arXiv subjects

Thach V. Bui

Publications and source records attributed to Thach V. Bui.

13 recordsLinked to original sources

arXiv2025

Matrix Completion in Group Testing: Bounds and Simulations

The goal of group testing is to identify a small number of defective items within a large population. In the non-adaptive setting, tests are designed in advance and represented by a measurement matrix $\mM$, where rows correspond to tests and columns to items. A test is positive if it includes at least one defective item. Traditionally, $\mM$ remains fixed during both testing and recovery. In this work, we address the case where some entries of $\mM$ are missing, yielding a missing measurement matrix $\mG$. Our aim is to reconstruct $\mM$ from $\mG$ using available samples and their outcome vectors. The above problem can be considered as a problem intersected between Boolean matrix factorization and matrix completion, called the matrix completion in group testing (MCGT) problem, as follows. Given positive integers $t,s,n$, let $\mY:=(y_{ij}) \in \{0, 1\}^{t \times s}$, $\mM:=(m_{ij}) \in \{0,1\}^{t \times n}$, $\mX:=(x_{ij}) \in \{0,1\}^{n \times s}$, and matrix $\mG \in \{0,1 \}^{t \times n}$ be a matrix generated from matrix $\mM$ by erasing some entries in $\mM$. Suppose $\mY:=\mM \odot \mX$, where an entry $y_{ij}:=\bigvee_{k=1}^n (m_{ik}\wedge x_{kj})$, and $\wedge$ and $\vee$ are AND and OR operators. Unlike the problem in group testing whose objective is to find $\mX$ when given $\mM$ and $\mY$, our objective is to recover $\mM$ given $\mY,\mX$, and $\mG$. We first prove that the MCGT problem is NP-complete. Next, we show that certain rows with missing entries aid recovery while others do not. For Bernoulli measurement matrices, we establish that larger $s$ increases the higher the probability that $\mM$ can be recovered. We then instantiate our bounds for specific decoding algorithms and validate them through simulations, demonstrating superiority over standard matrix completion and Boolean matrix factorization methods.

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arXiv2024

Efficient designs for threshold group testing without gap

Given $d$ defective items in a population of $n$ items with $d \ll n$, in threshold group testing without gap, the outcome of a test on a subset of items is positive if the subset has at least $u$ defective items and negative otherwise, where $1 \leq u \leq d$. The basic goal of threshold group testing is to quickly identify the defective items via a small number of tests. In non-adaptive design, all tests are designed independently and can be performed in parallel. The decoding time in the non-adaptive state-of-the-art work is a polynomial of $(d/u)^u (d/(d-u))^{d - u}, d$, and $\log{n}$. In this work, we present a novel design that significantly reduces the number of tests and the decoding time to polynomials of $\min\{u^u, (d - u)^{d - u}\}, d$, and $\log{n}$. In particular, when $u$ is a constant, the number of tests and the decoding time are $O(d^3 (\log^2{n}) \log{(n/d)} )$ and $O\big(d^3 (\log^2{n}) \log{(n/d)} + d^2 (\log{n}) \log^3{(n/d)} \big)$, respectively. For a special case when $u = 2$, with non-adaptive design, the number of tests and the decoding time are $O(d^3 (\log{n}) \log{(n/d)} )$ and $O(d^2 (\log{n} + \log^2{(n/d)}) )$, respectively. Moreover, with 2-stage design, the number of tests and the decoding time are $O(d^2 \log^2{(n/d)} )$.

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arXiv2023

Concomitant Group Testing

In this paper, we introduce a variation of the group testing problem capturing the idea that a positive test requires a combination of multiple ``types'' of item. Specifically, we assume that there are multiple disjoint \emph{semi-defective sets}, and a test is positive if and only if it contains at least one item from each of these sets. The goal is to reliably identify all of the semi-defective sets using as few tests as possible, and we refer to this problem as \textit{Concomitant Group Testing} (ConcGT). We derive a variety of algorithms for this task, focusing primarily on the case that there are two semi-defective sets. Our algorithms are distinguished by (i) whether they are deterministic (zero-error) or randomized (small-error), and (ii) whether they are non-adaptive, fully adaptive, or have limited adaptivity (e.g., 2 or 3 stages). Both our deterministic adaptive algorithm and our randomized algorithms (non-adaptive or limited adaptivity) are order-optimal in broad scaling regimes of interest, and improve significantly over baseline results that are based on solving a more general problem as an intermediate step (e.g., hypergraph learning).

