Information theoretic and algorithmic aspects algorithms for GT and - - PowerPoint PPT Presentation

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Information theoretic and algorithmic aspects algorithms for GT and - - PowerPoint PPT Presentation

Jan 07, 2023 297 likes •937 views

Information- theory and Information theoretic and algorithmic aspects algorithms for GT and QGT of binary and quantitative group testing in the Oliver Gebhard sublinear regime Basic setup Binary group testing Oliver Gebhard Quantitative

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Information theoretic and algorithmic aspects

  • f binary and quantitative group testing in the

sublinear regime

Oliver Gebhard

Goethe-University Frankfurt gebhard@math.uni-frankfurt.de Joint work with

  • A. Coja-Oghlan, M.Hahn-Klimroth, P. Loick

and M.Hahn-Klimroth, D. Kaaser, P. Loick

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Overview

1 Basic setup 2 Binary group testing 3

Quantitative group testing

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Basic setup

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Basic setup

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Basic setup

Goal:Are we able to identify the sick individuals?

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Basic setup

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Basic setup

  • Goal:Are we able to reduce the number of tests?
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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Basic setup

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Basic setup

Binary: Is a sick individual contained? Quantitative: Number of sick individuals?

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Basic setup

Options to choose the underlying testing procedure

1 Number of stages 2 Pooling procedure

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Basic setup

Options to choose the underlying testing procedure

1 Number of stages 2 Pooling procedure

To Do: Rigorous Analysis of the choice

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Basic setup

n Number of tests

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Basic setup

n Number of tests Information Theoretic Bound

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Basic setup

n Number of tests Information Theoretic Bound Algorithmic Bound

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Binary group testing

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Binary group testing

x1 x2 x3 x4 x5 x6 x7 a1 a2 a3 a4 a5

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Binary group testing: Information Theory

Counting Bound implies m > k ⋅ log2(n/k).

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Binary group testing: Information Theory

Counting Bound implies m > k ⋅ log2(n/k). Baldassini et al: Adaptive testing strategies achieve this bound.

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Binary group testing: Information Theory

Counting Bound implies m > k ⋅ log2(n/k). Baldassini et al: Adaptive testing strategies achieve this bound. Question: Is non-adaptive group testing able to achieve the bound as well?

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Binary group testing: Information Theory

Assume a constant weight testing scheme

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Binary group testing: Information Theory

Assume a constant weight testing scheme The underlying graph structure:

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Binary group testing: Information Theory

Assume a constant weight testing scheme The underlying graph structure:

Bipartite Factor Graph Fixed variable nopde degree

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Binary group testing: Information Theory

Assume a constant weight testing scheme The underlying graph structure:

Bipartite Factor Graph Fixed variable nopde degree

Theorem 1 [CGHL18] Let minf =

nθlog(n) min{1, 1−θ

θ log(2)}log(2)

1 m < (1 − ǫ)minf : No algorithm exists to output the right

configuration for the constant weight pooling

2 m > (1 + ǫ)minf : There exist an algorithm, which outputs

the right configuration w.h.p.

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Binary group testing: Information Theory

Lower Bound: Derive the value m∗, below which infected/uninfected individuals occur that may swap status without harming the test result

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Binary group testing: Information Theory

Lower Bound: Derive the value m∗, below which infected/uninfected individuals occur that may swap status without harming the test result Upper Bound: Derive the value m∗∗ above which no satisfying assignment beside the original one exists

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Binary group testing: Information Theory

Lower Bound: Derive the value m∗, below which infected/uninfected individuals occur that may swap status without harming the test result Upper Bound: Derive the value m∗∗ above which no satisfying assignment beside the original one exists

1 Small overlap argument: Analyse probability that

configuration with certain overlap fulfills the test result

2 Large overlap argument: Analyse probability that positive

(negative) tests stay positive (negative)

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Binary group testing: Information Theory

Lower Bound: Derive the value m∗, below which infected/uninfected individuals occur that may swap status without harming the test result Upper Bound: Derive the value m∗∗ above which no satisfying assignment beside the original one exists

1 Small overlap argument: Analyse probability that

configuration with certain overlap fulfills the test result

2 Large overlap argument: Analyse probability that positive

(negative) tests stay positive (negative)

m∗,m∗∗ set conditions to derive the minf (n,θ) as stated

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Binary group testing: Algorithmic Aspect

Allemann: multi-stage algorithm at the predicted lower bound

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Binary group testing: Algorithmic Aspect

Allemann: multi-stage algorithm at the predicted lower bound Aldridge, Scarlett et. al.: Sub-optimal non-adaptive strategies available

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Binary group testing: Algorithmic Aspect

Allemann: multi-stage algorithm at the predicted lower bound Aldridge, Scarlett et. al.: Sub-optimal non-adaptive strategies available Most promising algorithms: SCOMP, Definite Defective

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Binary group testing: Algorithmic Aspect

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Binary group testing: Algorithmic Aspect

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Binary group testing: Algorithmic Aspect

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Binary group testing: Algorithmic Aspect

x1 x2 x3 x4 x5 x6 x7 Truth x1 x2 x3 x4 x5 x6 x7 DD-Estimation

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Binary group testing: Algorithmic Aspect

x1 x2 x3 x4 x5 x6 x7 x8 Truth x1 x2 x3 x4 x5 x6 x7 x8 a1 a2 a3 a4 a5 a6 SCOMP-Start

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Binary group testing: Algorithmic Aspect

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Binary group testing: Algorithmic Aspect

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Binary group testing: Algorithmic Aspect

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Binary group testing: Algorithmic Aspect

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Binary group testing: Algorithmic Aspect

x1 x2 x3 x4 x5 x6 x7 x8 Truth x1 x2 x3 x4 x5 x6 x7 x8 SCOMP-Estimation

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Binary group testing: Algorithmic Aspect

Conjecture: SCOMP outperforms the Definite Defective

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Binary group testing: Algorithmic Aspect

Conjecture: SCOMP outperforms the Definite Defective We refute the conjecture by showing that the algorithms fail at the same point

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Binary group testing: Algorithmic Aspect

Conjecture: SCOMP outperforms the Definite Defective We refute the conjecture by showing that the algorithms fail at the same point Theorem 2 [CGHL18] Let malg(n,θ) =

k log(n/k) min{1, 1−θ

θ log2 2},0 < θ < 1 and ǫ > 0:

For m < (1 − ǫ)malg(n,θ), both SCOMP and DD fail to output σ w.h.p.

