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m<n

m. ?. n. m<n. Compressive sensing. ?. m. ?. n. k. k ≤ m<n. Robust compressive sensing. ?. e. z. y=A( x+ z )+ e. Approximate sparsity. Measurement noise. Apps: 1. Compression. W( x + z ). x + z. BW( x + z ). = A( x + z ).

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m<n

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  1. m ? n m<n

  2. Compressive sensing ? m ? n k k ≤ m<n

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

  4. 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, Cambridge University Press, 2012. 

  5. Apps: 2. Network tomography Weiyu Xu; Mallada, E.; Ao Tang; , "Compressive sensing over graphs," INFOCOM, 2011 M. Cheraghchi, A. Karbasi, S. Mohajer, V.Saligrama: Graph-Constrained Group Testing. IEEE Transactions on Information Theory 58(1): 248-262 (2012)

  6. Apps: 3. 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). ACM, New York, NY, USA, 563-578.

  7. Apps: 4. One-pixel camera http://dsp.rice.edu/sites/dsp.rice.edu/files/cs/cscam.gif

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

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

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

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

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

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

  14. SHO(rt)-FA(st)O(k) meas., O(k) steps

  15. SHO(rt)-FA(st)O(k) meas., O(k) steps

  16. SHO(rt)-FA(st)O(k) meas., O(k) steps

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

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

  19. 1. Graph-Matrix

  20. 2. (Most) x-expansion ≥2|S| |S|

  21. 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

  22. 4. Matrix

  23. Encoding – Recap. 0 1 0 1 0

  24. Decoding – Initialization

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

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

  27. Decoding – Leaf Check(1-Passed)

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

  29. Decoding – Recap. 0 0 0 0 0 0 0 0 1 0 ? ? ?

  30. Decoding – Recap. 0 1 0 1 0

  31. Noise/approx. sparsity

  32. Meas/phase error

  33. Correlated phase meas.

  34. Correlated phase meas.

  35. Correlated phase meas.

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