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arXiv2021

Improved algorithms for non-adaptive group testing with consecutive positives

The goal of group testing is to efficiently identify a few specific items, called positives, in a large population of items via tests. A test is an action on a subset of items which returns positive if the subset contains at least one positive and negative otherwise. In non-adaptive group testing, all tests are fixed in advance and can be performed in parallel. In this work, we consider non-adaptive group testing with consecutive positives in which the items are linearly ordered and the positives are consecutive in that order. We present two contributions here. The first is the direct use of a binary code to construct measurement matrices compared to the use of Gray code in the state-of-the-art work, which is a rearrangement of the binary code, when the maximum number of consecutive positives is known. This leads to a reduction in decoding time in practice. The second one is efficient designs to identify positives when the number of consecutive positives is known. To the best of our knowledge, this setting has not been surveyed yet. Our simulations verify the efficiency of our proposed designs. In particular, it only requires up to $300$ tests to identify up to $100$ positives in a set of $2^{32} \approx 4.3\mathrm{B}$ items in less than $300$ nanoseconds. When the maximum number of consecutive positives is known, the simulations validate the superiority of our proposed design in decoding compared to the state-of-the-art work. Moreover, when the number of consecutive positives is known, the number of tests and the decoding time are almost reduced half.

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arXiv2021

Improved non-adaptive algorithms for threshold group testing with a gap

The basic goal of threshold group testing is to identify up to $d$ defective items among a population of $n$ items, where $d$ is usually much smaller than $n$. The outcome of a test on a subset of items is positive if the subset has at least $u$ defective items, negative if it has up to $\ell$ defective items, where $0\leq\ell 0$, i.e., threshold group testing with a gap. Note that the results presented here are also applicable to the case $g = 0$; however, the results are not as efficient as those in related work. Currently, a few reported studies have investigated test designs and decoding algorithms for identifying defective items. Most of the previous studies have not been feasible because there are numerous constraints on their problem settings or the decoding complexities of their proposed schemes are relatively large. Therefore, it is compulsory to reduce the number of tests as well as the decoding complexity, i.e., the time for identifying the defective items, for achieving practical schemes. The work presented here makes five contributions. The first is a more accurate theorem for a non-adaptive algorithm for threshold group testing proposed by Chen and Fu. The second is an improvement in the construction of disjunct matrices, which are the main tools for tackling (threshold) group testing and other tasks such as constructing cover-free families or learning hidden graphs. The third and fourth contributions are a reduced exact upper bound on the number of tests and a reduced asymptotic bound on the decoding time for identifying defective items in a noisy setting on test outcomes. The fifth contribution is a simulation on the number of tests of the resulting improvements for previous work and the proposed theorems.

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arXiv2019

Efficient and error-tolerant schemes for non-adaptive complex group testing and its application in complex disease genetics

The goal of combinatorial group testing is to efficiently identify up to $d$ defective items in a large population of $n$ items, where $d \ll n$. Defective items satisfy certain properties while the remaining items in the population do not. To efficiently identify defective items, a subset of items is pooled and then tested. In this work, we consider complex group testing (CmplxGT) in which a set of defective items consists of subsets of positive items (called \textit{positive complexes}). CmplxGT is classified into two categories: classical CmplxGT (CCmplxGT) and generalized CmplxGT (GCmplxGT). In CCmplxGT, the outcome of a test on a subset of items is positive if the subset contains at least one positive complex, and negative otherwise. In GCmplxGT, the outcome of a test on a subset of items is positive if the subset has a certain number of items of some positive complex, and negative otherwise. For CCmplxGT, we present a scheme that efficiently identifies all positive complexes in time $t \times \mathrm{poly}(d, \ln{n})$ in the presence of erroneous outcomes, where $t$ is a predefined parameter. As $d \ll n$, this is significantly better than the currently best time of $\mathrm{poly}(t) \times O(n \ln{n})$. Moreover, in specific cases, the number of tests in our proposed scheme is smaller than previous work. For GCmplxGT, we present a scheme that efficiently identifies all positive complexes. These schemes are directly applicable in various areas such as complex disease genetics, molecular biology, and learning a hidden graph.