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Binary group testing: Algorithmic Aspect

DD work iff every infected individual is in at least one test with only uninfected individuals that are themselves in at least one test with only other uninfected individuals Additional SCOMP step is a Greedy Vertex Cover Show w.h.p.: Local structure of infected/ uninfected individuals in remaining graph look the same SCOMP fails in first step w.h.p. ⇒ SCOMP and DD have the same algorithmic threshold

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Quantitative group testing

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Quantitative group testing

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Quantitative group testing

Counting bound: (k + 1)m ≥ (n

k).

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Quantitative group testing

Counting bound: (k + 1)m ≥ (n

k).

Alaoui et. al.: For the linear case Information Theoretic phase transition and efficient algorithm at lower bound established

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Quantitative group testing

Counting bound: (k + 1)m ≥ (n

k).

Alaoui et. al.: For the linear case Information Theoretic phase transition and efficient algorithm at lower bound established For sub-linear regime: Information Theory not entirely understood (Djackov) and only sub-optimal algorithms available (Karimi et. al.) .

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Quantitative group testing

Counting bound: (k + 1)m ≥ (n

k).

Alaoui et. al.: For the linear case Information Theoretic phase transition and efficient algorithm at lower bound established For sub-linear regime: Information Theory not entirely understood (Djackov) and only sub-optimal algorithms available (Karimi et. al.) . Contribution:

1 Establish sharp phase transition in the sublinear regime

(i.e. we show the achievability).

2 Introduce a Greedy Algorithm that outperforms the best

known one in certain sparsity levels

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Quantitative group testing: Information Theory

Theorem 3 [GHKL19] Suppose that 0 < θ < 1,k = nθ and ǫ > 0 and let minf = 21 − θ θ k

1 For m < (1 − ǫ)minf (n,θ): No algorithm that outputs σ

exists

2 For m > (1 + ǫ)minf (n,θ): An algorithm exists that

  • utputs σ with high probability
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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Quantitative group testing: Information Theory

Derive a m∗ s.t. for m > m∗ no second satisfying assignment exists

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Quantitative group testing: Information Theory

Derive a m∗ s.t. for m > m∗ no second satisfying assignment exists Analyse it for small and high overlaps

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Quantitative group testing: Information Theory

Derive a m∗ s.t. for m > m∗ no second satisfying assignment exists Analyse it for small and high overlaps

1 Large overlap: Analyse changes via Balls and Bins 2 Small overlap: Analyse changes as returning random walk

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Quantitative group testing: Information Theory

Derive a m∗ s.t. for m > m∗ no second satisfying assignment exists Analyse it for small and high overlaps

1 Large overlap: Analyse changes via Balls and Bins 2 Small overlap: Analyse changes as returning random walk

Analysis establishes conditions for minf

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Quantitative group testing: Algorithmic aspect

Maximum-Neighborhood Algorithm [GHKL19] Input: G,y,k Output: Estimation ˜ ⃗ σ for ⃗ σ.

  • 1. For every xi for i ∈ [n] calculate Ψi = ∑j∈∂xi yj
  • 2. Set Ψ′

i ∶= Ψi ⋅ m/2 ∆i

  • 3. Order the individuals i in decreasing order due to Ψ′

i

  • 4. Declare the first k ordered individuals as infected, declare

the other individuals as healthy

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Quantitative group testing: Algorithmic aspect

Maximum-Neighborhood Algorithm [GHKL19] Input: G,y,k Output: Estimation ˜ ⃗ σ for ⃗ σ.

  • 1. For every xi for i ∈ [n] calculate Ψi = ∑j∈∂xi yj
  • 2. Set Ψ′

i ∶= Ψi ⋅ m/2 ∆i

  • 3. Order the individuals i in decreasing order due to Ψ′

i

  • 4. Declare the first k ordered individuals as infected, declare

the other individuals as healthy Theorem 4 [GHKL19] Define mMN(n,θ) = 1+

√ θ 1− √ θk log(n/k). The MN-Algorithm

  • utputs the correct configuration w.h.p. if m > mMN
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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Quantitative group testing: Algorithmic aspect

Seperate the distribution of Ψi for uninfected and infected individuals.

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Quantitative group testing: Algorithmic aspect

Seperate the distribution of Ψi for uninfected and infected individuals. Union bound over all k infected and n − k uninfected individuals.

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Quantitative group testing: Algorithmic aspect

Seperate the distribution of Ψi for uninfected and infected individuals. Union bound over all k infected and n − k uninfected individuals. Applying the Chernoff Bound and optimizing w.r.t. the seperating-parameter gives the (sufficient) condition on mMN.

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Quantitative group testing: Algorithmic aspect

Recovery success rate n = 1000 individuals. 1000 rounds of simulation. Dashed lines: Asymptotic prediction of the required number of tests.

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Information- theory and algorithms for GT and QGT Oliver Gebhard Basic setup Binary group testing Quantitative group testing

Thank you for your attention