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arxiv: 2606.11353 · v1 · pith:CMUPMM6Fnew · submitted 2026-06-09 · 💻 cs.IT · math.IT· math.PR

An Information-Theoretic Analysis of Threshold Group Testing

Remco van der Hofstad , Noela M\"uller , Connor Riddlesden This is my paper

Pith reviewed 2026-06-27 11:23 UTC · model grok-4.3

classification 💻 cs.IT math.ITmath.PR
keywords threshold group testinginformation-theoretic boundsphase transitionnon-adaptive testingconstant-column designdefective prevalencesparse recovery
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The pith

Threshold group testing requires c k log(n/k) non-adaptive tests where c depends on defect prevalence and threshold value.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives a sharp information-theoretic phase transition for the minimal number of tests in non-adaptive threshold group testing under the constant-column design. It shows this transition occurs at c_inf^TGT k log(n/k) tests, with the constant expressed in terms of the fraction of defectives and the threshold. The value of the constant matches classical group testing when defectives are rare but is smaller at higher prevalence, indicating fewer tests are needed. Evidence is given that threshold group testing becomes strictly harder than classical group testing when defectives form a positive fraction of all items. The upper bound rests on an analytic assumption verified only for threshold equal to 2.

Core claim

In the constant-column design for non-adaptive noiseless threshold group testing, there is a sharp information-theoretic transition at c_inf^TGT k log(n/k) tests, with c_inf^TGT a function of the defective prevalence and the threshold value. The upper bound holds under an analytic assumption verified for threshold 2. For small prevalence this matches classical group testing, while higher prevalence yields a reduction in tests due to the threshold. When the proportion of defectives is bounded below by a positive constant, threshold group testing requires strictly more tests than classical group testing.

What carries the argument

The threshold constant c_inf^TGT that sets the location of the sharp phase transition for the number of tests required in threshold group testing.

If this is right

  • In the low-prevalence regime threshold group testing and classical group testing require the same order of tests.
  • At moderate prevalence the threshold produces a strict reduction in the number of tests needed.
  • When the fraction of defectives is bounded away from zero, threshold group testing demands more tests than the classical case.
  • The phase transition result is specific to the constant-column test design.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The analytic assumption used for the upper bound may be provable for thresholds larger than 2.
  • The same constant-column analysis could be extended to the noisy observation model.
  • Applications such as pooled testing with variable infection rates could benefit from choosing thresholds to operate in the reduced-test regime.

Load-bearing premise

The upper bound on the number of tests assumes an analytic condition whose validity is only checked explicitly for a threshold value of 2.

What would settle it

A calculation of the mutual information for threshold value 3 that yields a different leading constant from the predicted c_inf^TGT, or a numerical check showing the transition point deviates from c k log(n/k).

Figures

Figures reproduced from arXiv: 2606.11353 by Connor Riddlesden, Noela M\"uller, Remco van der Hofstad.

Figure 1
Figure 1. Figure 1: Graphical representation of a pooling scheme for [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Left: The difference between the information-theoretically optimal constant for [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: illustrates the behaviour of the function and its first five derivatives, high￾lighting why an analytical argument is not straightforward. 1.0 1.2 1.4 1.6 1.8 2.0 [1, 2] 0.0 0.2 0.4 0.6 0.8 1.0 1.2 h( ) r = 2.0 r = 2.2 r = 2.4 r = 2.6 r = 2.8 r = 3.0 1.0 1.2 1.4 1.6 1.8 2.0 [1, 2] 2 0 2 4 6 h 0( ) r = 2.0 r = 2.2 r = 2.4 r = 2.6 r = 2.8 r = 3.0 1.0 1.2 1.4 1.6 1.8 2.0 [1, 2] 20 10 0 10 20 h 00( ) r = 2.0 r… view at source ↗
read the original abstract

We study the Threshold Group Testing (TGT) problem in the noiseless and non-adaptive setting, where the objective is to exactly recover a sparse binary vector from pooled tests, using as few tests as possible. In TGT, each test applied to a subset of items returns a positive outcome if the number of 1's (defective items) in that subset meets or exceeds a specified threshold, and has a negative outcome otherwise. We investigate how the complexity of TGT compares to that of Classical Group Testing (CGT), corresponding to the special case of the threshold equal to one, and analyse the impact of increasing the threshold on the required number of tests. Our main contribution is the derivation of a sharp information-theoretic phase transition at $c_{\mathrm{inf}}^{\mathrm{TGT}}k\log(n/k)$ (non-adaptive) tests for TGT within the constant-column test design. The threshold constant $c_{\mathrm{inf}}^{\mathrm{TGT}}$ is expressed as a function of the prevalence of defectives and the threshold value. Our upper bound is derived under an analytic assumption, and we verify that this assumption is satisfied for a threshold value of 2. The value of $c_{\mathrm{inf}}^{\mathrm{TGT}}$ reveals that TGT on the constant-column design has the same information-theoretic behaviour as CGT in the low-prevalence regime. Yet, strikingly, at higher prevalences, the threshold leads to a significant reduction in the number of tests. On the other hand, we provide evidence that when the asymptotic proportion of defective items is positive, TGT actually becomes strictly harder than CGT (excluding trivial reductions).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript analyzes non-adaptive noiseless threshold group testing (TGT) in the constant-column design. It derives a sharp information-theoretic phase transition at c_inf^TGT k log(n/k) tests, with c_inf^TGT a function of defective prevalence and threshold t. The lower bound holds generally; the matching upper bound is obtained under an analytic assumption verified only for t=2. TGT matches classical group testing (CGT) at low prevalence, requires fewer tests at higher prevalences, and is strictly harder than CGT when the defective proportion is positive.

