This article presents a new polynomial parameterized sigmoid called SIGTRON, which is an extended asymmetric sigmoid with Perceptron, and its companion convex model called SIGTRON-imbalanced classification (SIC) model that employs a virtual SIGTRON-induced convex loss function. In contrast to the conventional $\pi$-weighted cost-sensitive learning model, the SIC model does not have an external $\pi$-weight on the loss function but has internal parameters in the virtual SIGTRON-induced loss function. As a consequence, when the given training dataset is close to the well-balanced condition, we show that the proposed SIC model is more adaptive to variations of the dataset, such as the inconsistency of the scale-class-imbalance ratio between the training and test datasets. This adaptation is achieved by creating a skewed hyperplane equation. Additionally, we present a quasi-Newton optimization(L-BFGS) framework for the virtual convex loss by developing an interval-based bisection line search. Empirically, we have observed that the proposed approach outperforms $\pi$-weighted convex focal loss and balanced classifier LIBLINEAR(logistic regression, SVM, and L2SVM) in terms of test classification accuracy with $51$ two-class and $67$ multi-class datasets. In binary classification problems, where the scale-class-imbalance ratio of the training dataset is not significant but the inconsistency exists, a group of SIC models with the best test accuracy for each dataset (TOP$1$) outperforms LIBSVM(C-SVC with RBF kernel), a well-known kernel-based classifier.
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Complex system design problems, such as those involved in aerospace engineering, require the use of numerically costly simulation codes in order to predict the performance of the system to be designed. In this context, these codes are often embedded into an optimization process to provide the best design while satisfying the design constraints. Recently, new approaches, called Quality-Diversity, have been proposed in order to enhance the exploration of the design space and to provide a set of optimal diversified solutions with respect to some feature functions. These functions are interesting to assess trade-offs. Furthermore, complex design problems often involve mixed continuous, discrete, and categorical design variables allowing to take into account technological choices in the optimization problem. Existing Bayesian Quality-Diversity approaches suited for intensive high-fidelity simulations are not adapted to mixed variables constrained optimization problems. In order to overcome these limitations, a new Quality-Diversity methodology based on mixed variables Bayesian optimization strategy is proposed in the context of limited simulation budget. Using adapted covariance models and dedicated enrichment strategy for the Gaussian processes in Bayesian optimization, this approach allows to reduce the computational cost up to two orders of magnitude, with respect to classical Quality-Diversity approaches while dealing with discrete choices and the presence of constraints. The performance of the proposed method is assessed on a benchmark of analytical problems as well as on two aerospace system design problems highlighting its efficiency in terms of speed of convergence. The proposed approach provides valuable trade-offs for decision-markers for complex system design.
We ask whether there exists a function or measure that (1) minimizes a given convex functional or risk and (2) satisfies a symmetry property specified by an amenable group of transformations. Examples of such symmetry properties are invariance, equivariance, or quasi-invariance. Our results draw on old ideas of Stein and Le Cam and on approximate group averages that appear in ergodic theorems for amenable groups. A class of convex sets known as orbitopes in convex analysis emerges as crucial, and we establish properties of such orbitopes in nonparametric settings. We also show how a simple device called a cocycle can be used to reduce different forms of symmetry to a single problem. As applications, we obtain results on invariant kernel mean embeddings and a Monge-Kantorovich theorem on optimality of transport plans under symmetry constraints. We also explain connections to the Hunt-Stein theorem on invariant tests.
We develop new multilevel Monte Carlo (MLMC) methods to estimate the expectation of the smallest eigenvalue of a stochastic convection-diffusion operator with random coefficients. The MLMC method is based on a sequence of finite element (FE) discretizations of the eigenvalue problem on a hierarchy of increasingly finer meshes. For the discretized, algebraic eigenproblems we use both the Rayleigh quotient (RQ) iteration and implicitly restarted Arnoldi (IRA), providing an analysis of the cost in each case. By studying the variance on each level and adapting classical FE error bounds to the stochastic setting, we are able to bound the total error of our MLMC estimator and provide a complexity analysis. As expected, the complexity bound for our MLMC estimator is superior to plain Monte Carlo. To improve the efficiency of the MLMC further, we exploit the hierarchy of meshes and use coarser approximations as starting values for the eigensolvers on finer ones. To improve the stability of the MLMC method for convection-dominated problems, we employ two additional strategies. First, we consider the streamline upwind Petrov--Galerkin formulation of the discrete eigenvalue problem, which allows us to start the MLMC method on coarser meshes than is possible with standard FEs. Second, we apply a homotopy method to add stability to the eigensolver for each sample. Finally, we present a multilevel quasi-Monte Carlo method that replaces Monte Carlo with a quasi-Monte Carlo (QMC) rule on each level. Due to the faster convergence of QMC, this improves the overall complexity. We provide detailed numerical results comparing our different strategies to demonstrate the practical feasibility of the MLMC method in different use cases. The results support our complexity analysis and further demonstrate the superiority over plain Monte Carlo in all cases.
Cycloids are particular Petri nets for modelling processes of actions or events. They belong to the fundaments of Petri's general systems theory and have very different interpretations, ranging from Einstein's relativity theory and elementary information processing gates to the modelling of interacting sequential processes. This article contains previously unpublished proofs of cycloid properties using linear algebra.
