This severely limits their scalability and sample complexity.

If a major disaster like Fukushima or Chornobyl ever happens again, the world would know almost straight away, thanks to an array of government and DIY radiation-monitoring programs running globally. Why Don't Norwegians Hate Tesla Like the Rest of Europe Does? November's Tesla registrations were down in France, Sweden, Denmark, and Germany. Norway, however, is bucking the trend--thanks to a tax incentive system that will soon be rolled back.

This paper considers the problem of testing for latent structure in large symmetric data matrices. The goal here is to develop statistically principled methodology that is flexible in its applicability, computationally efficient, and insensitive to extreme data variation, thereby overcoming limitations facing existing approaches. To do so, we introduce and systematically study certain symmetric matrices, called Wilcoxon--Wigner random matrices, whose entries are normalized rank statistics derived from an underlying independent and identically distributed sample of absolutely continuous random variables. These matrices naturally arise as the matricization of one-sample problems in statistics and conceptually lie at the interface of nonparametrics, multivariate analysis, and data reduction. Among our results, we establish that the leading eigenvalue and corresponding eigenvector of Wilcoxon--Wigner random matrices admit asymptotically Gaussian fluctuations with explicit centering and scaling terms. These asymptotic results enable rigorous parameter-free and distribution-free spectral methodology for addressing two hypothesis testing problems, namely community detection and principal submatrix detection. Numerical examples illustrate the performance of the proposed approach. Throughout, our findings are juxtaposed with existing results based on the spectral properties of independent entry symmetric random matrices in signal-plus-noise data settings.

In this work, we investigate the existence and effect of percolation in training deep Neural Networks (NNs) with dropout. Dropout methods are regularisation techniques for training NNs, first introduced by G. Hinton et al. (2012). These methods temporarily remove connections in the NN, randomly at each stage of training, and update the remaining subnetwork with Stochastic Gradient Descent (SGD). The process of removing connections from a network at random is similar to percolation, a paradigm model of statistical physics. If dropout were to remove enough connections such that there is no path between the input and output of the NN, then the NN could not make predictions informed by the data. We study new percolation models that mimic dropout in NNs and characterise the relationship between network topology and this path problem. The theory shows the existence of a percolative effect in dropout. We also show that this percolative effect can cause a breakdown when training NNs without biases with dropout; and we argue heuristically that this breakdown extends to NNs with biases.

In decentralized federated learning (DFL), the presence of abnormal clients, often caused by noisy or poisoned data, can significantly disrupt the learning process and degrade the overall robustness of the model. Previous methods on this issue often require a sufficiently large number of normal neighboring clients or prior knowledge of reliable clients, which reduces the practical applicability of DFL. To address these limitations, we develop here a novel adaptive DFL (aDFL) approach for robust estimation. The key idea is to adaptively adjust the learning rates of clients. By assigning smaller rates to suspicious clients and larger rates to normal clients, aDFL mitigates the negative impact of abnormal clients on the global model in a fully adaptive way. Our theory does not put any stringent conditions on neighboring nodes and requires no prior knowledge. A rigorous convergence analysis is provided to guarantee the oracle property of aDFL. Extensive numerical experiments demonstrate the superior performance of the aDFL method.

Neural models have shown promise in solving NP-hard graph combinatorial optimization (CO) problems. Once trained, they offer fast inference and reasonably high-quality solutions for in-distribution testing instances, but they generally fall short in terms of absolute solution quality compared to classical search-based algorithms that are admittedly slower but offer optimality guarantee once search finishes. We propose a novel framework that combines the inference efficiency and exploratory power of neural models with the solution quality guarantee of search-based algorithms. In particular, we use parameterized algorithms (PAs) as the search component. PAs are dedicated to identifying easy instances of generally NP-hard problems, and allow for practically efficient search by exploiting structural simplicity (of the identified easy instances). Under our framework, we use parameterized analysis to identify the structurally hard parts of a CO instance. The neural model handles the hard parts by generating advisory signals based on its data-driven understanding. The PA-based search component then integrates the advisory signals to systematically and efficiently searches through the remaining structurally easy parts. Notably, our framework is agnostic to the choice of neural model and produces strictly better solutions than neural solvers alone. We examine our framework on multiple CO tasks. Empirical results show that it achieves superior solution quality, competitive with that of commercial solvers. Furthermore, by using the neural model only for exploratory advisory signals, our framework exhibits improved out-of-distribution generalization, addressing a key limitation of existing neural CO solvers.

Community detection, or network clustering, is used to identify latent community structure in networks. Due to the scarcity of labeled ground truth in real-world networks, evaluating these algorithms poses significant challenges. To address this, researchers use synthetic network generators that produce networks with ground-truth community labels. RECCS is one such algorithm that takes a network and its clustering as input and generates a synthetic network through a modular pipeline. Each generated ground truth cluster preserves key characteristics of the corresponding input cluster, including connectivity, minimum degree, and degree sequence distribution. The output consists of a synthetically generated network, and disjoint ground truth cluster labels for all nodes. In this paper, we present two enhanced versions: RECCS+ and RECCS++. RECCS+ maintains algorithmic fidelity to the original RECCS while introducing parallelization through an orchestrator that coordinates algorithmic components across multiple processes and employs multithreading. RECCS++ builds upon this foundation with additional algorithmic optimizations to achieve further speedup. Our experimental results demonstrate that RECCS+ and RECCS++ achieve speedups of up to 49x and 139x respectively on our benchmark datasets, with RECCS++'s additional performance gains involving a modest accuracy tradeoff. With this newfound performance, RECCS++ can now scale to networks with over 100 million nodes and nearly 2 billion edges.

A network, which is used to model interactions or communications among a set of agents or nodes, is arguably among one of the most common and important representations for modern complex data.

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