2. m ? n
m<n

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

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

5. 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,"in Compressed Sensing: Theory and Applications, Cambridge University Press, 2012. 

6. 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): (2012)

7. Apps: 3. Fast(er) Fourier Transform
H. Hassanieh, P. Indyk, D. Katabi, and E. Price. Nearly optimal sparse fourier transform. In Proceedings of the 44th symposium on Theory of Computing (STOC '12). ACM, New York, NY, USA,

8. Apps: 4. One-pixel camera

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

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

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

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

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

14. (Information-theoretically) order-optimal
Support Recovery

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

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

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

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

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

20. 1. Graph-Matrix

21. 2. (Most) x-expansion
|S|
≥2|S|

22. 3. “Many” leafs L+L’≥2|S| |S| ≥2|S| 3|S|≥L+2L’ L≥|S| L+L’≤3|S|

23. 4. Matrix

24. Encoding – Recap. 1

25. Decoding – Initialization

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

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

28. Decoding – Leaf Check(1-Passed)

29. Decoding – Step 4 (4-Passed/STOP)

30. Decoding – Recap. 1 ? ? ?

31. Decoding – Recap. 1

33. Noise/approx. sparsity

34. Meas/phase error

35. Correlated phase meas.

36. Correlated phase meas.

37. Correlated phase meas.