When generalizing schemes for real-valued data approximation or decomposition to data living in Riemannian manifolds, tangent space-based schemes are very attractive for the simple reason that these spaces are linear. An open challenge is to do this in such a way that the generalized scheme is applicable to general Riemannian manifolds, is global-geometry aware and is computationally feasible. Existing schemes have been unable to account for all three of these key factors at the same time. In this work, we take a systematic approach to developing a framework that is able to account for all three factors. First, we will restrict ourselves to the -- still general -- class of symmetric Riemannian manifolds and show how curvature affects general manifold-valued tensor approximation schemes. Next, we show how the latter observations can be used in a general strategy for developing approximation schemes that are also global-geometry aware. Finally, having general applicability and global-geometry awareness taken into account we restrict ourselves once more in a case study on low-rank approximation. Here we show how computational feasibility can be achieved and propose the curvature-corrected truncated higher-order singular value decomposition (CC-tHOSVD), whose performance is subsequently tested in numerical experiments with both synthetic and real data living in symmetric Riemannian manifolds with both positive and negative curvature.
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Feature selection is popular for obtaining small, interpretable, yet highly accurate prediction models. Conventional feature-selection methods typically yield one feature set only, which might not suffice in some scenarios. For example, users might be interested in finding alternative feature sets with similar prediction quality, offering different explanations of the data. In this article, we introduce alternative feature selection and formalize it as an optimization problem. In particular, we define alternatives via constraints and enable users to control the number and dissimilarity of alternatives. Next, we analyze the complexity of this optimization problem and show NP-hardness. Further, we discuss how to integrate conventional feature-selection methods as objectives. Finally, we evaluate alternative feature selection with 30 classification datasets. We observe that alternative feature sets may indeed have high prediction quality, and we analyze several factors influencing this outcome.
We study the Landau-de Gennes Q-tensor model of liquid crystals subjected to an electric field and develop a fully discrete numerical scheme for its solution. The scheme uses a convex splitting of the bulk potential, and we introduce a truncation operator for the Q-tensors to ensure well-posedness of the problem. We prove the stability and well-posedness of the scheme. Finally, making a restriction on the admissible parameters of the scheme, we show that up to a subsequence, solutions to the fully discrete scheme converge to weak solutions of the Q-tensor model as the time step and mesh are refined. We then present numerical results computed by the numerical scheme, among which, we show that it is possible to simulate the Fr\'eedericksz transition with this scheme.
High-dimensional data arises in numerous applications, and the rapidly developing field of geometric deep learning seeks to develop neural network architectures to analyze such data in non-Euclidean domains, such as graphs and manifolds. Recent work by Z. Wang, L. Ruiz, and A. Ribeiro has introduced a method for constructing manifold neural networks using the spectral decomposition of the Laplace Beltrami operator. Moreover, in this work, the authors provide a numerical scheme for implementing such neural networks when the manifold is unknown and one only has access to finitely many sample points. The authors show that this scheme, which relies upon building a data-driven graph, converges to the continuum limit as the number of sample points tends to infinity. Here, we build upon this result by establishing a rate of convergence that depends on the intrinsic dimension of the manifold but is independent of the ambient dimension. We also discuss how the rate of convergence depends on the depth of the network and the number of filters used in each layer.
Having efficient testing strategies is a core challenge that needs to be overcome for the release of automated driving. This necessitates clear requirements as well as suitable methods for testing. In this work, the requirements for perception modules are considered with respect to relevance. The concept of relevance currently remains insufficiently defined and specified. In this paper, we propose a novel methodology to overcome this challenge by exemplary application to collision safety in the highway domain. Using this general system and use case specification, a corresponding concept for relevance is derived. Irrelevant objects are thus defined as objects which do not limit the set of safe actions available to the ego vehicle under consideration of all uncertainties. As an initial step, the use case is decomposed into functional scenarios with respect to collision relevance. For each functional scenario, possible actions of both the ego vehicle and any other dynamic object are formalized as equations. This set of possible actions is constrained by traffic rules, yielding relevance criteria. As a result, we present a conservative estimation which dynamic objects are relevant for perception and need to be considered for a complete evaluation. The estimation provides requirements which are applicable for offline testing and validation of perception components. A visualization is presented for examples from the highD dataset, showing the plausibility of the results. Finally, a possibility for a future validation of the presented relevance concept is outlined.
