Group Testing with a Compressed Sensing Perspective€¦ · Based on Ref[2] , [M.Cherchagchi,...

IntroductionGroup Testing Simulations

Information theoretic model of group testingOther problems considered

Group Testing with a Compressed SensingPerspective

Kedar Tatwawadi

Guide: Prof. Sibi Raj Pillai

October 28, 2013

Kedar Tatwawadi Group Testing



Overview

References

1 IntroductionIntroduction to group testingMathematical modelling

2 Group Testing SimulationsDecoder designSimulation results

3 Information theoretic model of group testingProblem setupLower bound

4 Other problems considered




Introduction to group testingMathematical modelling

The Dorfman experiment

Group Testing problem originated with Dorfman’s [1943]experiment of detection of syphilis antigen during the World War.

Testing for every indivisual was expensive and time consuming.

Pool the blood samples of a group of indivisuals together andtested.

If the result was negative, the entire group was declaredhealthy. If not, then grouping can be performed in the secondstage to detect the affected indivisuals.





The 12-coin example

We have 12 gold coins amongst which one coin is counterfeit

Both adaptive as well as non-adaptive ways of solving theproblem





Mathematical Modelling

We have N samples , K of them defective,& M measurements

The Mathematical Model for non-adaptive GT

y = B(c)xwhere: B(c) : The M × N contact matrix for the M measurements

Example

y1y2yM

= 1 0 1 0 1 00 1 0 1 0 0

0 1 1 0 1 1

x1x2x3..xN

,





The Dilution noise

The Dilution noise results from inactivity of defective samplesleading to false negativeThis has serious effects as we cannot entirely declare a grouphealthy if it tests negativeSimplified dilution model: each defective sample , becomesinactive for a particular measurement with probability �independent of other samples

Figure: dilution model channel




Decoder designSimulation results

Group Testing Simulations

Based on Ref[2] , [M.Cherchagchi, M.Vetterli]

we use random matrices with parameter q for contact matrix,i.e. a sample can be included in a measurement withprobability q

Simple dilution model with flipping probability �

Minimum support distance based decoder





Understanding the min support decoder

We will understand the min support decoder for noiseless case. Formin support decoder to work, the matrix must be K -disjunct

K-disjunct property

A boolean matrix A with n columns is called K -disjunct if, forevery subset S ⊆ [n] of the columns with |S | ≤ K , and everyi /∈ S , for ai the i th column of matrix A, we have:∣∣∣supp(ai ) \ (⋃j∈S supp(aj))∣∣∣ > 0





The support distance Decoder

For noiseless case

For any column ci of the contact matrix B(c), the decoder verifies

the following:

|supp(ci ) \ supp(y)| = 0

The coordinate xi is decided to be nonzero if and only if theequality holds.

Decoder for noisy case

For, e = (1 + δ)Mq, any column ci of the contact matrix B(c), if

|supp(ci ) \ supp(y)| ≤ e

The coordinate xi is decided to be 1, else it is 0.





Simulation Results

For N = 100, 000 , K = 10, and q = 0.04, the simulation resultswere:





Simulation Results

For N = 100, 000 , K = 10, and q = 0.04, the number ofmeasurements for 99% success




Problem setupLower bound

Information theoretic model of Group Testing

Information theoretic models are useful as they provide lowerand achievable bounds on the number of measurements

V.Saligrama and G.Atia, Ref[1], have attempted this problemand have obtained achievability and lower bounds which areoptimal to a log factor.

We try to build a different model(by using typical decodingmethods instead of error exponents) and try to obtain tighterbounds





Problem setup

y1y2y3

= 1 0 1 0 1 00 1 0 1 0 0

0 1 1 0 1 1

x1x2x3..xn

,

We can model the problem as a channel coding problem withN users





Problem setup

1 N channel users, and K active users

2 Useri has codebook containing 2 codewords,[000...0] and acodeword ci

3 If useri is active, XMi = ci is transmitted , else X

Mi = [000...0]

4 In the noiseless case, the output codeword YM is the bit-wiseOR addition of the transmitted codewords.





Channel Model

Figure: channel model

Noisy group testing model : YM =N∨i=1Z(XMi ) +N

The decoder: g(YM) = ω̂, forω̂ ∈W where, W is the set ofall K − sets of active users