Significance. If the upper bound construction extends to general t, the result supplies a precise, design-specific characterization of TGT sample complexity and identifies regimes in which raising the threshold reduces the number of tests relative to CGT. The explicit dependence of the constant on prevalence and t, together with the comparison to the t=1 case, would be a useful addition to the group-testing literature.

major comments (2)
  1. [Abstract] Abstract / main contribution: the claim of a sharp phase transition at c_inf^TGT k log(n/k) for general thresholds requires matching upper and lower bounds. The upper bound is derived under an analytic assumption that the manuscript states is verified only for threshold value 2; no verification or counter-example is supplied for t>2. This directly affects the central claim that the constant is tight for arbitrary t.
  2. [Abstract] The statement that TGT 'has the same information-theoretic behaviour as CGT in the low-prevalence regime' and 'a significant reduction in the number of tests' at higher prevalences rests on the same upper-bound construction. Because the construction is conditional on the unverified assumption for t>2, the comparative claims are not yet established for general thresholds.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and for highlighting the scope of the upper-bound result. We address the two major comments below.

read point-by-point responses

  1. Referee: [Abstract] Abstract / main contribution: the claim of a sharp phase transition at c_inf^TGT k log(n/k) for general thresholds requires matching upper and lower bounds. The upper bound is derived under an analytic assumption that the manuscript states is verified only for threshold value 2; no verification or counter-example is supplied for t>2. This directly affects the central claim that the constant is tight for arbitrary t.

    Authors: We agree that the upper bound holds only under the stated analytic assumption, which the manuscript verifies solely for t=2. The information-theoretic lower bound is unconditional. Because no general verification or counter-example is currently available for t>2, the claim of a sharp (matching) phase transition is conditional for thresholds other than 2. We will revise the abstract and the statement of the main theorem to make this limitation explicit and will add a short discussion of the assumption's plausibility for small t>2 based on the explicit expressions already computed. revision: partial

  2. Referee: [Abstract] The statement that TGT 'has the same information-theoretic behaviour as CGT in the low-prevalence regime' and 'a significant reduction in the number of tests' at higher prevalences rests on the same upper-bound construction. Because the construction is conditional on the unverified assumption for t>2, the comparative claims are not yet established for general thresholds.

    Authors: The low-prevalence equivalence follows directly from the fact that c_inf^TGT approaches the CGT constant as prevalence tends to zero; this limit does not rely on the analytic assumption. The claimed reduction at higher prevalences, however, does depend on the value of c_inf^TGT obtained from the upper-bound construction and is therefore conditional for t>2. We will revise the abstract and the relevant discussion paragraph to separate the unconditional low-prevalence statement from the conditional higher-prevalence comparison. revision: partial

standing simulated objections not resolved
  • Verification (or counter-example) of the analytic assumption for the upper-bound construction when the threshold t exceeds 2

Circularity Check

0 steps flagged

No circularity: phase-transition constant derived from independent mutual-information analysis

full rationale

The claimed sharp threshold at c_inf^TGT k log(n/k) is obtained by direct computation of the information-theoretic quantities (mutual information or entropy rates) for the constant-column design as a function of prevalence and threshold t. No equation reduces the final constant to a fitted parameter, a self-citation chain, or a definition that presupposes the result. The analytic assumption required only for the upper-bound construction is stated separately and does not enter the expression for c_inf^TGT itself; verification for t=2 is an external check rather than a definitional step. The derivation therefore remains self-contained against the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The upper-bound derivation invokes an unspecified analytic assumption whose verification is only given for threshold 2; the information-theoretic lower bound presumably relies on standard entropy calculations for the constant-column design.

axioms (1)
  • ad hoc to paper Analytic assumption required for the upper bound on the number of tests
    Explicitly stated in abstract as the basis for the matching upper bound; only verified for threshold 2.

pith-pipeline@v0.9.1-grok · 5839 in / 1342 out tokens · 21196 ms · 2026-06-27T11:23:21.715227+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Algorithms for Threshold Group Testing

    cs.IT 2026-06 unverdicted novelty 6.0

    Develops a spatially coupled inference algorithm for threshold group testing that achieves exact recovery at the information-theoretic threshold with a simpler proof than prior methods.

Reference graph

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