Out-of-distribution (OOD) generalization in the graph domain is challenging due to complex distribution shifts and a lack of environmental contexts. Recent methods attempt to enhance graph OOD generalization by generating flat environments. However, such flat environments come with inherent limitations to capture more complex data distributions. Considering the DrugOOD dataset, which contains diverse training environments (e.g., scaffold, size, etc.), flat contexts cannot sufficiently address its high heterogeneity. Thus, a new challenge is posed to generate more semantically enriched environments to enhance graph invariant learning for handling distribution shifts. In this paper, we propose a novel approach to generate hierarchical semantic environments for each graph. Firstly, given an input graph, we explicitly extract variant subgraphs from the input graph to generate proxy predictions on local environments. Then, stochastic attention mechanisms are employed to re-extract the subgraphs for regenerating global environments in a hierarchical manner. In addition, we introduce a new learning objective that guides our model to learn the diversity of environments within the same hierarchy while maintaining consistency across different hierarchies. This approach enables our model to consider the relationships between environments and facilitates robust graph invariant learning. Extensive experiments on real-world graph data have demonstrated the effectiveness of our framework. Particularly, in the challenging dataset DrugOOD, our method achieves up to 1.29% and 2.83% improvement over the best baselines on IC50 and EC50 prediction tasks, respectively.
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To solve many problems on graphs, graph traversals are used, the usual variants of which are the depth-first search and the breadth-first search. Implementing a graph traversal we consequently reach all vertices of the graph that belong to a connected component. The breadth-first search is the usual choice when constructing efficient algorithms for finding connected components of a graph. Methods of simple iteration for solving systems of linear equations with modified graph adjacency matrices and with the properly specified right-hand side can be considered as graph traversal algorithms. These traversal algorithms, generally speaking, turn out to be non-equivalent neither to the depth-first search nor the breadth-first search. The example of such a traversal algorithm is the one associated with the Gauss-Seidel method. For an arbitrary connected graph, to visit all its vertices, the algorithm requires not more iterations than that is required for BFS. For a large number of instances of the problem, fewer iterations will be required.
Robust optimisation is a well-established framework for optimising functions in the presence of uncertainty. The inherent goal of this problem is to identify a collection of inputs whose outputs are both desirable for the decision maker, whilst also being robust to the underlying uncertainties in the problem. In this work, we study the multi-objective case of this problem. We identify that the majority of all robust multi-objective algorithms rely on two key operations: robustification and scalarisation. Robustification refers to the strategy that is used to account for the uncertainty in the problem. Scalarisation refers to the procedure that is used to encode the relative importance of each objective to a scalar-valued reward. As these operations are not necessarily commutative, the order that they are performed in has an impact on the resulting solutions that are identified and the final decisions that are made. The purpose of this work is to give a thorough exposition on the effects of these different orderings and in particular highlight when one should opt for one ordering over the other. As part of our analysis, we showcase how many existing risk concepts can be integrated into the specification and solution of a robust multi-objective optimisation problem. Besides this, we also demonstrate how one can principally define the notion of a robust Pareto front and a robust performance metric based on our ``robustify and scalarise'' methodology. To illustrate the efficacy of these new ideas, we present two insightful case studies which are based on real-world data sets.
The discretization of fluid-poromechanics systems is typically highly demanding in terms of computational effort. This is particularly true for models of multiphysics flows in the brain, due to the geometrical complexity of the cerebral anatomy - requiring a very fine computational mesh for finite element discretization - and to the high number of variables involved. Indeed, this kind of problems can be modeled by a coupled system encompassing the Stokes equations for the cerebrospinal fluid in the brain ventricles and Multiple-network Poro-Elasticity (MPE) equations describing the brain tissue, the interstitial fluid, and the blood vascular networks at different space scales. The present work aims to rigorously derive a posteriori error estimates for the coupled Stokes-MPE problem, as a first step towards the design of adaptive refinement strategies or reduced order models to decrease the computational demand of the problem. Through numerical experiments, we verify the reliability and optimal efficiency of the proposed a posteriori estimator and identify the role of the different solution variables in its composition.
Variance reduction for causal inference in the presence of network interference is often achieved through either outcome modeling, which is typically analyzed under unit-randomized Bernoulli designs, or clustered experimental designs, which are typically analyzed without strong parametric assumptions. In this work, we study the intersection of these two approaches and consider the problem of estimation in low-order outcome models using data from a general experimental design. Our contributions are threefold. First, we present an estimator of the total treatment effect (also called the global average treatment effect) in a low-degree outcome model when the data are collected under general experimental designs, generalizing previous results for Bernoulli designs. We refer to this estimator as the pseudoinverse estimator and give bounds on its bias and variance in terms of properties of the experimental design. Second, we evaluate these bounds for the case of cluster randomized designs with both Bernoulli and complete randomization. For clustered Bernoulli randomization, we find that our estimator is always unbiased and that its variance scales like the smaller of the variance obtained from a low-order assumption and the variance obtained from cluster randomization, showing that combining these variance reduction strategies is preferable to using either individually. For clustered complete randomization, we find a notable bias-variance trade-off mediated by specific features of the clustering. Third, when choosing a clustered experimental design, our bounds can be used to select a clustering from a set of candidate clusterings. Across a range of graphs and clustering algorithms, we show that our method consistently selects clusterings that perform well on a range of response models, suggesting that our bounds are useful to practitioners.
