Optimizing strongly convex functions subject to linear constraints is a fundamental problem with numerous applications. In this work, we propose a block (accelerated) randomized Bregman-Kaczmarz method that only uses a block of constraints in each iteration to tackle this problem. We consider a dual formulation of this problem in order to deal in an efficient way with the linear constraints. Using convex tools, we show that the corresponding dual function satisfies the Polyak-Lojasiewicz (PL) property, provided that the primal objective function is strongly convex and verifies additionally some other mild assumptions. However, adapting the existing theory on coordinate descent methods to our dual formulation can only give us sublinear convergence results in the dual space. In order to obtain convergence results in some criterion corresponding to the primal (original) problem, we transfer our algorithm to the primal space, which combined with the PL property allows us to get linear convergence rates. More specifically, we provide a theoretical analysis of the convergence of our proposed method under different assumptions on the objective and demonstrate in the numerical experiments its superior efficiency and speed up compared to existing methods for the same problem.
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The generalized additive Runge-Kutta (GARK) framework provides a powerful approach for solving additively partitioned ordinary differential equations. This work combines the ideas of symplectic GARK schemes and multirate GARK schemes to efficiently solve additively partitioned Hamiltonian systems with multiple time scales. Order conditions, as well as conditions for symplecticity and time-reversibility, are derived in the general setting of non-separable Hamiltonian systems. Investigations of the special case of separable Hamiltonian systems are also carried out. We show that particular partitions may introduce stability issues, and discuss partitions that enable an implicit-explicit integration leading to improved stability properties. Higher-order symplectic multirate GARK schemes based on advanced composition techniques are discussed. The performance of the schemes is demonstrated by means of the Fermi-Pasta-Ulam problem.
Representing a polygon using a set of simple shapes has numerous applications in different use-case scenarios. We consider the problem of covering the interior of a rectilinear polygon with holes by a set of area-weighted, axis-aligned rectangles such that the total weight of the rectangles in the cover is minimized. Already the unit-weight case is known to be NP-hard and the general problem has, to the best of our knowledge, not been studied experimentally before. We show a new basic property of optimal solutions of the weighted problem. This allows us to speed up existing algorithms for the unit-weight case, obtain an improved ILP formulation for both the weighted and unweighted problem, and develop several approximation algorithms and heuristics for the weighted case. All our algorithms are evaluated in a large experimental study on 186 837 polygons combined with six cost functions, which provides evidence that our algorithms are both fast and yield close-to-optimal solutions in practice.
Inverse problems, which are related to Maxwell's equations, in the presence of nonlinear materials is a quite new topic in the literature. The lack of contributions in this area can be ascribed to the significant challenges that such problems pose. Retrieving the spatial behaviour of some unknown physical property, from boundary measurements, is a nonlinear and highly ill-posed problem even in the presence of linear materials. Furthermore, this complexity grows exponentially in the presence of nonlinear materials. In the tomography of linear materials, the Monotonicity Principle (MP) is the foundation of a class of non-iterative algorithms able to guarantee excellent performances and compatibility with real-time applications. Recently, the MP has been extended to nonlinear materials under very general assumptions. Starting from the theoretical background for this extension, we develop a first real-time inversion method for the inverse obstacle problem in the presence of nonlinear materials. The proposed method is intendend for all problems governed by the quasilinear Laplace equation, i.e. static problems involving nonlinear materials. In this paper, we provide some preliminary results which give the foundation of our method and some extended numerical examples.
Deep learning models are known to suffer from the problem of bias, and researchers have been exploring methods to address this issue. However, most of these methods require prior knowledge of the bias and are not always practical. In this paper, we focus on a more practical setting with no prior information about the bias. Generally, in this setting, there are a large number of bias-aligned samples that cause the model to produce biased predictions and a few bias-conflicting samples that do not conform to the bias. If the training data is limited, the influence of the bias-aligned samples may become even stronger on the model predictions, and we experimentally demonstrate that existing debiasing techniques suffer severely in such cases. In this paper, we examine the effects of unknown bias in small dataset regimes and present a novel approach to mitigate this issue. The proposed approach directly addresses the issue of the extremely low occurrence of bias-conflicting samples in limited data settings through the synthesis of hybrid samples that can be used to reduce the effect of bias. We perform extensive experiments on several benchmark datasets and experimentally demonstrate the effectiveness of our proposed approach in addressing any unknown bias in the presence of limited data. Specifically, our approach outperforms the vanilla, LfF, LDD, and DebiAN debiasing methods by absolute margins of 10.39%, 9.08%, 8.07%, and 9.67% when only 10% of the Corrupted CIFAR-10 Type 1 dataset is available with a bias-conflicting sample ratio of 0.05.
