arXiv subjects

Noela Müller

Publications and source records attributed to Noela Müller.

17 recordsLinked to original sources

arXiv2026

An Information-Theoretic Analysis of Threshold Group Testing

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).

cs.IT↗

arXiv2026

Algorithms for Threshold Group Testing

We study the Threshold Group Testing (TGT) problem without a gap in the noiseless, non-adaptive setting, where the goal is to exactly recover a sparse binary vector from pooled test outcomes using as few tests as possible. In TGT, a test applied to a subset of items returns a positive outcome if the number of defective items in the subset reaches a prescribed threshold, and a negative outcome otherwise. Under the assumption of an analytic condition, TGT has been shown to undergo a sharp information-theoretic phase transition for exact recovery on the class of constant-column test designs. In this paper, we develop an efficient inference algorithm that achieves exact recovery with high probability using the minimum number of non-adaptive tests that are needed for the constant-column design, thereby matching the information-theoretic threshold of a natural benchmark test design. Our approach is based on a spatially coupled test design and admits a significantly simpler analysis than existing algorithms for related group testing problems. In particular, unlike previous methods for binary group testing, our algorithm does not rely on the analysis of intricate weighted sums. This leads to a more straightforward proof technique, while still allowing near-optimal performance guarantees.

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arXiv2025

Sharp Thresholds for Factors in Random Graphs

Let $F$ be a graph on $r$ vertices and let $G$ be a graph on $n$ vertices. Then an $F$-factor in $G$ is a subgraph of $G$ composed of $n/r$ vertex-disjoint copies of $F$, if $r$ divides $n$. In other words, an $F$-factor yields a partition of the $n$ vertices of $G$. The study of such $F$-factors in the Erdős-Rényi random graph dates back to Erdős himself. Decades later, in 2008, Johansson, Kahn and Vu established the thresholds for the existence of an $F$-factor for strictly 1-balanced $F$ -- up to the leading constant. The sharp thresholds, meaning the leading constants, were obtained only recently by Riordan and Heckel, but only for complete graphs $F=K_r$ and for so-called nice graphs. Their results rely on sophisticated couplings that utilize the recent, celebrated solution of Shamir's problem by Kahn. We extend the couplings by Riordan and Heckel to any strictly 1-balanced $F$ and thereby obtain the sharp threshold for the existence of an $F$-factor. In particular, we confirm the thirty year old conjecture by Rucínski that this sharp threshold indeed coincides with the sharp threshold for the disappearance of the last vertices which are not contained in a copy of $F$.

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arXiv2025

The asymptotic rank of adjacency matrices of weighted configuration models over arbitrary fields

We study the asymptotic rank of adjacency matrices of a large class of edge-weighted configuration models. Here, the weight of a (multi-)edge can be any fixed non-zero element from an arbitrary field, as long as it is independent of the (multi-)graph. Our main result demonstrates that the asymptotic behavior of the normalized rank of the adjacency matrix neither depends on the fixed edge-weights, nor on which field they are chosen from. Our approach relies on a novel adaptation of the component exploration method of \cite{janson2009new}, which enables the application of combinatorial techniques from \cite{coja2022rank, HofMul25}.

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arXiv2024

Random $2$-SAT: The set of atoms of the limiting empirical marginal distribution

We show that the set of atoms of the limiting empirical marginal distribution in the random $2$-SAT model is $\mathbb Q \cap (0,1)$, for all clause-to-variable densities up to the satisfiability threshold. While for densities up to $1/2$, the measure is purely discrete, we additionally establish the existence of a nontrivial continuous part for any density in $(1/2, 1)$. Our proof is based on the construction of a random variable with the correct distribution as the the root marginal of a multi-type Galton-Watson tree, along with a subsequent analysis of the resulting almost sure recursion.

math.PR↗

arXiv2024

The hitting time of nice factors

Consider the random $u$-uniform hypergraph (or $u$-graph) process on $n$ vertices, where $n$ is divisible by $r>u\ge 2$. It was recently shown that with high probability, as soon as every vertex is covered by a copy of the complete $u$-graph $K_r$, it also contains a $K_r$-factor (RSA, Vol. 65 II, Sept. 2024). The hitting time result is obtained using a process coupling, which is based on the proof of the corresponding sharp threshold result (RSA, Vol. 61 IV, Dec. 2022). The latter, however, was not only derived for complete $u$-graphs, but for a broader class of so-called nice $u$-graphs. The purpose of this article is to extend the process coupling for complete $u$-graphs to the full scope of the sharp threshold result: nice $u$-graphs. As a byproduct, we obtain the extension of the hitting time result to nice $u$-graphs. Since the relevant combinatorial bounds in the proof for the $K_r$-case cannot be generalized, we introduce new arguments that do not only apply to nice u-graphs, but will be relevant for the broader class of strictly 1-balanced u-graphs. Further, we show how the remainder of the process coupling for the $K_r$-case can be utilized in a black-box manner for any u-graph. These advances pave the way for future generalizations.

