A sequential solver for differential-algebraic equations with embedded optimization criteria (DAEOs) was developed to take advantage of the theoretical work done by Deussen et al. Solvers of this type separate the optimization problem from the differential equation and solve each individually. The new solver relies on the reduction of a DAEO to a sequence of differential inclusions separated by jump events. These jump events occur when the global solution to the optimization problem jumps to a new value. Without explicit treatment, these events will reduce the order of convergence of the integration step to one. The solver implements a "local optimizer tracking" procedure to detect and correct these jump events. Local optimizer tracking is much less expensive than running a deterministic global optimizer at every time step. This preserves the order of convergence of the integrator component without sacrificing performance to perform deterministic global optimization at every time step. The newly developed solver produces correct solutions to DAEOs and runs much faster than sequential DAEO solvers that rely only on global optimization.
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Tensor decomposition is a powerful tool for extracting physically meaningful latent factors from multi-dimensional nonnegative data, and has been an increasing interest in a variety of fields such as image processing, machine learning, and computer vision. In this paper, we propose a sparse nonnegative Tucker decomposition and completion method for the recovery of underlying nonnegative data under noisy observations. Here the underlying nonnegative data tensor is decomposed into a core tensor and several factor matrices with all entries being nonnegative and the factor matrices being sparse. The loss function is derived by the maximum likelihood estimation of the noisy observations, and the $\ell_0$ norm is employed to enhance the sparsity of the factor matrices. We establish the error bound of the estimator of the proposed model under generic noise scenarios, which is then specified to the observations with additive Gaussian noise, additive Laplace noise, and Poisson observations, respectively. Our theoretical results are better than those by existing tensor-based or matrix-based methods. Moreover, the minimax lower bounds are shown to be matched with the derived upper bounds up to logarithmic factors. Numerical examples on both synthetic and real-world data sets demonstrate the superiority of the proposed method for nonnegative tensor data completion.
We present a constructive universal approximation theorem for learning machines equipped with joint-group-equivariant feature maps, called the joint-equivariant machines, based on the group representation theory. "Constructive" here indicates that the distribution of parameters is given in a closed-form expression known as the ridgelet transform. Joint-group-equivariance encompasses a broad class of feature maps that generalize classical group-equivariance. Particularly, fully-connected networks are not group-equivariant but are joint-group-equivariant. Our main theorem also unifies the universal approximation theorems for both shallow and deep networks. Until this study, the universality of deep networks has been shown in a different manner from the universality of shallow networks, but our results discuss them on common ground. Now we can understand the approximation schemes of various learning machines in a unified manner. As applications, we show the constructive universal approximation properties of four examples: depth-$n$ joint-equivariant machine, depth-$n$ fully-connected network, depth-$n$ group-convolutional network, and a new depth-$2$ network with quadratic forms whose universality has not been known.
Bilevel optimization problems, which are problems where two optimization problems are nested, have more and more applications in machine learning. In many practical cases, the upper and the lower objectives correspond to empirical risk minimization problems and therefore have a sum structure. In this context, we propose a bilevel extension of the celebrated SARAH algorithm. We demonstrate that the algorithm requires $\mathcal{O}((n+m)^{\frac12}\varepsilon^{-1})$ oracle calls to achieve $\varepsilon$-stationarity with $n+m$ the total number of samples, which improves over all previous bilevel algorithms. Moreover, we provide a lower bound on the number of oracle calls required to get an approximate stationary point of the objective function of the bilevel problem. This lower bound is attained by our algorithm, making it optimal in terms of sample complexity.
A computed approximation of the solution operator to a system of partial differential equations (PDEs) is needed in various areas of science and engineering. Neural operators have been shown to be quite effective at predicting these solution generators after training on high-fidelity ground truth data (e.g. numerical simulations). However, in order to generalize well to unseen spatial domains, neural operators must be trained on an extensive amount of geometrically varying data samples that may not be feasible to acquire or simulate in certain contexts (e.g., patient-specific medical data, large-scale computationally intensive simulations.) We propose that in order to learn a PDE solution operator that can generalize across multiple domains without needing to sample enough data expressive enough for all possible geometries, we can train instead a latent neural operator on just a few ground truth solution fields diffeomorphically mapped from different geometric/spatial domains to a fixed reference configuration. Furthermore, the form of the solutions is dependent on the choice of mapping to and from the reference domain. We emphasize that preserving properties of the differential operator when constructing these mappings can significantly reduce the data requirement for achieving an accurate model due to the regularity of the solution fields that the latent neural operator is training on. We provide motivating numerical experimentation that demonstrates an extreme case of this consideration by exploiting the conformal invariance of the Laplacian
We provide a pipeline for calculating, managing and visualising correlations and other pairwise scores for numerical and categorical data. We present a uniform interface for calculating a plethora of pairwise scores and a new tidy data structure for managing the results. We also provide new visualisations which simultaneously show multiple and/or grouped pairwise scores. The visualisations are far richer than a traditional heatmap of correlation scores, as they help identify relationships with categorical variables, numeric variable pairs with non-linear associations or those which exhibit Simpson's paradox. These methods are available in our R package bullseye.
