The problem of combining p-values is an old and fundamental one, and the classic assumption of independence is often violated or unverifiable in many applications. There are many well-known rules that can combine a set of arbitrarily dependent p-values (for the same hypothesis) into a single p-value. We show that essentially all these existing rules can be strictly improved when the p-values are exchangeable, or when external randomization is allowed (or both). For example, we derive randomized and/or exchangeable improvements of well known rules like "twice the median" and "twice the average", as well as geometric and harmonic means. Exchangeable p-values are often produced one at a time (for example, under repeated tests involving data splitting), and our rules can combine them sequentially as they are produced, stopping when the combined p-values stabilize. Our work also improves rules for combining arbitrarily dependent p-values, since the latter becomes exchangeable if they are presented to the analyst in a random order. The main technical advance is to show that all existing combination rules can be obtained by calibrating the p-values to e-values (using an $\alpha$-dependent calibrator), averaging those e-values, converting to a level-$\alpha$ test using Markov's inequality, and finally obtaining p-values by combining this family of tests; the improvements are delivered via recent randomized and exchangeable variants of Markov's inequality.
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Predicting quantum operator matrices such as Hamiltonian, overlap, and density matrices in the density functional theory (DFT) framework is crucial for understanding material properties. Current methods often focus on individual operators and struggle with efficiency and scalability for large systems. Here we introduce a novel deep learning model, SLEM (Strictly Localized Equivariant Message-passing) for predicting multiple quantum operators, that achieves state-of-the-art accuracy while dramatically improving computational efficiency. SLEM's key innovation is its strict locality-based design, constructing local, equivariant representations for quantum tensors while preserving physical symmetries. This enables complex many-body dependence without expanding the effective receptive field, leading to superior data efficiency and transferability. Using an innovative SO(2) convolution technique, SLEM reduces the computational complexity of high-order tensor products and is therefore capable of handling systems requiring the $f$ and $g$ orbitals in their basis sets. We demonstrate SLEM's capabilities across diverse 2D and 3D materials, achieving high accuracy even with limited training data. SLEM's design facilitates efficient parallelization, potentially extending DFT simulations to systems with device-level sizes, opening new possibilities for large-scale quantum simulations and high-throughput materials discovery.
We study the problem of computing the value function from a discretely-observed trajectory of a continuous-time diffusion process. We develop a new class of algorithms based on easily implementable numerical schemes that are compatible with discrete-time reinforcement learning (RL) with function approximation. We establish high-order numerical accuracy as well as the approximation error guarantees for the proposed approach. In contrast to discrete-time RL problems where the approximation factor depends on the effective horizon, we obtain a bounded approximation factor using the underlying elliptic structures, even if the effective horizon diverges to infinity.
Divergence constraints are present in the governing equations of many physical phenomena, and they usually lead to a Poisson equation whose solution typically is the main bottleneck of many simulation codes. Algebraic Multigrid (AMG) is arguably the most powerful preconditioner for Poisson's equation, and its effectiveness results from the complementary roles played by the smoother, responsible for damping high-frequency error components, and the coarse-grid correction, which in turn reduces low-frequency modes. This work presents several strategies to make AMG more compute-intensive by leveraging reflection, translational and rotational symmetries, often present in academic and industrial configurations. The best-performing method, AMGR, is based on a multigrid reduction framework that introduces an aggressive coarsening to the multigrid hierarchy, reducing the memory footprint, setup and application costs of the top-level smoother. While preserving AMG's excellent convergence, AMGR allows replacing the standard sparse matrix-vector product with the more compute-intensive sparse matrix-matrix product, yielding significant accelerations. Numerical experiments on industrial CFD applications demonstrated up to 70% speed-ups when solving Poisson's equation with AMGR instead of AMG. Additionally, strong and weak scalability analyses revealed no significant degradation.
Subsampling is one of the popular methods to balance statistical efficiency and computational efficiency in the big data era. Most approaches aim at selecting informative or representative sample points to achieve good overall information of the full data. The present work takes the view that sampling techniques are recommended for the region we focus on and summary measures are enough to collect the information for the rest according to a well-designed data partitioning. We propose a multi-resolution subsampling strategy that combines global information described by summary measures and local information obtained from selected subsample points. We show that the proposed method will lead to a more efficient subsample-based estimator for general large-scale classification problems. Some asymptotic properties of the proposed method are established and connections to existing subsampling procedures are explored. Finally, we illustrate the proposed subsampling strategy via simulated and real-world examples.
We investigate the set of invariant idempotent probabilities for countable idempotent iterated function systems (IFS) defined in compact metric spaces. We demonstrate that, with constant weights, there exists a unique invariant idempotent probability. Utilizing Secelean's approach to countable IFSs, we introduce partially finite idempotent IFSs and prove that the sequence of invariant idempotent measures for these systems converges to the invariant measure of the original countable IFS. We then apply these results to approximate such measures with discrete systems, producing, in the one-dimensional case, data series whose Higuchi fractal dimension can be calculated. Finally, we provide numerical approximations for two-dimensional cases and discuss the application of generalized Higuchi dimensions in these scenarios.