The broad class of multivariate unified skew-normal (SUN) distributions has been recently shown to possess important conjugacy properties. When used as priors for the vector of parameters in general probit, tobit, and multinomial probit models, these distributions yield posteriors that still belong to the SUN family. Although such a core result has led to important advancements in Bayesian inference and computation, its applicability beyond likelihoods associated with fully-observed, discretized, or censored realizations from multivariate Gaussian models remains yet unexplored. This article covers such an important gap by proving that the wider family of multivariate unified skew-elliptical (SUE) distributions, which extends SUNs to more general perturbations of elliptical densities, guarantees conjugacy for broader classes of models, beyond those relying on fully-observed, discretized or censored Gaussians. Such a result leverages the closure under linear combinations, conditioning and marginalization of SUE to prove that this family is conjugate to the likelihood induced by general multivariate regression models for fully-observed, censored or dichotomized realizations from skew-elliptical distributions. This advancement enlarges the set of models that enable conjugate Bayesian inference to general formulations arising from elliptical and skew-elliptical families, including the multivariate Student's t and skew-t, among others.
In this work, N\'ed\'elec elements on locally refined meshes with hanging nodes are considered. A crucial aspect is the orientation of the hanging edges and faces. For non-orientable meshes, no solution or implementation has been available to date. The problem statement and corresponding algorithms are described in great detail. As a model problem, the time-harmonic Maxwell's equations are adopted because N\'ed\'elec elements constitute their natural discretization. The algorithms and implementation are demonstrated through two numerical examples on different uniformly and adaptively refined meshes. The implementation is performed within the finite element library deal.II.
The homogenization procedure developed here is conducted on a laminate with periodic space-time modulation on the fine scale: at leading order, this modulation creates convection in the low-wavelength regime if both parameters are modulated. However, if only one parameter is modulated, which is more realistic, this convective term disappears and one recovers a standard diffusion equation with effective homogeneous parameters; this does not describe the non-reciprocity and the propagation of the field observed from exact dispersion diagrams. This inconsistency is corrected here by considering second-order homogenization which results in a non-reciprocal propagation term that is proved to be non-zero for any laminate and verified via numerical simulation. The same methodology is also applied to the case when the density is modulated in the heat equation, leading therefore to a corrective advective term which cancels out non-reciprocity at the leading order but not at the second order.
Motivated by the recent successful application of physics-informed neural networks (PINNs) to solve Boltzmann-type equations [S. Jin, Z. Ma, and K. Wu, J. Sci. Comput., 94 (2023), pp. 57], we provide a rigorous error analysis for PINNs in approximating the solution of the Boltzmann equation near a global Maxwellian. The challenge arises from the nonlocal quadratic interaction term defined in the unbounded domain of velocity space. Analyzing this term on an unbounded domain requires the inclusion of a truncation function, which demands delicate analysis techniques. As a generalization of this analysis, we also provide proof of the asymptotic preserving property when using micro-macro decomposition-based neural networks.
A common method for estimating the Hessian operator from random samples on a low-dimensional manifold involves locally fitting a quadratic polynomial. Although widely used, it is unclear if this estimator introduces bias, especially in complex manifolds with boundaries and nonuniform sampling. Rigorous theoretical guarantees of its asymptotic behavior have been lacking. We show that, under mild conditions, this estimator asymptotically converges to the Hessian operator, with nonuniform sampling and curvature effects proving negligible, even near boundaries. Our analysis framework simplifies the intensive computations required for direct analysis.
Hashing has been widely used in approximate nearest search for large-scale database retrieval for its computation and storage efficiency. Deep hashing, which devises convolutional neural network architecture to exploit and extract the semantic information or feature of images, has received increasing attention recently. In this survey, several deep supervised hashing methods for image retrieval are evaluated and I conclude three main different directions for deep supervised hashing methods. Several comments are made at the end. Moreover, to break through the bottleneck of the existing hashing methods, I propose a Shadow Recurrent Hashing(SRH) method as a try. Specifically, I devise a CNN architecture to extract the semantic features of images and design a loss function to encourage similar images projected close. To this end, I propose a concept: shadow of the CNN output. During optimization process, the CNN output and its shadow are guiding each other so as to achieve the optimal solution as much as possible. Several experiments on dataset CIFAR-10 show the satisfying performance of SRH.
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