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arXiv2019

A framework for generalized group testing with inhibitors and its potential application in neuroscience

The main goal of group testing with inhibitors (GTI) is to efficiently identify a small number of defective items and inhibitor items in a large set of items. A test on a subset of items is positive if the subset satisfies some specific properties. Inhibitor items cancel the effects of defective items, which often make the outcome of a test containing defective items negative. Different GTI models can be formulated by considering how specific properties have different cancellation effects. This work introduces generalized GTI (GGTI) in which a new type of items is added, i.e., hybrid items. A hybrid item plays the roles of both defectives items and inhibitor items. Since the number of instances of GGTI is large (more than 7 million), we introduce a framework for classifying all types of items non-adaptively, i.e., all tests are designed in advance. We then explain how GGTI can be used to classify neurons in neuroscience. Finally, we show how to realize our proposed scheme in practice.

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arXiv2019

Sublinear decoding schemes for non-adaptive group testing with inhibitors

Identification of up to $d$ defective items and up to $h$ inhibitors in a set of $n$ items is the main task of non-adaptive group testing with inhibitors. To efficiently reduce the cost of this Herculean task, a subset of the $n$ items is formed and then tested. This is called \textit{group testing}. A test outcome on a subset of items is positive if the subset contains at least one defective item and no inhibitors, and negative otherwise. We present two decoding schemes for efficiently identifying the defective items and the inhibitors in the presence of $e$ erroneous outcomes in time $\mathsf{poly}(d, h, e, \log_2{n})$, which is sublinear to the number of items $n$. This decoding complexity significantly improves the state-of-the-art schemes in which the decoding time is linear to the number of items $n$, i.e., $\mathsf{poly}(d, h, e, n)$. Moreover, each column of the measurement matrices associated with the proposed schemes can be nonrandomly generated in polynomial order of the number of rows. As a result, one can save space for storing them. Simulation results confirm our theoretical analysis. When the number of items is sufficiently large, the decoding time in our proposed scheme is smallest in comparison with existing work. In addition, when some erroneous outcomes are allowed, the number of tests in the proposed scheme is often smaller than the number of tests in existing work.

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arXiv2019

Improved encoding and decoding for non-adaptive threshold group testing

The goal of threshold group testing is to identify up to $d$ defective items among a population of $n$ items, where $d$ is usually much smaller than $n$. A test is positive if it has at least $u$ defective items and negative otherwise. Our objective is to identify defective items in sublinear time the number of items, e.g., $\mathrm{poly}(d, \ln{n}),$ by using the number of tests as low as possible. In this paper, we reduce the number of tests to $O \left( h \times \frac{d^2 \ln^2{n}}{\mathsf{W}^2(d \ln{n})} \right)$ and the decoding time to $O \left( \mathrm{dec}_0 \times h \right),$ where $\\mathrm{dec}_0 = O \left( \frac{d^{3.57} \ln^{6.26}{n}}{\mathsf{W}^{6.26}(d \ln{n})} \right) + O \left( \frac{d^6 \ln^4{n}}{\mathsf{W}^4(d \ln{n})} \right)$, $h = O\left( \frac{d_0^2 \ln{\frac{n}{d_0}}}{(1-p)^2} \right)$ , $d_0 = \max\{u, d - u \}$, $p \in [0, 1),$ and $\mathsf{W}(x) = Θ\left( \ln{x} - \ln{\ln{x}} \right).$ If the number of tests is increased to $O\left( h \times \frac{d^2\ln^3{n}}{\mathsf{W}^2(d \ln{n})} \right),$ the decoding complexity is reduced to $O \left(\mathrm{dec}_1 \times h \right),$ where $\mathrm{dec}_1 = \max \left\{ \frac{d^2 \ln^3{n}}{\mathsf{W}^2(d \ln{n})}, \frac{ud \ln^4{n}}{\mathsf{W}^3(d \ln{n})} \right\}.$ Moreover, our proposed scheme is capable of handling errors in test outcomes.

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arXiv2018

Efficient (nonrandom) construction and decoding for non-adaptive group testing

The task of non-adaptive group testing is to identify up to $d$ defective items from $N$ items, where a test is positive if it contains at least one defective item, and negative otherwise. If there are $t$ tests, they can be represented as a $t \times N$ measurement matrix. We have answered the question of whether there exists a scheme such that a larger measurement matrix, built from a given $t\times N$ measurement matrix, can be used to identify up to $d$ defective items in time $O(t \log_2{N})$. In the meantime, a $t \times N$ nonrandom measurement matrix with $t = O \left(\frac{d^2 \log_2^2{N}}{(\log_2(d\log_2{N}) - \log_2{\log_2(d\log_2{N})})^2} \right)$ can be obtained to identify up to $d$ defective items in time $\mathrm{poly}(t)$. This is much better than the best well-known bound, $t = O \left( d^2 \log_2^2{N} \right)$. For the special case $d = 2$, there exists an efficient nonrandom construction in which at most two defective items can be identified in time $4\log_2^2{N}$ using $t = 4\log_2^2{N}$ tests. Numerical results show that our proposed scheme is more practical than existing ones, and experimental results confirm our theoretical analysis. In particular, up to $2^{7} = 128$ defective items can be identified in less than $16$s even for $N = 2^{100}$.

