms

  4. Matrix Compression Methods

    In a line of similar efforts...
    Here: sketching
  5. Outline

  6. Approximate Matrix Product

  7. Streaming Matrix Updates and Pass Efficiency

  8. Algorithm Bounds

  9. Why this works: the sign matrix `S`

  10. Expected Error and a Tail Estimate

  11. Relation to Johnson-Lindenstrauss

  12. JL `=>` Matrix Product Estimate

  13. JL and Matrix Product

  14. Related Work

    This JL-based algorithm is due to Sarlós [S06], who gave two algorithms for product:
  15. Lower Bound on Space

  16. A Moment Bound Implies the Tail Estimate

  17. The Moment Bound, Roughly

    To bound `bb E [||Lambda|| ^{2p}]`:
  18. Regression

  19. Regression Analysis

  20. Regression Analysis, cont.

  21. Best Low-Rank Approximation

  22. Best Low-Rank Approximation and `S^TA`

  23. Low-Rank Approximation : Using Regression

  24. Best Low-Rank Approximation:
    Two Pass Algorithm

  25. Nearly Best Nearly-Low-Rank Approximation

  26. Nearly Best Nearly-Low-Rank Algorithm

  27. Nearly Best Low-Rank Approximation

    Still haven't found a good rank `k` matrix
  28. Concluding Remarks