We study the problem of clustering networks whose nodes have imputed or physical positions in a single dimension, for example prestige hierarchies or the similarity dimension of hyperbolic embeddings. Existing algorithms, such as the critical gap method and other greedy strategies, only offer approximate solutions to this problem. Here, we introduce a dynamic programming approach that returns provably optimal solutions in polynomial time -- O(n^2) steps -- for a broad class of clustering objectives. We demonstrate the algorithm through applications to synthetic and empirical networks and show that it outperforms existing heuristics by a significant margin, with a similar execution time.
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We present a complete numerical analysis for a general discretization of a coupled flow-mechanics model in fractured porous media, considering single-phase flows and including frictionless contact at matrix-fracture interfaces, as well as nonlinear poromechanical coupling. Fractures are described as planar surfaces, yielding the so-called mixed- or hybrid-dimensional models. Small displacements and a linear elastic behavior are considered for the matrix. The model accounts for discontinuous fluid pressures at matrix-fracture interfaces in order to cover a wide range of normal fracture conductivities. The numerical analysis is carried out in the Gradient Discretization framework, encompassing a large family of conforming and nonconforming discretizations. The convergence result also yields, as a by-product, the existence of a weak solution to the continuous model. A numerical experiment in 2D is presented to support the obtained result, employing a Hybrid Finite Volume scheme for the flow and second-order finite elements ($\mathbb P_2$) for the mechanical displacement coupled with face-wise constant ($\mathbb P_0$) Lagrange multipliers on fractures, representing normal stresses, to discretize the contact conditions.
This study presents a Bayesian regression framework to model the relationship between scalar outcomes and brain functional connectivity represented as symmetric positive definite (SPD) matrices. Unlike many proposals that simply vectorize the connectivity predictors thereby ignoring their matrix structures, our method respects the Riemannian geometry of SPD matrices by modelling them in a tangent space. We perform dimension reduction in the tangent space, relating the resulting low-dimensional representations with the responses. The dimension reduction matrix is learnt in a supervised manner with a sparsity-inducing prior imposed on a Stiefel manifold to prevent overfitting. Our method yields a parsimonious regression model that allows uncertainty quantification of the estimates and identification of key brain regions that predict the outcomes. We demonstrate the performance of our approach in simulation settings and through a case study to predict Picture Vocabulary scores using data from the Human Connectome Project.
The growing reliance on online services underscores the crucial role of recommendation systems, especially on social media platforms seeking increased user engagement. This study investigates how recommendation systems influence the impact of personal behavioral traits on social network dynamics. It explores the interplay between homophily, users' openness to novel ideas, and recommendation-driven exposure to new opinions. Additionally, the research examines the impact of recommendation systems on the diversity of newly generated ideas, shedding light on the challenges and opportunities in designing effective systems that balance the exploration of new ideas with the risk of reinforcing biases or filtering valuable, unconventional concepts.
We introduce the higher-order refactoring problem, where the goal is to compress a logic program by discovering higher-order abstractions, such as map, filter, and fold. We implement our approach in Stevie, which formulates the refactoring problem as a constraint optimisation problem. Our experiments on multiple domains, including program synthesis and visual reasoning, show that refactoring can improve the learning performance of an inductive logic programming system, specifically improving predictive accuracies by 27% and reducing learning times by 47%. We also show that Stevie can discover abstractions that transfer to multiple domains.
A discrete spatial lattice can be cast as a network structure over which spatially-correlated outcomes are observed. A second network structure may also capture similarities among measured features, when such information is available. Incorporating the network structures when analyzing such doubly-structured data can improve predictive power, and lead to better identification of important features in the data-generating process. Motivated by applications in spatial disease mapping, we develop a new doubly regularized regression framework to incorporate these network structures for analyzing high-dimensional datasets. Our estimators can easily be implemented with standard convex optimization algorithms. In addition, we describe a procedure to obtain asymptotically valid confidence intervals and hypothesis tests for our model parameters. We show empirically that our framework provides improved predictive accuracy and inferential power compared to existing high-dimensional spatial methods. These advantages hold given fully accurate network information, and also with networks which are partially misspecified or uninformative. The application of the proposed method to modeling COVID-19 mortality data suggests that it can improve prediction of deaths beyond standard spatial models, and that it selects relevant covariates more often.
