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) 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. 

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)

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, 563-578.

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

y=A(x+z)+e

y=A(x+z)+e

y=A(x+z)+e

y=A(x+z)+e

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

(Information-theoretically) order-optimal Support Recovery

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

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

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

1. Graph-Matrix A d=3 n ck

1. Graph-Matrix A d=3 n ck

1. Graph-Matrix

2. (Most) x-expansion |S| ≥2|S|

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

4. Matrix

Encoding – Recap. 1

Decoding – Initialization

Decoding – Leaf Check(2-Failed-ID)

Decoding – Leaf Check (4-Failed-VER)

Decoding – Leaf Check(1-Passed)

Decoding – Step 4 (4-Passed/STOP)

Decoding – Recap. 1 ? ? ?

Decoding – Recap. 1

Noise/approx. sparsity

Meas/phase error

Correlated phase meas.

Correlated phase meas.

Correlated phase meas.