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We extend the typical forcing of M. M\"uller and derive conditions on the forcing frame for which generic expansions preserve injective/bijective pigeonhole principle for polynomial-time computable graphs of functions. Applying this machinery, we show that the bounded arithmetic theory $\forall \textsf{T}^1_2(\textsf{PV}(\alpha))$ augmented by the polynomial-time injective pigeonhole principle does not prove the linear ordering, tournament, and dual weak pigeonhole principles.

We investigate diffusion models to solve the Traveling Salesman Problem. Building on the recent DIFUSCO and T2TCO approaches, we propose IDEQ. IDEQ improves the quality of the solutions by leveraging the constrained structure of the state space of the TSP. Another key component of IDEQ consists in replacing the last stages of DIFUSCO curriculum learning by considering a uniform distribution over the Hamiltonian tours whose orbits by the 2-opt operator converge to the optimal solution as the training objective. Our experiments show that IDEQ improves the state of the art for such neural network based techniques on synthetic instances. More importantly, our experiments show that IDEQ performs very well on the instances of the TSPlib, a reference benchmark in the TSP community: it closely matches the performance of the best heuristics, LKH3, being even able to obtain better solutions than LKH3 on 2 instances of the TSPlib defined on 1577 and 3795 cities. IDEQ obtains 0.3% optimality gap on TSP instances made of 500 cities, and 0.5% on TSP instances with 1000 cities. This sets a new SOTA for neural based methods solving the TSP. Moreover, IDEQ exhibits a lower variance and better scales-up with the number of cities with regards to DIFUSCO and T2TCO.

Data sharing is ubiquitous in the metaverse, which adopts blockchain as its foundation. Blockchain is employed because it enables data transparency, achieves tamper resistance, and supports smart contracts. However, securely sharing data based on blockchain necessitates further consideration. Ciphertext-policy attribute-based encryption (CP-ABE) is a promising primitive to provide confidentiality and fine-grained access control. Nonetheless, authority accountability and key abuse are critical issues that practical applications must address. Few studies have considered CP-ABE key confidentiality and authority accountability simultaneously. To our knowledge, we are the first to fill this gap by integrating non-interactive zero-knowledge (NIZK) proofs into CP-ABE keys and outsourcing the verification process to a smart contract. To meet the decentralization requirement, we incorporate a decentralized CP-ABE scheme into the proposed data sharing system. Additionally, we provide an implementation based on smart contract to determine whether an access control policy is satisfied by a set of CP-ABE keys. We also introduce an open incentive mechanism to encourage honest participation in data sharing. Hence, the key abuse issue is resolved through the NIZK proof and the incentive mechanism. We provide a theoretical analysis and conduct comprehensive experiments to demonstrate the feasibility and efficiency of the data sharing system. Based on the proposed accountable approach, we further illustrate an application in GameFi, where players can play to earn or contribute to an accountable DAO, fostering a thriving metaverse ecosystem.

Cross-sectional incidence estimation based on recency testing has become a widely used tool in HIV research. Recently, this method has gained prominence in HIV prevention trials to estimate the "placebo" incidence that participants might experience without preventive treatment. The application of this approach faces challenges due to non-representative sampling, as individuals aware of their HIV-positive status may be less likely to participate in screening for an HIV prevention trial. To address this, a recent phase 3 trial excluded individuals based on whether they have had a recent HIV test. To the best of our knowledge, the validity of this approach has yet to be studied. In our work, we investigate the performance of cross-sectional HIV incidence estimation when excluding individuals based on prior HIV tests in realistic trial settings. We develop a statistical framework that incorporates a testing-based criterion and possible non-representative sampling. We introduce a metric we call the effective mean duration of recent infection (MDRI) that mathematically quantifies bias in incidence estimation. We conduct an extensive simulation study to evaluate incidence estimator performance under various scenarios. Our findings reveal that when screening attendance is affected by knowledge of HIV status, incidence estimators become unreliable unless all individuals with recent HIV tests are excluded. Additionally, we identified a trade-off between bias and variability: excluding more individuals reduces bias from non-representative sampling but in many cases increases the variability of incidence estimates. These findings highlight the need for caution when applying testing-based criteria and emphasize the importance of refining incidence estimation methods to improve the design and evaluation of future HIV prevention trials.