Robust Markov decision processes (MDPs) are used for applications of dynamic optimization in uncertain environments and have been studied extensively. Many of the main properties and algorithms of MDPs, such as value iteration and policy iteration, extend directly to RMDPs. Surprisingly, there is no known analog of the MDP convex optimization formulation for solving RMDPs. This work describes the first convex optimization formulation of RMDPs under the classical sa-rectangularity and s-rectangularity assumptions. By using entropic regularization and exponential change of variables, we derive a convex formulation with a number of variables and constraints polynomial in the number of states and actions, but with large coefficients in the constraints. We further simplify the formulation for RMDPs with polyhedral, ellipsoidal, or entropy-based uncertainty sets, showing that, in these cases, RMDPs can be reformulated as conic programs based on exponential cones, quadratic cones, and non-negative orthants. Our work opens a new research direction for RMDPs and can serve as a first step toward obtaining a tractable convex formulation of RMDPs.
Generalized Additive Runge-Kutta schemes have shown to be a suitable tool for solving ordinary differential equations with additively partitioned right-hand sides. This work develops symplectic GARK schemes for additively partitioned Hamiltonian systems. In a general setting, we derive conditions for symplecticness, as well as symmetry and time-reversibility. We show how symplectic and symmetric schemes can be constructed based on schemes which are only symplectic, or only symmetric. Special attention is given to the special case of partitioned schemes for Hamiltonians split into multiple potential and kinetic energies. Finally we show how symplectic GARK schemes can leverage different time scales and evaluation costs for different potentials, and provide efficient numerical solutions by using different order for these parts.
Video moment retrieval is a challenging task requiring fine-grained interactions between video and text modalities. Recent work in image-text pretraining has demonstrated that most existing pretrained models suffer from information asymmetry due to the difference in length between visual and textual sequences. We question whether the same problem also exists in the video-text domain with an auxiliary need to preserve both spatial and temporal information. Thus, we evaluate a recently proposed solution involving the addition of an asymmetric co-attention network for video grounding tasks. Additionally, we incorporate momentum contrastive loss for robust, discriminative representation learning in both modalities. We note that the integration of these supplementary modules yields better performance compared to state-of-the-art models on the TACoS dataset and comparable results on ActivityNet Captions, all while utilizing significantly fewer parameters with respect to baseline.
We design a deterministic particle method for the solution of the spatially homogeneous Landau equation with uncertainty. The deterministic particle approximation is based on the reformulation of the Landau equation as a formal gradient flow on the set of probability measures, whereas the propagation of uncertain quantities is computed by means of a sg representation of each particle. This approach guarantees spectral accuracy in uncertainty space while preserving the fundamental structural properties of the model: the positivity of the solution, the conservation of invariant quantities, and the entropy production. We provide a regularity results for the particle method in the random space. We perform the numerical validation of the particle method in a wealth of test cases.
Neural operators have been validated as promising deep surrogate models for solving partial differential equations (PDEs). Despite the critical role of boundary conditions in PDEs, however, only a limited number of neural operators robustly enforce these conditions. In this paper we introduce semi-periodic Fourier neural operator (SPFNO), a novel spectral operator learning method, to learn the target operators of PDEs with non-periodic BCs. This method extends our previous work (arXiv:2206.12698), which showed significant improvements by employing enhanced neural operators that precisely satisfy the boundary conditions. However, the previous work is associated with Gaussian grids, restricting comprehensive comparisons across most public datasets. Additionally, we present numerical results for various PDEs such as the viscous Burgers' equation, Darcy flow, incompressible pipe flow, and coupled reactiondiffusion equations. These results demonstrate the computational efficiency, resolution invariant property, and BC-satisfaction behavior of proposed model. An accuracy improvement of approximately 1.7X-4.7X over the non-BC-satisfying baselines is also achieved. Furthermore, our studies on SOL underscore the significance of satisfying BCs as a criterion for deep surrogate models of PDEs.
In survival analysis, complex machine learning algorithms have been increasingly used for predictive modeling. Given a collection of features available for inclusion in a predictive model, it may be of interest to quantify the relative importance of a subset of features for the prediction task at hand. In particular, in HIV vaccine trials, participant baseline characteristics are used to predict the probability of infection over the intended follow-up period, and investigators may wish to understand how much certain types of predictors, such as behavioral factors, contribute toward overall predictiveness. Time-to-event outcomes such as time to infection are often subject to right censoring, and existing methods for assessing variable importance are typically not intended to be used in this setting. We describe a broad class of algorithm-agnostic variable importance measures for prediction in the context of survival data. We propose a nonparametric efficient estimation procedure that incorporates flexible learning of nuisance parameters, yields asymptotically valid inference, and enjoys double-robustness. We assess the performance of our proposed procedure via numerical simulations and analyze data from the HVTN 702 study to inform enrollment strategies for future HIV vaccine trials.
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