Compressive sensing meets group testing: LP decoding for non-linear (disjunctive) measurements Chun Lam Chan, Sidharth Jaggi and Samar Agnihotri The Chinese University of Hong Kong Venkatesh Saligrama Boston University

2 n-d d Lower bound: OMP: What’s known BP: Compressive sensing

3 n-d d Group testing: q 1 q Lower bound: Noisy Combinatorial OMP: What’s known This work: Noisy Combinatorial BP: …[CCJS11]

4 Group-testing model p=1/D [CCJS11]

5 CBP-LP relaxation weight positive tests negative tests

6 NCBP-LP “Slack”/noise variables Minimum distance decoding

7 “Perturbation analysis” 1.For all (“Conservation of mass”) 2. LP change under a single ρ i (Case analysis) 3. LP change under all n(n-d) ρ i s (Chernoff/union bounds) 4. LP change under all (∞) perturbations (Convexity) (5.) If d unknown but bounded, try ‘em all (“Info thry”)

8 1. Perturbation vectors NCBLP feasible set x ρiρi ρjρj dn-d

9 2. LP value change with ONE perturbation vector x

10 3. LP value change with EACH (n(n-d)) perturbation vector Union boundChernoff bound Prob error < x

11 4. LP value change under ALL (∞) perturbations x Prob error < Convexity of min LP = x

12 (5.) NCBP-LPs Information-theoretic argument – just a single d “works”.

13 Bonus: NCBP-SLPs ONLY negative tests ONLY positive tests

14

Noiseless CBP 15 n-d d

Noiseless CBP 16 n-d d Discard

Noiseless CBP 17  Sample g times to form a group n-d d

Noiseless CBP 18  Sample g times to form a group n-d d

Noiseless CBP 19  Sample g times to form a group n-d d

Noiseless CBP 20  Sample g times to form a group n-d d

Noiseless CBP 21  Sample g times to form a group  Total non-defective items drawn: n-d d

Noiseless CBP 22  Sample g times to form a group  Total non-defective items drawn:  Coupon collection: n-d d

Noiseless CBP 23  Sample g times to form a group  Total non-defective items drawn:  Coupon collection:  Conclusion: n-d d

Noisy CBP 24 n-d d

Noisy CBP 25 n-d d

Noisy CBP 26 n-d d

Noisy CBP 27 n-d d

Noiseless COMP 28

Noiseless COMP 29

Noiseless COMP 30

Noiseless COMP 31

Noiseless COMP 32

Noisy COMP 33

Noisy COMP 34

Noisy COMP 35

Noisy COMP 36

Noisy COMP 37

Noisy COMP 38

Noisy COMP 39

Simulations 40

Simulations 41

Summary 42  With small error,

Noiseless COMP x My

x My x9x9 01 → Noiseless COMP 44

Noiseless COMP x My x7x7 11 →

Noiseless COMP x My x4x4 01 →

Noiseless COMP x My x4x4 00x7x7 10x9x9 (a)01 → 1(b)11 → 1(c)01 →

Noisy COMP x My ν ŷ →

Noisy COMP x My ν ŷ → x3x3 10 →

Noisy COMP x My ν ŷ → x2x2 10 →

Noisy COMP x My ν ŷ → x7x7 10 →

Noisy COMP x My ν ŷ → x2x2 01x3x3 01x7x7 (a)10 → 1(b)10 → 1(c)10 →

Noisy COMP x My ν ŷ → x2x2 01x3x3 01x7x7 (a)10 → 1(b)10 → 1(c)10 →