We derive a priori and a posteriori error estimates for the discontinuous Galerkin (dG) approximation of the time-harmonic Maxwell's equations. Specifically, we consider an interior penalty dG method, and establish error estimates that are valid under minimal regularity assumptions and involving constants that do not depend on the frequency for sufficiently fine meshes. The key result of our a priori error analysis is that the dG solution is asymptotically optimal in an augmented energy norm that contains the dG stabilization. Specifically, up to a constant that tends to one as the mesh is refined, the dG solution is as accurate as the best approximation in the same norm. The main insight is that the quantities controlling the smallness of the mesh size are essentially those already appearing in the conforming setting. We also show that for fine meshes, the inf-sup stability constant is as good as the continuous one up to a factor two. Concerning the a posteriori analysis, we consider a residual-based error estimator under the assumption of piecewise constant material properties. We derive a global upper bound and local lower bounds on the error with constants that (i) only depend on the shape-regularity of the mesh if it is sufficiently refined and (ii) are independent of the stabilization bilinear form.

Large Language Models (LLMs) are increasingly explored as knowledge bases (KBs), yet current evaluation methods focus too narrowly on knowledge retention, overlooking other crucial criteria for reliable performance. In this work, we rethink the requirements for evaluating reliable LLM-as-KB usage and highlight two essential factors: factuality, ensuring accurate responses to seen and unseen knowledge, and consistency, maintaining stable answers to questions about the same knowledge. We introduce UnseenQA, a dataset designed to assess LLM performance on unseen knowledge, and propose new criteria and metrics to quantify factuality and consistency, leading to a final reliability score. Our experiments on 26 LLMs reveal several challenges regarding their use as KBs, underscoring the need for more principled and comprehensive evaluation.

In this work, we study the problem of aggregation in the context of Bayesian Federated Learning (BFL). Using an information geometric perspective, we interpret the BFL aggregation step as finding the barycenter of the trained posteriors for a pre-specified divergence metric. We study the barycenter problem for the parametric family of $\alpha$-divergences and, focusing on the standard case of independent and Gaussian distributed parameters, we recover the closed-form solution of the reverse Kullback-Leibler barycenter and develop the analytical form of the squared Wasserstein-2 barycenter. Considering a non-IID setup, where clients possess heterogeneous data, we analyze the performance of the developed algorithms against state-of-the-art (SOTA) Bayesian aggregation methods in terms of accuracy, uncertainty quantification (UQ), model calibration (MC), and fairness. Finally, we extend our analysis to the framework of Hybrid Bayesian Deep Learning (HBDL), where we study how the number of Bayesian layers in the architecture impacts the considered performance metrics. Our experimental results show that the proposed methodology presents comparable performance with the SOTA while offering a geometric interpretation of the aggregation phase.

In the recent years, Physics Informed Neural Networks (PINNs) have received strong interest as a method to solve PDE driven systems, in particular for data assimilation purpose. This method is still in its infancy, with many shortcomings and failures that remain not properly understood. In this paper we propose a natural gradient approach to PINNs which contributes to speed-up and improve the accuracy of the training. Based on an in depth analysis of the differential geometric structures of the problem, we come up with two distinct contributions: (i) a new natural gradient algorithm that scales as $\min(P^2S, S^2P)$, where $P$ is the number of parameters, and $S$ the batch size; (ii) a mathematically principled reformulation of the PINNs problem that allows the extension of natural gradient to it, with proved connections to Green's function theory.

In this work, we present a parametric finite element approximation of two-phase Navier-Stokes flow with viscoelasticity. The free boundary problem is given by the viscoelastic Navier-Stokes equations in the two fluid phases, connected by jump conditions across the interface. The elasticity in the fluids is characterised using the Oldroyd-B model with possible stress diffusion. The model was originally introduced to approximate fluid-structure interaction problems between an incompressible Newtonian fluid and a hyperelastic neo-Hookean solid, which are possible limit cases of the model. We approximate a variational formulation of the model with an unfitted finite element method that uses piecewise linear parametric finite elements. The two-phase Navier-Stokes-Oldroyd-B system in the bulk regions is discretised in a way that guarantees unconditional solvability and stability for the coupled bulk-interface system. Good volume conservation properties for the two phases are observed in the case where the pressure approximation space is enriched with the help of an XFEM function. We show the applicability of our method with some numerical results.

Human-in-the-loop aims to train an accurate prediction model with minimum cost by integrating human knowledge and experience. Humans can provide training data for machine learning applications and directly accomplish some tasks that are hard for computers in the pipeline with the help of machine-based approaches. In this paper, we survey existing works on human-in-the-loop from a data perspective and classify them into three categories with a progressive relationship: (1) the work of improving model performance from data processing, (2) the work of improving model performance through interventional model training, and (3) the design of the system independent human-in-the-loop. Using the above categorization, we summarize major approaches in the field, along with their technical strengths/ weaknesses, we have simple classification and discussion in natural language processing, computer vision, and others. Besides, we provide some open challenges and opportunities. This survey intends to provide a high-level summarization for human-in-the-loop and motivates interested readers to consider approaches for designing effective human-in-the-loop solutions.