The evaluation of text-generative vision-language models is a challenging yet crucial endeavor. By addressing the limitations of existing Visual Question Answering (VQA) benchmarks and proposing innovative evaluation methodologies, our research seeks to advance our understanding of these models' capabilities. We propose a novel VQA benchmark based on well-known visual classification datasets which allows a granular evaluation of text-generative vision-language models and their comparison with discriminative vision-language models. To improve the assessment of coarse answers on fine-grained classification tasks, we suggest using the semantic hierarchy of the label space to ask automatically generated follow-up questions about the ground-truth category. Finally, we compare traditional NLP and LLM-based metrics for the problem of evaluating model predictions given ground-truth answers. We perform a human evaluation study upon which we base our decision on the final metric. We apply our benchmark to a suite of vision-language models and show a detailed comparison of their abilities on object, action, and attribute classification. Our contributions aim to lay the foundation for more precise and meaningful assessments, facilitating targeted progress in the exciting field of vision-language modeling.
The Crank-Nicolson (CN) method is a well-known time integrator for evolutionary partial differential equations (PDEs) arising in many real-world applications. Since the solution at any time depends on the solution at previous time steps, the CN method is inherently difficult to parallelize. In this paper, we consider a parallel method for the solution of evolutionary PDEs with the CN scheme. Using an all-at-once approach, we can solve for all time steps simultaneously using a parallelizable over time preconditioner within a standard iterative method. Due to the diagonalization of the proposed preconditioner, we can prove that most eigenvalues of preconditioned matrices are equal to 1 and the others lie in the set: $\left\{z\in\mathbb{C}: 1/(1 + \alpha) < |z| < 1/(1 - \alpha)~{\rm and}~\Re{\rm e}(z) > 0\right\}$, where $0 < \alpha < 1$ is a free parameter. Besides, the efficient implementation of the proposed preconditioner is described. Given certain conditions, we prove that the preconditioned GMRES method exhibits a mesh-independent convergence rate. Finally, we will verify both theoretical findings and the efficacy of the proposed preconditioner via numerical experiments on financial option pricing PDEs.
We propose a hybrid iterative method based on MIONet for PDEs, which combines the traditional numerical iterative solver and the recent powerful machine learning method of neural operator, and further systematically analyze its theoretical properties, including the convergence condition, the spectral behavior, as well as the convergence rate, in terms of the errors of the discretization and the model inference. We show the theoretical results for the frequently-used smoothers, i.e. Richardson (damped Jacobi) and Gauss-Seidel. We give an upper bound of the convergence rate of the hybrid method w.r.t. the model correction period, which indicates a minimum point to make the hybrid iteration converge fastest. Several numerical examples including the hybrid Richardson (Gauss-Seidel) iteration for the 1-d (2-d) Poisson equation are presented to verify our theoretical results, and also reflect an excellent acceleration effect. As a meshless acceleration method, it is provided with enormous potentials for practice applications.
Mendelian randomization (MR) is an instrumental variable (IV) approach to infer causal relationships between exposures and outcomes with genome-wide association studies (GWAS) summary data. However, the multivariable inverse-variance weighting (IVW) approach, which serves as the foundation for most MR approaches, cannot yield unbiased causal effect estimates in the presence of many weak IVs. To address this problem, we proposed the MR using Bias-corrected Estimating Equation (MRBEE) that can infer unbiased causal relationships with many weak IVs and account for horizontal pleiotropy simultaneously. While the practical significance of MRBEE was demonstrated in our parallel work (Lorincz-Comi (2023)), this paper established the statistical theories of multivariable IVW and MRBEE with many weak IVs. First, we showed that the bias of the multivariable IVW estimate is caused by the error-in-variable bias, whose scale and direction are inflated and influenced by weak instrument bias and sample overlaps of exposures and outcome GWAS cohorts, respectively. Second, we investigated the asymptotic properties of multivariable IVW and MRBEE, showing that MRBEE outperforms multivariable IVW regarding unbiasedness of causal effect estimation and asymptotic validity of causal inference. Finally, we applied MRBEE to examine myopia and revealed that education and outdoor activity are causal to myopia whereas indoor activity is not.
We show that Edge Multiway Cut (also called Multiterminal Cut) and Node Multiway Cut are NP-complete on graphs of maximum degree $3$ (also known as subcubic graphs). This improves on a previous degree bound of $11$. Our NP-completeness result holds even for subcubic graphs that are planar.
We provide a new theoretical framework for the variable-step deferred correction (DC) methods based on the well-known BDF2 formula. By using the discrete orthogonal convolution kernels, some high-order BDF2-DC methods are proven to be stable on arbitrary time grids according to the recent definition of stability (SINUM, 60: 2253-2272). It significantly relaxes the existing step-ratio restrictions for the BDF2-DC methods (BIT, 62: 1789-1822). The associated sharp error estimates are established by taking the numerical effects of the starting approximations into account, and they suggest that the BDF2-DC methods have no aftereffect, that is, the lower-order starting scheme for the BDF2 scheme will not cause a loss in the accuracy of the high-order BDF2-DC methods. Extensive tests on the graded and random time meshes are presented to support the new theory.
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