We propose a way to maintain strong consistency and facilitate error analysis in the context of dissipation-based WENO stabilization for continuous and discontinuous Galerkin discretizations of conservation laws. Following Kuzmin and Vedral (J. Comput. Phys. 487:112153, 2023) and Vedral (arXiv preprint arXiv:2309.12019), we use WENO shock detectors to determine appropriate amounts of low-order artificial viscosity. In contrast to existing WENO methods, our approach blends candidate polynomials using residual-based nonlinear weights. The shock-capturing terms of our stabilized Galerkin methods vanish if residuals do. This enables us to achieve improved accuracy compared to weakly consistent alternatives. As we show in the context of steady convection-diffusion-reaction (CDR) equations, nonlinear local projection stabilization terms can be included in a way that preserves the coercivity of local bilinear forms. For the corresponding Galerkin-WENO discretization of a CDR problem, we rigorously derive a priori error estimates. Additionally, we demonstrate the stability and accuracy of the proposed method through one- and two-dimensional numerical experiments for hyperbolic conservation laws and systems thereof. The numerical results for representative test problems are superior to those obtained with traditional WENO schemes, particularly in scenarios involving shocks and steep gradients.
Common statistical measures of uncertainty such as $p$-values and confidence intervals quantify the uncertainty due to sampling, that is, the uncertainty due to not observing the full population. However, sampling is not the only source of uncertainty. In practice, distributions change between locations and across time. This makes it difficult to gather knowledge that transfers across data sets. We propose a measure of instability that quantifies the distributional instability of a statistical parameter with respect to Kullback-Leibler divergence, that is, the sensitivity of the parameter under general distributional perturbations within a Kullback-Leibler divergence ball. In addition, we quantify the instability of parameters with respect to directional or variable-specific shifts. Measuring instability with respect to directional shifts can be used to detect the type of shifts a parameter is sensitive to. We discuss how such knowledge can inform data collection for improved estimation of statistical parameters under shifted distributions. We evaluate the performance of the proposed measure on real data and show that it can elucidate the distributional instability of a parameter with respect to certain shifts and can be used to improve estimation accuracy under shifted distributions.
We propose an extremely versatile approach to address a large family of matrix nearness problems, possibly with additional linear constraints. Our method is based on splitting a matrix nearness problem into two nested optimization problems, of which the inner one can be solved either exactly or cheaply, while the outer one can be recast as an unconstrained optimization task over a smooth real Riemannian manifold. We observe that this paradigm applies to many matrix nearness problems of practical interest appearing in the literature, thus revealing that they are equivalent in this sense to a Riemannian optimization problem. We also show that the objective function to be minimized on the Riemannian manifold can be discontinuous, thus requiring regularization techniques, and we give conditions for this to happen. Finally, we demonstrate the practical applicability of our method by implementing it for a number of matrix nearness problems that are relevant for applications and are currently considered very demanding in practice. Extensive numerical experiments demonstrate that our method often greatly outperforms its predecessors, including algorithms specifically designed for those particular problems.
Reduced basis methods for approximating the solutions of parameter-dependant partial differential equations (PDEs) are based on learning the structure of the set of solutions - seen as a manifold ${\mathcal S}$ in some functional space - when the parameters vary. This involves investigating the manifold and, in particular, understanding whether it is close to a low-dimensional affine space. This leads to the notion of Kolmogorov $N$-width that consists of evaluating to which extent the best choice of a vectorial space of dimension $N$ approximates ${\mathcal S}$ well enough. If a good approximation of elements in ${\mathcal S}$ can be done with some well-chosen vectorial space of dimension $N$ -- provided $N$ is not too large -- then a ``reduced'' basis can be proposed that leads to a Galerkin type method for the approximation of any element in ${\mathcal S}$. In many cases, however, the Kolmogorov $N$-width is not so small, even if the parameter set lies in a space of small dimension yielding a manifold with small dimension. In terms of complexity reduction, this gap between the small dimension of the manifold and the large Kolmogorov $N$-width can be explained by the fact that the Kolmogorov $N$-width is linear while, in contrast, the dependency in the parameter is, most often, non-linear. There have been many contributions aiming at reconciling these two statements, either based on deterministic or AI approaches. We investigate here further a new paradigm that, in some sense, merges these two aspects: the nonlinear compressive reduced basisapproximation. We focus on a simple multiparameter problem and illustrate rigorously that the complexity associated with the approximation of the solution to the parameter dependant PDE is directly related to the number of parameters rather than the Kolmogorov $N$-width.
Prediction models are used to predict an outcome based on input variables. Missing data in input variables often occurs at model development and at prediction time. The missForestPredict R package proposes an adaptation of the missForest imputation algorithm that is fast, user-friendly and tailored for prediction settings. The algorithm iteratively imputes variables using random forests until a convergence criterion (unified for continuous and categorical variables and based on the out-of-bag error) is met. The imputation models are saved for each variable and iteration and can be applied later to new observations at prediction time. The missForestPredict package offers extended error monitoring, control over variables used in the imputation and custom initialization. This allows users to tailor the imputation to their specific needs. The missForestPredict algorithm is compared to mean/mode imputation, linear regression imputation, mice, k-nearest neighbours, bagging, miceRanger and IterativeImputer on eight simulated datasets with simulated missingness (48 scenarios) and eight large public datasets using different prediction models. missForestPredict provides competitive results in prediction settings within short computation times.
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