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For several types of information relations, the induced rough sets system RS does not form a lattice but only a partially ordered set. However, by studying its Dedekind-MacNeille completion DM(RS), one may reveal new important properties of rough set structures. Building upon D. Umadevi's work on describing joins and meets in DM(RS), we previously investigated pseudo-Kleene algebras defined on DM(RS) for reflexive relations. This paper delves deeper into the order-theoretic properties of DM(RS) in the context of reflexive relations. We describe the completely join-irreducible elements of DM(RS) and characterize when DM(RS) is a spatial completely distributive lattice. We show that even in the case of a non-transitive reflexive relation, DM(RS) can form a Nelson algebra, a property generally associated with quasiorders. We introduce a novel concept, the core of a relational neighborhood, and use it to provide a necessary and sufficient condition for DM(RS) to determine a Nelson algebra.

This paper demonstrates numerically and experimentally that it is possible to tailor flexural band gaps in the low-frequency regime by appropriate choice of cutout characteristics. The finite element method is used to obtain the numerical dispersion relation and band gaps. The influence of the cutout's shape, size, and location on the band gap is systematically studied. The study demonstrates that the cutout should pass through the center of the unit cell, and a large aspect ratio is required to introduce flexural band gaps in the low-frequency regime. This is validated by experiments on a finite plate with 3 $\times$ 3 unit cells.

In industrial contexts, effective workforce allocation is crucial for operational efficiency. This paper presents an ongoing project focused on developing a decision-making tool designed for workforce allocation, emphasising the explainability to enhance its trustworthiness. Our objective is to create a system that not only optimises the allocation of teams to scheduled tasks but also provides clear, understandable explanations for its decisions, particularly in cases where the problem is infeasible. By incorporating human-in-the-loop mechanisms, the tool aims to enhance user trust and facilitate interactive conflict resolution. We implemented our approach on a prototype tool/digital demonstrator intended to be evaluated on a real industrial scenario both in terms of performance and user acceptability.

Gradient descent is one of the most widely used iterative algorithms in modern statistical learning. However, its precise algorithmic dynamics in high-dimensional settings remain only partially understood, which has therefore limited its broader potential for statistical inference applications. This paper provides a precise, non-asymptotic distributional characterization of gradient descent iterates in a broad class of empirical risk minimization problems, in the so-called mean-field regime where the sample size is proportional to the signal dimension. Our non-asymptotic state evolution theory holds for both general non-convex loss functions and non-Gaussian data, and reveals the central role of two Onsager correction matrices that precisely characterize the non-trivial dependence among all gradient descent iterates in the mean-field regime. Although the Onsager correction matrices are typically analytically intractable, our state evolution theory facilitates a generic gradient descent inference algorithm that consistently estimates these matrices across a broad class of models. Leveraging this algorithm, we show that the state evolution can be inverted to construct (i) data-driven estimators for the generalization error of gradient descent iterates and (ii) debiased gradient descent iterates for inference of the unknown signal. Detailed applications to two canonical models--linear regression and (generalized) logistic regression--are worked out to illustrate model-specific features of our general theory and inference methods.

Background: The standard regulatory approach to assess replication success is the two-trials rule, requiring both the original and the replication study to be significant with effect estimates in the same direction. The sceptical p-value was recently presented as an alternative method for the statistical assessment of the replicability of study results. Methods: We compare the statistical properties of the sceptical p-value and the two-trials rule. We illustrate the performance of the different methods using real-world evidence emulations of randomized, controlled trials (RCTs) conducted within the RCT DUPLICATE initiative. Results: The sceptical p-value depends not only on the two p-values, but also on sample size and effect size of the two studies. It can be calibrated to have the same Type-I error rate as the two-trials rule, but has larger power to detect an existing effect. In the application to the results from the RCT DUPLICATE initiative, the sceptical p-value leads to qualitatively similar results than the two-trials rule, but tends to show more evidence for treatment effects compared to the two-trials rule. Conclusion: The sceptical p-value represents a valid statistical measure to assess the replicability of study results and is especially useful in the context of real-world evidence emulations.

In this paper, we consider a class of non-convex and non-smooth sparse optimization problems, which encompass most existing nonconvex sparsity-inducing terms. We show the second-order optimality conditions only depend on the nonzeros of the stationary points. We propose two damped iterative reweighted algorithms including the iteratively reweighted $\ell_1$ algorithm (DIRL$_1$) and the iteratively reweighted $\ell_2$ (DIRL$_2$) algorithm, to solve these problems. For DIRL$_1$, we show the reweighted $\ell_1$ subproblem has support identification property so that DIRL$_1$ locally reverts to a gradient descent algorithm around a stationary point. For DIRL$_2$, we show the solution map of the reweighted $\ell_2$ subproblem is differentiable and Lipschitz continuous everywhere. Therefore, the map of DIRL$_1$ and DIRL$_2$ and their inverse are Lipschitz continuous, and the strict saddle points are their unstable fixed points. By applying the stable manifold theorem, these algorithms are shown to converge only to local minimizers with randomly initialization when the strictly saddle point property is assumed.

This paper is concerned with a Bayesian approach to testing hypotheses in statistical inverse problems. Based on the posterior distribution $\Pi \left(\cdot |Y = y\right)$, we want to infer whether a feature $\langle\varphi, u^\dagger\rangle$ of the unknown quantity of interest $u^\dagger$ is positive. This can be done by the so-called maximum a posteriori test. We provide a frequentistic analysis of this test's properties such as level and power, and prove that it is a regularized test in the sense of Kretschmann et al. (2024). Furthermore we provide lower bounds for its power under classical spectral source conditions in case of Gaussian priors. Numerical simulations illustrate its superior performance both in moderately and severely ill-posed situations.

We consider the problem of causal inference based on observational data (or the related missing data problem) with a binary or discrete treatment variable. In that context, we study inference for the counterfactual density functions and contrasts thereof, which can provide more nuanced information than counterfactual means and the average treatment effect. We impose the shape-constraint of log-concavity, a type of unimodality constraint, on the counterfactual densities, and then develop doubly robust estimators of the log-concave counterfactual density based on augmented inverse-probability weighted pseudo-outcomes. We provide conditions under which the estimator is consistent in various global metrics. We also develop asymptotically valid pointwise confidence intervals for the counterfactual density functions and differences and ratios thereof, which serve as a building block for more comprehensive analyses of distributional differences. We also present a method for using our estimator to implement density confidence bands.

Two sequential estimators are proposed for the odds p/(1-p) and log odds log(p/(1-p)) respectively, using independent Bernoulli random variables with parameter p as inputs. The estimators are unbiased, and guarantee that the variance of the estimation error divided by the true value of the odds, or the variance of the estimation error of the log odds, are less than a target value for any p in (0,1). The estimators are close to optimal in the sense of Wolfowitz's bound.

This paper does not describe a working system. Instead, it presents a single idea about representation which allows advances made by several different groups to be combined into an imaginary system called GLOM. The advances include transformers, neural fields, contrastive representation learning, distillation and capsules. GLOM answers the question: How can a neural network with a fixed architecture parse an image into a part-whole hierarchy which has a different structure for each image? The idea is simply to use islands of identical vectors to represent the nodes in the parse tree. If GLOM can be made to work, it should significantly improve the interpretability of the representations produced by transformer-like systems when applied to vision or language