We consider a lognormal diffusion process having a multisigmoidal logistic mean, useful to model the evolution of a population which reaches the maximum level of the growth after many stages. Referring to the problem of statistical inference, two procedures to find the maximum likelihood estimates of the unknown parameters are described. One is based on the resolution of the system of the critical points of the likelihood function, and the other is on the maximization of the likelihood function with the simulated annealing algorithm. A simulation study to validate the described strategies for finding the estimates is also presented, with a real application to epidemiological data. Special attention is also devoted to the first-passage-time problem of the considered diffusion process through a fixed boundary.
Most existing neural network-based approaches for solving stochastic optimal control problems using the associated backward dynamic programming principle rely on the ability to simulate the underlying state variables. However, in some problems, this simulation is infeasible, leading to the discretization of state variable space and the need to train one neural network for each data point. This approach becomes computationally inefficient when dealing with large state variable spaces. In this paper, we consider a class of this type of stochastic optimal control problems and introduce an effective solution employing multitask neural networks. To train our multitask neural network, we introduce a novel scheme that dynamically balances the learning across tasks. Through numerical experiments on real-world derivatives pricing problems, we prove that our method outperforms state-of-the-art approaches.
We study the problem of learning unknown parameters in stochastic interacting particle systems with polynomial drift, interaction and diffusion functions from the path of one single particle in the system. Our estimator is obtained by solving a linear system which is constructed by imposing appropriate conditions on the moments of the invariant distribution of the mean field limit and on the quadratic variation of the process. Our approach is easy to implement as it only requires the approximation of the moments via the ergodic theorem and the solution of a low-dimensional linear system. Moreover, we prove that our estimator is asymptotically unbiased in the limits of infinite data and infinite number of particles (mean field limit). In addition, we present several numerical experiments that validate the theoretical analysis and show the effectiveness of our methodology to accurately infer parameters in systems of interacting particles.
We study the problem of parameter estimation for large exchangeable interacting particle systems when a sample of discrete observations from a single particle is known. We propose a novel method based on martingale estimating functions constructed by employing the eigenvalues and eigenfunctions of the generator of the mean field limit, where the law of the process is replaced by the (unique) invariant measure of the mean field dynamics. We then prove that our estimator is asymptotically unbiased and asymptotically normal when the number of observations and the number of particles tend to infinity, and we provide a rate of convergence towards the exact value of the parameters. Finally, we present several numerical experiments which show the accuracy of our estimator and corroborate our theoretical findings, even in the case the mean field dynamics exhibit more than one steady states.
The computational demands of modern AI have spurred interest in optical neural networks (ONNs) which offer the potential benefits of increased speed and lower power consumption. However, current ONNs face various challenges,most significantly a limited calculation precision (typically around 4 bits) and the requirement for high-resolution signal format converters (digital-to-analogue conversions (DACs) and analogue-to-digital conversions (ADCs)). These challenges are inherent to their analog computing nature and pose significant obstacles in practical implementation. Here, we propose a digital-analog hybrid optical computing architecture for ONNs, which utilizes digital optical inputs in the form of binary words. By introducing the logic levels and decisions based on thresholding, the calculation precision can be significantly enhanced. The DACs for input data can be removed and the resolution of the ADCs can be greatly reduced. This can increase the operating speed at a high calculation precision and facilitate the compatibility with microelectronics. To validate our approach, we have fabricated a proof-of-concept photonic chip and built up a hybrid optical processor (HOP) system for neural network applications. We have demonstrated an unprecedented 16-bit calculation precision for high-definition image processing, with a pixel error rate (PER) as low as $1.8\times10^{-3}$ at an signal-to-noise ratio (SNR) of 18.2 dB. We have also implemented a convolutional neural network for handwritten digit recognition that shows the same accuracy as the one achieved by a desktop computer. The concept of the digital-analog hybrid optical computing architecture offers a methodology that could potentially be applied to various ONN implementations and may intrigue new research into efficient and accurate domain-specific optical computing architectures for neural networks.
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