compressive sensing SHO-FA: Robust compressive sensing with order-optimal complexity, measurements, and bits 1 Mayank Bakshi, Sidharth Jaggi, Sheng Cai and Minghua Chen The Chinese University of Hong Kong FasterHigherStronger order-optimal complexity, measurements, and bits with Robust SHO-FA:

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

Robust compressive sensing Approximate sparsity Measurement noise 3 ? Random

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Decoding complexity # of measurements ° RS’60 ° TG’07 ° CM’06 ° C’08 ° IR’08 ° SBB’06 ° GSTV’06 ° MV’12,KP’12 ° DJM’11  This work Lower bound

6 SHO(rt)-FA(st) Good Bad Good Bad

High-Level Overview n ck k= n ck k=2

High-Level Overview n ck k=2 How to find the leaf nodes and utilize the leaf nodes to do decoding How to guarantee the existence of leaf node

Left-regular Bipartite Graph n ck d=3 9 A Q1: How to guarantee the existence of leaf node?

Existence of leaf nodes 10 e.g., existence of 2-core in d-uniform hypergraph M. T. Goodrich and M. Mitzenmacher, “Invertible bloom lookup tables,” ArXiv.org e-Print archive, arXiv: [cs.DS], Sharp transition Q1: How to guarantee the existence of leaf node?

Existence of “Many” leafs ≥2|S| |S| L+L’≥2|S| 3|S|≥L+2L’ (L+L’)/(L+2L’) ≥2/3 11 L/(L+L’) ≥1/2 Q1: How to guarantee the existence of leaf node?

Bipartite Graph → Sensing Matrix n ck d=3 12 A Distinct weights Q2: How to find the leaf nodes and utilize the leaf nodes to do decoding?

Bipartite Graph → Sensing Matrix 13 n ck A Q2: How to find the leaf nodes and utilize the leaf nodes to do decoding?

14 Encoding Q2: How to find the leaf nodes and utilize the leaf nodes to do decoding?

15 Q2: How to find the leaf nodes and utilize the leaf nodes to do decoding? Decoding

Decoding – First Iteration 16

Decoding – Second Iteration 17 Verification Measurements

Decoding – Third Iteration 18

Decoding – Fourth Iteration 19

SHO-FA v.s. Pick-Up-Sticks 20 Peeling process: Iterative Decoding Observation: Identification Check: VerificationPicking up a “top” stick: Leaf-based decoding

Robust Compressive Sensing 21 Phase error Propagation error …… Pawar, Sameer and Ramchandran, Kannan, “A Hybrid DFT-LDPC Framework for Fast and Robust Compressive Sensing”

Truncated Reconstruction 22 Threshold

Correlated Measurements 23 Phase quantization

Correlated Measurements (First bit) 24 Phase quantization

Correlated Measurements (Second bit) 25

Correlated Measurements (Third bit) 26

Additional Properties Other works – Group Testing – Network Tomography Reduce the number of measurements – Combine Identification and verification More noise models Sparse in different bases Database query ……

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2-core in d-uniform hypergraph The 2-core is the largest sub-hypergraph that has minimum degree at least 2. The standard “peeling process” finds the 2- core: while there exists a vertex with degree 1, delete it and the corresponding hyperedges. hyperedge Node degree 1 29

(Almost) S(x)-expansion ≥2|S| |S| 30 n ck