Noisy group testing model

Table: Parameter comparison

Group Testing Channel-coding problem

N number of samples number of usersK number of defective samples number of active usersM number of measurements size of the codewords transmittedq parameter for contact matrix construction parameter for codeword construction� dilution noise-flipping probability Z − channel 1→ 0 probability

Figure: dilution model channelKedar Tatwawadi Group Testing




Lower bound

Theorem (Lower bound)

For successful decoding of the K − set of active users, the lowerbound on M is :

M ≥ log(NK)

I (Y ;X1,X2,...,XN)

Existing Lower bounds

The lower bound given by [V.Saligrama],Ref[1]

M0 ≥ maxi :(S1,S2)∈S

log(N−K+i

i

)I (XS1 ;Y ,XS2)

(1)

S1 and S2 are partitions of size i and K − i , respectively of theK − set of active users





Proof arguments

If we know some more information about the distribution ofthe elements in K − set, then the number of measurementswill decrease

For example: K − i active users are from the first N1 usersand i from the remaining N − N1 users, then lower boundshould decrease

Lower bound is equal tolog(N−K+ii )I (XS1 ;Y ,XS2 )

, for N1 = K − i





Lower bound for different models

Noiseless group testing

The lower bound obtained is:

I (X1,X2, ...,XN ;Y ) = Hb((1− q)K ),M ≈ ΘKlog(N

K)

Noisy group testing

define: p = �q + (1− q)

I (X1,X2, ...,XN ;Y ) = H(Y )− H(Y |X1,X2, ...,XN)

≈ KpK log( 1p

)− Kq�pK−1log(1�

)

M ≈ Θ(K2log(N)

(1− �)2)




Other problems considered

Effect of different/inaccurate dilution model

Most of the times, the simplified dilution rule, each samplecan be inactive independently with probability � is used

Try to analyze the inaccuracies due to such a model ( forexample, consider the dorfman blood sampling example)

Different channel coding model necessary

Finding a subset of healthy samples

Provide a set of a given number of healthy samples

Finds application in many cases( eg: in the spectrum holesearch problem in cognitive radio networks)

C.Murthy & A.Sharma ,Ref[6] have given a informationtheoretic model for this problem





Graph-constrained group testing

M.cheragchi, V.Saligrama in Ref[7] have discussed grouptesting when the measurements are subjected to graphicalconstraints.

Practical relevance: Finding which link is down in a connectedsensor network.

Formulating an information theoretic model for the problem,and obtaining bounds

Structured sparsity in compressed sensing

Analyzing the effect of group sparsity, only one in a subsettypes of sparsities

Applications in Multi-user detection problems with delay, etc




Thank You

Thank You




References I

1 G. Atia and V. Saligrama Boolean compressed sensing andnoisy group testing 2009, arxiv 0907.1061

2 R. Dorfman ”The detection of defective members of largepopulations” Ann. Math. Statist., vol. 14, pp. 436-440, 1943

3 M. Cheraghchi , A. Hormati , A. Karbasi and M. Vetterli”Compressed sensing with probabilistic measurements: Agroup testing solution” Proc. Allerton Conf. Commun.,Contr. Computat. (UIUC), 2009

4 Leonardo Baldassini, Oliver Johnson, Matthew Aldridge: Thecapacity of adaptive group testing. ISIT 2013: 2676-2680




References II

5 Limits on Support Recovery of Sparse Signals viaMultiple-Access Communication Techniques, Yuzhe Jin,Young-Han Kim, Bhaskar Rao, IEEE Trans. on InformationTheory, Dec. 2011.

6 Abhay Sharma, Chandra R. Murthy: On Finding a Subset ofHealthy Individuals from a Large Population. CoRRabs/1307.8240 (2013)

7 Graph-Constrained Group Testing: M.cherchagchi,V.Saligrama,2011


IntroductionIntroduction to group testingMathematical modelling

Group Testing SimulationsDecoder designSimulation results

Information theoretic model of group testingProblem setupLower bound


Group Testing with a Compressed Sensing Perspective€¦ · Based on Ref[2] , [M.Cherchagchi,...

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