**1. Introduction**

Group Testing (GT) is a underdetermined problem in [1], and numerous methods have been developed to solve their problems. GT has become relevant in various problems including probabilistic approaches. The expansion of compressive sensing goes back to the fundamental idea of GT because it is an effort to find sparse signals [2,3]. Recently, academia has begun using the GT method as on vital approach to finding confirmed COVID-19 cases, showing this field's potential importance in these uncertain times [4,5]

The first study for GT was proposed by Dorfman [1]. The background to the emergence of GT is that a large project was conducted in the United States to find soldiers with syphilis during World War II. Syphilis testing of inviduals involves taking a blood sample, then analyzing that to produce a positive or negative result for syphilis in that patient. The syphilis testing carried out at the time was very inefficient since it took a lot of time and money to test all the soldiers one by one [3]. After all, if *N* soldiers are individually tested for syphilis, *N* tests are required. Note that the number of soldiers infected with syphilis is very small compared to the total number of soldiers. That is why it is probably inefficient to test every soldier for syphilis one by one, and why the GT technique emerged. The initial GT model was performed in the following way [1]. Several soldiers' blood samples were randomly selected, and the blood was put into a pool and mixed. Then, the blood pool was checked to see if it activated to syphilis or not. A positive result indicates that at least one of the soldiers in the pool was infected with syphilis. A negative result, on the other hand, indicates that all soldiers in the pool were free of syphilis. GT is attractive because the number of tests can be drastically reduced in the case of fewer soldiers infected with

**Citation:** Seong, J.-T. Theoretical Bounds on the Number of Tests in Noisy Threshold Group Testing Frameworks. *Mathematics* **2022**, *10*, 2508. https://doi.org/10.3390/ math10142508

Academic Editors: Gurami Tsitsiashvili and Alexander Bochkov

Received: 9 June 2022 Accepted: 15 July 2022 Published: 19 July 2022

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syphilis. After these beginnings, GT has mainly been studied with two different approaches, each forming a field of research of its own. One these fields is how to generate GT models. That is, it is a method of selecting samples to be included in one test pool. The second area is to reconstruct defective samples with as few tests as possible. GT loses its benefits if the requirement for a large number of retests leads to as many tests as the number of tests for individual screening.

For GT, various models have been proposed in consideration of how the test results express positive and negative results and the presence or absence of noise. In general, GT's test results told us to see if the pool under being tested contains one or more defective samples. That is, a positive or negative result indicates whether at least one of the defective samples in the pool are present. The model called quantitative GT [3] is a generalization framework of GT. The test result of quantitative GT indicates the number of defective samples in the test pool. There is also another GT model called Threshold Group Testing (TGT) [6]. In the TGT model, a test result of a pool is positive or negative as in conventional GT schemes. However, unlike the conventional GT model, the positive result occurs only when the number of defective samples in the pool is greater than a given threshold. Otherwise, the test outcome is negative. The TGT model is used because it can represent situations in which the test result can be different depending on whether it is high or low, such as the COVID-19 virus concentration. A modified GT model in which measurement noise causes false negatives or false positives is also considered.

TGT problems have been dealt with in various fields such as construction of TGT models [7], theoretical analysis of performance [8], and efficient model design [9,10]. However, there have been no studies so far to quantify how much measurement noise affects performance of TGT models. In this paper, we consider a Noisy Threshold Group Testing (NTGT) model. We provide guidelines for designing a NTGT model that is robust and reliable to measurement noise. To this end, a lower bound on the number of tests is derived using Fano's inequality. We show the trade-off relationship between the sparsity of the group matrix and the defective rate of the signal. And we obtain an upper bound on the probability of an error using the MAP decoding method. We show necessary and sufficient conditions on the number of tests required for finding a set of given defective samples using the lower and upper bounds.

#### **2. Related Work**

We look through previous studies and their significance to GT. Then, we will classify each type of problem related to current approaches to GT and consider the issues surrounding these problems. The study of GT first began in 1943 [1]. Dorfman made an effort to find a small number of syphilis-infected soldiers. Dorfman performed the GT with the following procedure. When testing for syphilis, all the soldiers were divided into various groups that were equal in size, then individual testing was only performed on soldiers from the groups that had recoded positive test results. In [1], the optimal group size for a given total number of samples and defective rate was summarized and presented. Later, Sterrett improved the performance by slightly modifying the existing GT method [11]. The main idea of Sterrett's approach is that once the first positive result is obtained, the remaining untested individuals are put in one large grouped and tested. Other than that, there is no difference between Sterrett's method and Dorfman's. If there is a low infection rate, Sterrett's method is more efficient because most of the samples are normal. A more general GT has been presented in [12], in which several algorithms were developed for finding defective samples when no infection rate exists. The paper [12] also provided a link between information theory and GT, introduced a new application of GT, and discussed the generalization of GT.