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arXiv2024

The number of random 2-SAT solutions is asymptotically log-normal

We prove that throughout the satisfiable phase, the logarithm of the number of satisfying assignments of a random 2-SAT formula satisfies a central limit theorem. This implies that the log of the number of satisfying assignments exhibits fluctuations of order $\sqrt n$, with $n$ the number of variables. The formula for the variance can be evaluated effectively. By contrast, for numerous other random constraint satisfaction problems the typical fluctuations of the logarithm of the number of solutions are {\em bounded} throughout all or most of the satisfiable regime.

cs.DM↗

arXiv2023

On a Near-Optimal \& Efficient Algorithm for the Sparse Pooled Data Problem

The pooled data problem asks to identify the unknown labels of a set of items from condensed measurements. More precisely, given $n$ items, assume that each item has a label in $\cbc{0,1,\ldots, d}$, encoded via the ground-truth $\SIGMA$. We call the pooled data problem sparse if the number of non-zero entries of $\SIGMA$ scales as $k \sim n^θ$ for $θ\in (0,1)$. The information that is revealed about $\SIGMA$ comes from pooled measurements, each indicating how many items of each label are contained in the pool. The most basic question is to design a pooling scheme that uses as few pools as possible, while reconstructing $\SIGMA$ with high probability. Variants of the problem and its combinatorial ramifications have been studied for at least 35 years. However, the study of the modern question of \emph{efficient} inference of the labels has suggested a statistical-to-computational gap of order $\log n$ in the minimum number of pools needed for theoretically possible versus efficient inference. In this article, we resolve the question whether this $\log n$-gap is artificial or of a fundamental nature by the design of an efficient algorithm, called \algoname, based upon a novel pooling scheme on a number of pools very close to the information-theoretic threshold.

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arXiv2023

The hitting time of clique factors

In a recent paper, Kahn gave the strongest possible, affirmative, answer to Shamir's problem, which had been open since the late 1970s: Let $r \ge 3 $ and let $n$ be divisible by $r$. Then, in the random $r$-uniform hypergraph process on $n$ vertices, as soon as the last isolated vertex disappears, a perfect matching emerges. In the present work, we transfer this hitting time result to the setting of clique factors in the random graph process: At the time that the last vertex joins a copy of the complete graph $K_r$, the random graph process contains a $K_r$-factor. Our proof draws on a novel sequence of couplings, extending techniques of Riordan and the first author. An analogous result is proved for clique factors in the $s$-uniform hypergraph process ($s \ge 3$).

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arXiv2023

The rank of sparse symmetric matrices over arbitrary fields

Let $\FF$ be an arbitrary field and $(\bm{G}_{n,d/n})_n$ be a sequence of sparse weighted Erdős-Rényi random graphs on $n$ vertices with edge probability $d/n$, where weights from $\FF \setminus\{0\}$ are assigned to the edges according to a fixed matrix $J_n$. We show that the normalised rank of the adjacency matrix of $(\bm{G}_{n,d/n})_n$ converges in probability to a constant, and derive the limiting expression. Our result shows that for the general class of sparse symmetric matrices under consideration, the asymptotics of the normalised rank are independent of the edge weights and even the field, in the sense that the limiting constant for the general case coincides with the one previously established for adjacency matrices of sparse (non-weighted) Erdős-Rényi matrices over $\RR$ from \cite{bordenave2011rank}. Our proof, which is purely combinatorial in its nature, is based on an intricate extension of the novel perturbation approach from \cite{coja2022rank} to the symmetric setting.

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arXiv2022

Near optimal efficient decoding from pooled data

Consider $n$ items, each of which is characterised by one of $d+1$ possible features in $\{0, \ldots, d\}$. We study the inference task of learning these types by queries on subsets, or pools, of the items that only reveal a form of coarsened information on the features - in our case, the sum of all the features in the pool. This is a realistic scenario in situations where one has memory or technical constraints in the data collection process, or where the data is subject to anonymisation. Related prominent problems are the quantitative group testing problem, of which it is a generalisation, as well as the compressed sensing problem, of which it is a special case. In the present article, we are interested in the minimum number of queries needed to efficiently infer the labels, if one of the features, say $0$, is dominant in the sense that the number $k$ of non-zero features among the items is much smaller than $n$. It is known that in this case, all features can be recovered in exponential time by using no more than $O(k)$ queries. However, so far, all \textit{efficient} inference algorithms required at least $Ω(k\ln n)$ queries, and it was unknown whether this gap is artificial or of a fundamental nature. Here we show that indeed, the previous gap between the information-theoretic and computational bounds is not inherent to the problem by providing an efficient algorithm that succeeds with high probability and employs no more than $O(k)$ measurements. This also solves a long standing open question for the quantitative group testing problem.

