Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1503.03597

Nonadaptive group testing with random set of defectives

Arya Mazumdar
View original source ↗Read source document ↗Export metadata ↓

Abstract

In a group testing scheme, a set of tests is designed to identify a small number $t$ of defective items that are present among a large number $N$ of items. Each test takes as input a group of items and produces a binary output indicating whether any defective item is present in the group. In a non-adaptive scheme designing a testing scheme is equivalent to the construction of a disjunct matrix, an $M \times N$ binary matrix where the union of supports of any $t$ columns does not contain the support of any other column. In this paper we consider the scenario where defective items are random and follow simple probability distributions. In particular we consider the cases where 1) each item can be defective independently with probability $\frac{t}{N}$ and 2) each $t$-set of items can be defective with uniform probability. In both cases our aim is to design a testing matrix that successfully identifies the set of defectives with high probability. Both of these models have been studied in the literature before and it is known that $O(t\log N)$ tests are necessary as well as sufficient (via random coding) in both cases. Our main focus is explicit deterministic construction of the test matrices amenable to above scenarios. One of the most popular ways of constructing test matrices relies on \emph{constant-weight error-correcting codes} and their minimum distance. We go beyond the minimum distance analysis and connect the average distance of a constant weight code to the parameters of the resulting test matrix. With our relaxed requirements, we show that using explicit constant-weight codes (e.g., based on algebraic geometry codes) we may achieve a number of tests equal to $O(t \frac{\log^2 N}{ \log t})$ for both the first and the second cases.

Explore related subjects

cs.ITmath.IT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Arya Mazumdar. 2016-10-04. Nonadaptive group testing with random set of defectives. https://doi.org/10.1109/tit.2016.2613870

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Radiance-Field Guided Pretraining: Scaling Localization Models with Unlabeled Wireless Signals

Radio frequency (RF)-based indoor localization offers significant promise for applications such as indoor navigation, augmented reality, and pervasive computing. While deep learning has greatly enhanced localization accuracy and robustness, existing localization models still face major challenges in cross-scene generalization due to their reliance on scene-specific labeled data. To address this, we introduce Radiance-Field Reinforced Pretraining (RFRP). This novel self-supervised pretraining framework couples a large localization model (LM) with a neural radio-frequency radiance field (RF-NeRF) in an asymmetrical autoencoder architecture. In this design, the LM encodes received RF spectra into latent, position-relevant representations, while the RF-NeRF decodes them to reconstruct the original spectra. This alignment between input and output enables effective representation learning using large-scale, unlabeled RF data, which can be collected continuously with minimal effort. To this end, we collected RF samples at 7,327,321 positions across 100 diverse scenes using four common wireless technologies--RFID, BLE, WiFi, and IIoT. Data from 75 scenes were used for training, and the remaining 25 for evaluation. Experimental results show that the RFRP-pretrained LM reduces localization error by over 40% compared to non-pretrained models and by 21% compared to those pretrained using supervised learning.

cs.IT↗
arXiv2026

Uniform Recovery of Structured Signals from Nonlinear Observations: Improved Error Rates

Consider the recovery of structured signals from nonlinear observations. Under Gaussian matrix and a large class of unknown nonlinear link functions, Plan and Vershynin (2016) showed that generalized Lasso achieves accurate nonuniform recovery of a fixed signal. More recently, Genzel and Stollenwerk (2023) showed that generalized Lasso is indeed capable of accurately recovering all structured signals. However, in some canonical settings with discontinuous link functions, their uniform recovery error rate is essentially slower than the nonuniform one. Specifically, in the recovery of $n$-dimensional $k$-sparse vectors from $m$ measurements, generalized Lasso with a perfectly tuned $\ell_1$ constraint achieves nonuniform error rate $ O(\sqrt{k\log(en/k)/m})$, while the uniform error rate of Genzel and Stollenwerk is no faster than $O((k\log(en/k)/m)^{1/4})$. In this paper, we narrow this gap by establishing improved uniform recovery guarantees under piecewise Lipschitz link functions with well-separated jump discontinuities. We analyze a projected gradient descent (PGD) algorithm whose projection can be onto a convex set or a cone, and our results for the PGD with a convex set are also valid for the generalized Lasso. In sparse recovery, the improved uniform error rates match the nonuniform rate $O(\sqrt{k\log(en/k)/m})$ up to logarithmic factors. Under the sign link function, we further show that iterative hard thresholding (a specific instance of the PGD) achieves uniform recovery error rate $O(\sqrt{k\log(en/k)/m})$, matching the nonuniform rate up to a universal constant. Technically, the uniform guarantees for the PGD are obtained by showing that the gradient maps satisfy the restricted approximate invertibility condition uniformly over all signals. We demonstrate that this is a general approach to uniform recovery under nonlinear observations.

cs.IT↗
arXiv2026

Recursive overlap Bernoulli distributions and an entropy concavity conjecture

We introduce a family of recursively generated finite probability distributions obtained from left and right embeddings with overlaps. The construction interpolates between the classical binomial distribution and the non-overlapping Bernoulli product distribution. We derive explicit formulas for the expectation, variance, and the generating function of higher moments, and formulate a conjecture asserting that the Shannon entropy is concave. The conjecture is proved in the two extremal cases and supported by symbolic computations for numerous overlap sequences.

cs.IT↗