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  1. Fast and robust sparse recovery Mayank Bakshi INC, CUHK New Algorithms and Applications Sheng Cai Eric Chan Mohammad Jahangoshahi Sidharth Jaggi Venkatesh Saligrama Minghua Chen The Chinese University of Hong Kong The Institute of Network Coding

  2. Fast and robust sparse recovery m ? n Measurement Measurement output k m<n Reconstruct x Unknown x

  3. A. Compressive sensing ? m ? n k k ≤ m<n

  4. A. Robust compressive sensing ? e z y=A(x+z)+e Approximate sparsity Measurement noise

  5. Computerized Axial Tomography (CAT scan)

  6. B. Tomography y = Tx Estimate x given y and T

  7. B. Network Tomography • Transform T: • Network connectivity matrix (known a priori) • Measurements y: • End-to-end packet delays • Infer x: • Link/node congestion Hopefully “k-sparse” Compressive sensing? • Challenge: • Matrix T “fixed” • Can only take “some” • types of measurements

  8. n-d C. Robust group testing d q 0 1 q 0 1 For Pr(error)< ε , Lower bound: What’s known …[CCJS11] Noisy Combinatorial OMP:

  9. A. Robust compressive sensing ? e z y=A(x+z)+e Approximate sparsity Measurement noise

  10. Apps: 1. Compression W(x+z) x+z BW(x+z) = A(x+z) M.A. Davenport, M.F. Duarte, Y.C. Eldar, and G. Kutyniok, "Introduction to Compressed Sensing,"inCompressed Sensing: Theory and Applications, 2012

  11. Apps: 2. Fast(er) Fourier Transform H. Hassanieh, P. Indyk, D. Katabi, and E. Price. Nearly optimal sparse fourier transform. InProceedings of the 44th symposium on Theory of Computing (STOC '12).

  12. Apps: 3. One-pixel camera http://dsp.rice.edu/sites/dsp.rice.edu/files/cs/cscam.gif

  13. y=A(x+z)+e

  14. y=A(x+z)+e

  15. y=A(x+z)+e

  16. y=A(x+z)+e

  17. y=A(x+z)+e (Information-theoretically) order-optimal

  18. (Information-theoretically) order-optimal • Support Recovery

  19. SHO-FA:SHO(rt)-FA(st)

  20. O(k) measurements, O(k) time

  21. 1. Graph-Matrix A d=3 ck n

  22. 1. Graph-Matrix A d=3 ck n

  23. 1. Graph-Matrix

  24. 2. (Most) x-expansion ≥2|S| |S|

  25. 3. “Many” leafs L+L’≥2|S| ≥2|S| |S| 3|S|≥L+2L’ L≥|S| L+L’≤3|S| L/(L+L’) ≥1/2 L/(L+L’) ≥1/3

  26. 4. Matrix

  27. Encoding – Recap. 0 1 0 1 0

  28. Decoding – Initialization

  29. Decoding – Leaf Check(2-Failed-ID)

  30. Decoding – Leaf Check (4-Failed-VER)

  31. Decoding – Leaf Check(1-Passed)

  32. Decoding – Step 4 (4-Passed/STOP)

  33. Decoding – Recap. 0 0 0 0 0 0 0 0 1 0 ? ? ?

  34. Decoding – Recap. 0 1 0 1 0

  35. Noise/approx. sparsity

  36. Meas/phase error

  37. Correlated phase meas.

  38. Correlated phase meas.

  39. Correlated phase meas.

  40. Network Tomography • Goal: Infer network characteristics (edge or node delay) • Difficulties: • Edge-by-edge (or node-by node) monitoring too slow • Inaccessible nodes

  41. Network Tomography • Goal: Infer network characteristics (edge or node delay) • Difficulties: • Edge-by-edge (or node-by node) monitoring too slow • Inaccessible nodes • Network Tomography: • with very fewend-to-end measurements • quickly • for arbitrary network topology

  42. B. Network Tomography • Transform T: • Network connectivity matrix • (known a priori) • Measurements y: • End-to-end packet delays • Infer x: • Link/node congestion Hopefully “k-sparse” Our algorithm: FRANTIC Compressive sensing? • Challenge: • Matrix T “fixed” • Can only take “some” • types of measurements • Fast Reference-based Algorithm for Network Tomography vIa Compressive sensing • Idea: • “Mimic” random matrix

  43. SHO-FA A d=3 ck n

  44. 1. Integer valued CS [BJCC12] “SHO-FA-INT” T

  45. 2. Better mimicking of desired T

  46. Node delay estimation

  47. Node delay estimation

  48. Node delay estimation

  49. Edge delay estimation