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arXiv2018

Efficiently Decodable Non-Adaptive Threshold Group Testing

We consider non-adaptive threshold group testing for identification of up to $d$ defective items in a set of $n$ items, where a test is positive if it contains at least $2 \leq u \leq d$ defective items, and negative otherwise. The defective items can be identified using $t = O \left( \left( \frac{d}{u} \right)^u \left( \frac{d}{d - u} \right)^{d-u} \left(u \log{\frac{d}{u}} + \log{\frac{1}ε} \right) \cdot d^2 \log{n} \right)$ tests with probability at least $1 - ε$ for any $ε> 0$ or $t = O \left( \left( \frac{d}{u} \right)^u \left( \frac{d}{d -u} \right)^{d - u} d^3 \log{n} \cdot \log{\frac{n}{d}} \right)$ tests with probability 1. The decoding time is $t \times \mathrm{poly}(d^2 \log{n})$. This result significantly improves the best known results for decoding non-adaptive threshold group testing: $O(n\log{n} + n \log{\frac{1}ε})$ for probabilistic decoding, where $ε> 0$, and $O(n^u \log{n})$ for deterministic decoding.

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arXiv2018

Efficient Decoding Schemes for Noisy Non-Adaptive Group Testing when Noise Depends on Number of Items in Test

The goal of non-adaptive group testing is to identify at most $d$ defective items from $N$ items, in which a test of a subset of $N$ items is positive if it contains at least one defective item, and negative otherwise. However, in many cases, especially in biological screening, the outcome is unreliable due to biochemical interaction; i.e., \textit{noise.} Consequently, a positive result can change to a negative one (false negative) and vice versa (false positive). In this work, we first consider the dilution effect in which \textit{the degree of noise depends on the number of items in the test}. Two efficient schemes are presented for identifying the defective items in time linearly to the number of tests needed. Experimental results validate our theoretical analysis. Specifically, setting the error precision of 0.001 and $d\leq16$, our proposed algorithms always identify all defective items in less than 7 seconds for $N=2^{33}\approx 9$ billion.

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arXiv2017

Non-Adaptive Group Testing Framework based on Concatenation Code

We consider an efficiently decodable non-adaptive group testing (NAGT) problem that meets theoretical bounds. The problem is to find a few specific items (at most $d$) satisfying certain characteristics in a colossal number of $N$ items as quickly as possible. Those $d$ specific items are called \textit{defective items}. The idea of NAGT is to pool a group of items, which is called \textit{a test}, then run a test on them. If the test outcome is \textit{positive}, there exists at least one defective item in the test, and if it is \textit{negative}, there exists no defective items. Formally, a binary $t \times N$ measurement matrix $\mathcal{M} = (m_{ij})$ is the representation for $t$ tests where row $i$ stands for test $i$ and $m_{ij} = 1$ if and only if item $j$ belongs to test $i$. There are three main objectives in NAGT: minimize the number of tests $t$, construct matrix $\mathcal{M}$, and identify defective items as quickly as possible. In this paper, we present a strongly explicit construction of $\mathcal{M}$ for when the number of defective items is at most 2, with the number of tests $t \simeq 16 \log{N} = O(\log{N})$. In particular, we need only $K \simeq N \times 16\log{N} = O(N\log{N})$ bits to construct such matrices, which is optimal. Furthermore, given these $K$ bits, any entry in the matrix can be constructed in time $O \left(\ln{N}/ \ln{\ln{N}} \right)$. Moreover, $\mathcal{M}$ can be decoded with high probability in time $O\left( \frac{\ln^2{N}}{\ln^2{\ln{N}}} \right)$. When the number of defective items is greater than 2, we present a scheme that can identify at least $(1-ε)d$ defective items with $t \simeq 32 C(ε) d \log{N} = O(d \log{N})$ in time $O \left( d \frac{\ln^2{N}}{\ln^2{\ln{N}}} \right)$ for any close-to-zero $ε$, where $C(ε)$ is a constant that depends only on $ε$.

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