Multiport network theory (MNT) is a powerful analytical tool for modeling and optimizing complex systems based on circuit models. We present an overview of current research on the application of MNT to the development of electromagnetically consistent models for programmable metasurfaces, with focus on reconfigurable intelligent surfaces for wireless communications.
We present a simplified algorithm for solving the Negative-Weight Single-Source Shortest Paths (SSSP) problem, focusing on enhancing clarity and practicality over prior methods. Our algorithm uses graph diameter as a recursive parameter, offering greater robustness to the properties of the decomposed graph compared to earlier approaches. Additionally, we fully integrate negative-weight cycle finding into the algorithm by augmenting the Bellman-Ford/Dijkstra hybrid, eliminating the need for a separate cycle-finding procedure found in prior methods. Although the algorithm achieves no theoretical efficiency gains, it simplifies negative cycle finding and emphasizes design simplicity, making it more accessible for implementation and analysis. This work highlights the importance of robust parameterization and algorithmic simplicity in addressing the challenges of Negative-Weight SSSP.
Stiff systems of ordinary differential equations (ODEs) arise in a wide range of scientific and engineering disciplines and are traditionally solved using implicit integration methods due to their stability and efficiency. However, these methods are computationally expensive, particularly for applications requiring repeated integration, such as parameter estimation, Bayesian inference, neural ODEs, physics-informed neural networks, and MeshGraphNets. Explicit exponential integration methods have been proposed as a potential alternative, leveraging the matrix exponential to address stiffness without requiring nonlinear solvers. This study evaluates several state-of-the-art explicit single-step exponential schemes against classical implicit methods on benchmark stiff ODE problems, analyzing their accuracy, stability, and scalability with step size. Despite their initial appeal, our results reveal that explicit exponential methods significantly lag behind implicit schemes in accuracy and scalability for stiff ODEs. The backward Euler method consistently outperformed higher-order exponential methods in accuracy at small step sizes, with none surpassing the accuracy of the first-order integrating factor Euler method. Exponential methods fail to improve upon first-order accuracy, revealing the integrating factor Euler method as the only reliable choice for repeated, inexpensive integration in applications such as neural ODEs and parameter estimation. This study exposes the limitations of explicit exponential methods and calls for the development of improved algorithms.
The existence of representative datasets is a prerequisite of many successful artificial intelligence and machine learning models. However, the subsequent application of these models often involves scenarios that are inadequately represented in the data used for training. The reasons for this are manifold and range from time and cost constraints to ethical considerations. As a consequence, the reliable use of these models, especially in safety-critical applications, is a huge challenge. Leveraging additional, already existing sources of knowledge is key to overcome the limitations of purely data-driven approaches, and eventually to increase the generalization capability of these models. Furthermore, predictions that conform with knowledge are crucial for making trustworthy and safe decisions even in underrepresented scenarios. This work provides an overview of existing techniques and methods in the literature that combine data-based models with existing knowledge. The identified approaches are structured according to the categories integration, extraction and conformity. Special attention is given to applications in the field of autonomous driving.
Multi-relation Question Answering is a challenging task, due to the requirement of elaborated analysis on questions and reasoning over multiple fact triples in knowledge base. In this paper, we present a novel model called Interpretable Reasoning Network that employs an interpretable, hop-by-hop reasoning process for question answering. The model dynamically decides which part of an input question should be analyzed at each hop; predicts a relation that corresponds to the current parsed results; utilizes the predicted relation to update the question representation and the state of the reasoning process; and then drives the next-hop reasoning. Experiments show that our model yields state-of-the-art results on two datasets. More interestingly, the model can offer traceable and observable intermediate predictions for reasoning analysis and failure diagnosis.
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