GT is classified based on types of defective sample distributions and decoding approaches. A probabilistic model uses the assumption that a defective sample is generated from a given probability distribution. On the other hand, the combinatorial model is an attempt to find defective samples without knowledge of probability distributions [13,14]. A typical example of this model is the minmax algorithm [15]. In [16], the results of improved performance in the combinatorial model were presented. Looking at other classes, the adaptive case is a model in which samples to be included in one pool are not independent of the results of previous tests. The samples to be used for the next round are changed each time based on the results of previous tests. Specifically, the method of selecting samples to be included in the next pool is optimized by using the results obtained from previous tests. Conversely, in the non-adaptive model, all tests are performed at the same time by a sample selection process defined in advance. So in this model, every test is independent of each other. This model offers the advantage of being able to test simultaneously regardless of the test order. When predetermined multiple steps are used, the non-adaptive model is extended to multi-stage models [1,17]. In fact, although the adaptive model has more constraints in GT design than the non-adaptive model, the adaptive model generally outperforms the non-adaptive model [3]. However, the recent research in [18] showed limitations in improving the performance of the adaptive model. Non-adaptive GTs are more efficient if all tests are being performed at the same time.

We now look at the significance of certain recent studies on noisy GT. The work in [19] showed the information-theoretic performance of GT with and without measurement noise. Several studies have recently showed interesting and significant performance. In [17], the proposed algorithms uses positive rates in the group to be included for each sample. In this case, if it is greater than the set value, the sample is considered as defective. This approach does not lead to optimal performance in all domains, but it follows a scaling law for a specific domain. In [19], there is separate testing for signals, and all of the group testing is carried out while still considering each sample. That is, although no individual testing is performed, samples use a binary value such as positive or negative. In the case of samples affected by symmetric noise, it was shown that the minimum number of tests reduces to a proportional to *K* log *N* of the optimal information-theoretic bounds for identifying any *K* defectives samples in a population of *N* samples [19].

In [20,21], for noisy addition, GT algorithms were presented using message passing and linear programming. Although it does not guarantee optimal performance for decoding complexity, the algorithm proposed in [22] is capable of realistic runtime in terms of that case of a large population. Although many studies have been performed on the noiseless version of GT models, it has been considered as an assumption that the test results are always pure. But this is not realistic. In addition, most of the noisy GT approaches to deal with measurement noise were performed by considering the symmetric noise model such as binary symmetric channel mentioned in channel coding theory. The symmetric noise model referred to in this paper assumes that the test results have the same probability of occurrence of false negatives and false positives. However, asymmetric noise models are more natural than symmetric ones in various applications. For example, data forensics in [23] is an example of using noisy GT models where it identifies to see if recoded files are changed.

#### **3. Noisy Threshold Group Testing Framework**

## *3.1. Problem Statement*

We define our NTGT problem. Let be the input **x** expressed as a binary vector of size *N*, **x** = (*<sup>x</sup>*1, *x*2, ··· , *xN*), **x** ∈ {0, <sup>1</sup>}*<sup>N</sup>*. For *i* ∈ [*N*], *xi* is the *i*-th element of **x**. *xi* is expressed in binary to identify either a defective sample or a normal sample. In other words, if the *i*-th sample is defective, *xi* = 1, otherwise *xi* = 0. Throughout this work, we assume that *xi* has the following probability,

$$\Pr(\mathbf{x}\_{i} = \boldsymbol{a}) = \begin{cases} 1 - \delta & \text{if } \boldsymbol{a} = 0, \\ \delta & \text{if } \boldsymbol{a} = 1, \end{cases} \tag{1}$$

where *δ* is the defective sample rate, and *α* is a dummy variable. In this case, the defective sample rate is less than 0.5, 0 < *δ* < 0.5, which is considered a small value for GT problems.

As mentioned earlier, one of the key points in the GT problems is to determine which samples to participate in a pool. In this paper, samples to be included in the pool are

selected using a non-adaptive model. We use a matrix as a more concise way to define the samples to be included in the pool. Let be the group matrix which has *M* rows and *N* columns as denoted **A** ∈ {0, <sup>1</sup>}*<sup>M</sup>*×*N*, where *M* is the number of tests in the NTGT model. Note that we aim for a small *M* as the number of tests required to reconstruct the signal **x**. When the *j*-th test includes *i*-th sample *xi* and performs GT, it is expressed as *Aji* = 1. Otherwise, *Aji* = 0. Whether *i*-th sample is included in the *j*-th test and performs GT, is expressed as a binary value, i.e., 0 or 1, of each element *Aji* of the group matrix. Although the *d*-Separable matrix and the *d*-Disjunct matrix [3] were used to design the group matrix, the approach of randomly selecting the elements of the group matrix is also known to be a good design method [3]. For *i* ∈ [*N*] and *j* ∈ [*M*], *Aji* is identically independent distributed as follows:

$$\Pr\left(A\_{ji}=\kappa\right) = \begin{cases} 1-\gamma & \text{if } \kappa = 0, \\ \gamma & \text{if } \kappa = 1, \end{cases} \tag{2}$$

where *γ* denotes the sparsity of the group matrix and the range of *γ* is 0 < *γ* < 1. As *γ* increases, the density of the group matrix also increases. Conversely, as they ge<sup>t</sup> smaller, increasingly sparse group matrices are designed. It should be noted that the computational complexity of the GT framework also increases when a group matrix is constructed from a large *γ*. Therefore, it is necessary to design GT frameworks with as low as possible the sparsity of group matrices while improving the reconstruction performance. We will consider how the relationship between *δ* and *γ* affects the number of tests for signal reconstruction.

The reason we are considering the NTGT model is as follows. Consider a model that could be used for the diagnosis of COVID-19 infection. There are cases in which the COVID-19 test showed false positive or false negative results when the concentration of the virus was low or contaminated. The current diagnosis of COVID-19 infection is positive when the virus concentration is above a certain level. During the incubation period or early stage of infection, the virus concentration is low, and false negative results may be obtained. In addition, even if the COVID-19 infection is confirmed using a precise and accurate diagnostic method, the result is sometimes reversed due to unexpected measurement noise. Throughout this work, a NTGT model suited to these challenges is considered. In other words, we consider the best approach to a TGT scheme where positive and negative cases occur by the quatitative concentration, and we consider an additive noise model because measurement noise can reverse the results. In a recent study [24], for the diagnosis of COVID-19 infection, false positives and false negatives were reported to be between 0.1% and 4.5%, respectively. Next, we obtain lower and upper performance bounds on the NTGT model in Sections 4 and 5.

TGT is different from conventional GT models. In conventional GT, if at least one defective sample exists in one test, the output is positive without measurement noise. However, TGT is positive when there is a number of defective samples greater than the predefined threshold *T*. For example, *T* = 3 means that a positive result occurs only when there are at least three defective samples in the pool. Once there is only one defective sample in the pool, its result would be negative. In other words, the result in the pool becomes positive only when it is above *T* for TGT models, also whether it is negative or positive in the diagnosis of COVID-19 infection depends on whether the virus concentration is high or low. The conventional GT uses *T* = 1. The following (3) presents an output for a TGT model. Let *zj* be the result of the *j*-th test pool, which does not suffer from noise, where *zj* = 1 indicates a positive result and 0 for a negative result, *j* ∈ [*M*], **z** = (*<sup>z</sup>*1, *z*2, ··· , *zM*).

$$z\_{j} = \begin{cases} 0 & \text{if } \sum\_{i=1}^{N} A\_{ji} x\_{i} < T\_{\prime} \\ 1 & \text{if } \sum\_{i=1}^{N} A\_{ji} x\_{i} \ge T\_{\prime} \end{cases} \tag{3}$$

Through this paper, we consider the NTGT framework with measurement noise. Assume a model whose results can be flipped due to the measurement noise. *zj* is the pure

result of the pool test, and its result converts from positive to negative and vice versa due to additive noise. For the NTGT model, the additive noise is defined as follows:

$$\Pr(e\_j = a) = \begin{cases} 1 - \eta & \text{if } a = 0, \\ \eta & \text{if } a = 1, \end{cases} \tag{4}$$

where *η* is the measurement noise, and we assume all *ej* are independent of each other. Therefore, the *j*-th output *yj* in the NTGT model can be written as

$$y\_j = z\_j \oplus \mathfrak{e}\_j \tag{5}$$

where the symbol ⊕ denotes the logical operation XOR. We denote **y** = (*y*1, *y*2, ··· , *yM*) and **e** = (*<sup>e</sup>*1,*e*2, ··· ,*eM*).

Figure 1 shows an example of this NTGT. In this example, two samples out of ten are defective, which is realized from (1). As shown in Figure 1, the number of tests is 7, *M* = 7. The 7 × 10 group matrix is constructed by (2) mentioned above. For noiseless version, the vector **z** is (0, 0, 1, 0, 0, 0, 0) with *T* = 2. In the third test only, the number of defective samples becomes two, and the test result is positive. When additive noise is added as defined in (4), the output is **y** = (1, 0, 1, 0, 0, 0, <sup>0</sup>).

**Figure 1.** One example of NTGT where *M* = 7, *N* = 10, *T* = 2, the black boxes denotes 1 s, and white ones 0 s.

**x**