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arXiv2022

The full rank condition for sparse random matrices

We derive a sufficient condition for a sparse random matrix with given numbers of non-zero entries in the rows and columns having full row rank. The result covers both matrices over finite fields with independent non-zero entries and $\{0,1\}$-matrices over the rationals. The sufficient condition is generally necessary as well.

math.CO↗

arXiv2020

Inference and mutual information on random factor graphs

Random factor graphs provide a powerful framework for the study of inference problems such as decoding problems or the stochastic block model. Information-theoretically the key quantity of interest is the mutual information between the observed factor graph and the underlying ground truth around which the factor graph was created; in the stochastic block model, this would be the planted partition. The mutual information gauges whether and how well the ground truth can be inferred from the observable data. For a very general model of random factor graphs we verify a formula for the mutual information predicted by physics techniques. As an application we prove a conjecture about low-density generator matrix codes from [Montanari: IEEE Transactions on Information Theory 2005]. Further applications include phase transitions of the stochastic block model and the mixed $k$-spin model from physics.

cs.DM↗

arXiv2020

The random 2-SAT partition function

We show that throughout the satisfiable phase the normalised number of satisfying assignments of a random $2$-SAT formula converges in probability to an expression predicted by the cavity method from statistical physics. The proof is based on showing that the Belief Propagation algorithm renders the correct marginal probability that a variable is set to `true' under a uniformly random satisfying assignment.

math.PR↗

arXiv2019

Central limit theorem analogues for multicolour urn models

The asymptotic behaviour of a generalised Pólya--Eggenberger urn is well--known to depend on the spectrum of its replacement matrix: If its dominant eigenvalue $r$ is simple and no other eigenvalue is `large' in the sense that its real part is greater than $r/2$, the normalized urn composition is asymptotically normally distributed. However, if there is more than one large eigenvalue, the first few random draws have a non--negligible effect on the evolution of the urn process and almost sure random tendencies of order larger than $\sqrt{n}$ typically prevent a classical central limit theorem. In the present work, a central limit theorem analogue for the fluctuations of urn models with regard to random linear drift and random periodic growth of order larger than $\sqrt{n}$ is proved, covering the $m$-ary search tree and B-trees. The proof builds on an eigenspace decomposition of the process in order to separate components of different growth orders. By an accurately tailored adaption of martingale techniques to the components, their joint limiting behaviour is established and translated back to the urn process. Conveniently, the approach encompasses results on small urn models and therefore provides a unifying perspective on central limit theorems for certain urn models, irrespective of their spectrum.

math.PR↗

arXiv2019

Refined Asymptotics for the Composition of Cyclic Urns

A cyclic urn is an urn model for balls of types $0,\ldots,m-1$. The urn starts at time zero with an initial configuration. Then, in each time step, first a ball is drawn from the urn uniformly and independently from the past. If its type is $j$, it is then returned to the urn together with a new ball of type $j+1 \mod m$. The case $m=2$ is the well-known Friedman urn. The composition vector, i.e., the vector of the numbers of balls of each type after $n$ steps is, after normalization, known to be asymptotically normal for $2\le m\le 6$. For $m\ge 7$ the normalized composition vector is known not to converge. However, there is an almost sure approximation by a periodic random vector. In the present paper the asymptotic fluctuations around this periodic random vector are identified. We show that these fluctuations are asymptotically normal for all $7\le m\le 12$. For $m\ge 13$ we also find asymptotically normal fluctuations when normalizing in a more refined way. These fluctuations are of maximal dimension $m-1$ only when $6$ does not divide $m$. For $m$ being a multiple of $6$ the fluctuations are supported by a two-dimensional subspace.

math.PR↗

arXiv2018

The replica symmetric phase of random constraint satisfaction problems

Random constraint satisfaction problems play an important role in computer science and combinatorics. For example, they provide challenging benchmark instances for algorithms and they have been harnessed in probabilistic constructions of combinatorial structures with peculiar features. In an important contribution [Krzakala et al., PNAS 2007] physicists made several predictions on the precise location and nature of phase transitions in random constraint satisfaction problems. Specifically, they predicted that their satisfiability thresholds are quite generally preceded by several other thresholds that have a substantial impact both combinatorially and computationally. These include the condensation phase transition, where long-range correlations between variables emerge, and the reconstruction threshold. In this paper we prove these physics predictions for a broad class of random constraint satisfaction problems. Additionally, we obtain contiguity results that have implications on Bayesian inference tasks, a subject that has received a great deal of interest recently (e.g., [Banks et al., COLT 2016]).

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