1. Coresets and Sketches for High Dimensional Subspace Approximation Problems Morteza Monemizadeh TU Dortmund Joint work with: D. Feldman, C. Sohler, D. Woodruff SODA 2010 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA A AA A A A A A

2. Unbounded Precision Insertion-only Streaming: Head of stream Seen points Unseen points Input:

3. Subspace Problem Find a j-subspace F : Euclidean Distance

4. Subspace Approximation Find a j-subspace such that PTAS:

5. Simple Cases 1-median PCA/SVD Machine Learning LSI, PageRank, HIITS Collaborative Filtering, Recommendation Systems Clustering k-median

6. Simple Cases Linear regression Nonlinear regression Shape-fitting

7. Known Before Coresets (Har-Peled) Dynamic Programming (Arora, Mitchell) d =O(1): Low-dimensions d =O(n): High-dimensions Dimensionality Reduction (Indyk, Rabani, …) d =O(1): Low-dimensions

8. Simple PTAS PTAS: Centroid Set:

9. PTAS Weak Coreset: PTAS:

10. Tools Weak Coreset Centroid Set

11. Coreset Construction Assumptions: d=2, j=1 Fix a 1-subspace (line): Have a 1-subspace (line): GOAL:

12. 1 st Try Sampling u.a.r or even non-uniformly:

13. 2 nd Try

16. Chernoff Bounds

17. Recursion 0

18. 0

19. 0 (0,n) 0

20. 0

22. Strong Coreset Stream:

23. Centroid Set In time Centroid Set:

24. Centroid Set Construction

25. Bounded Precision Stream S: …., (i,j, -5), …, (i,j, +10), … : |S|=poly(n,M) A[i,j]-5A[i,j]+10 =A[n,d]

26. Bounded Precision 1-pass streaming algorithm Space: Time:

27. Open Problems Coreset size: Stream: PTAS: What other classes of Clustering?

28. Thanks!