A key challenge in Bayesian decentralized data fusion is the `rumor propagation' or `double counting' phenomenon, where previously sent data circulates back to its sender. It is often addressed by approximate methods like covariance intersection (CI) which takes a weighted average of the estimates to compute the bound. The problem is that this bound is not tight, i.e. the estimate is often over-conservative. In this paper, we show that by exploiting the probabilistic independence structure in multi-agent decentralized fusion problems a tighter bound can be found using (i) an expansion to the CI algorithm that uses multiple (non-monolithic) weighting factors instead of one (monolithic) factor in the original CI and (ii) a general optimization scheme that is able to compute optimal bounds and fully exploit an arbitrary dependency structure. We compare our methods and show that on a simple problem, they converge to the same solution. We then test our new non-monolithic CI algorithm on a large-scale target tracking simulation and show that it achieves a tighter bound and a more accurate estimate compared to the original monolithic CI.
Gaussian Partial Information Decomposition: Bias Correction and Application to High-dimensional Data
Recent advances in neuroscientific experimental techniques have enabled us to simultaneously record the activity of thousands of neurons across multiple brain regions. This has led to a growing need for computational tools capable of analyzing how task-relevant information is represented and communicated between several brain regions. Partial information decompositions (PIDs) have emerged as one such tool, quantifying how much unique, redundant and synergistic information two or more brain regions carry about a task-relevant message. However, computing PIDs is computationally challenging in practice, and statistical issues such as the bias and variance of estimates remain largely unexplored. In this paper, we propose a new method for efficiently computing and estimating a PID definition on multivariate Gaussian distributions. We show empirically that our method satisfies an intuitive additivity property, and recovers the ground truth in a battery of canonical examples, even at high dimensionality. We also propose and evaluate, for the first time, a method to correct the bias in PID estimates at finite sample sizes. Finally, we demonstrate that our Gaussian PID effectively characterizes inter-areal interactions in the mouse brain, revealing higher redundancy between visual areas when a stimulus is behaviorally relevant.
We consider the framework of penalized estimation where the penalty term is given by a real-valued polyhedral gauge, which encompasses methods such as LASSO (and many variants thereof such as the generalized LASSO), SLOPE, OSCAR, PACS and others. Each of these estimators can uncover a different structure or ``pattern'' of the unknown parameter vector. We define a general notion of patterns based on subdifferentials and formalize an approach to measure their complexity. For pattern recovery, we provide a minimal condition for a particular pattern to be detected by the procedure with positive probability, the so-called accessibility condition. Using our approach, we also introduce the stronger noiseless recovery condition. For the LASSO, it is well known that the irrepresentability condition is necessary for pattern recovery with probability larger than $1/2$ and we show that the noiseless recovery plays exactly the same role, thereby extending and unifying the irrepresentability condition of the LASSO to a broad class of penalized estimators. We show that the noiseless recovery condition can be relaxed when turning to thresholded penalized estimators, extending the idea of the thresholded LASSO: we prove that the accessibility condition is already sufficient (and necessary) for sure pattern recovery by thresholded penalized estimation provided that the signal of the pattern is large enough. Throughout the article, we demonstrate how our findings can be interpreted through a geometrical lens.
This paper is focused on the approximation of the Euler equations of compressible fluid dynamics on a staggered mesh. With this aim, the flow parameters are described by the velocity, the density and the internal energy. The thermodynamic quantities are described on the elements of the mesh, and thus the approximation is only in $L^2$, while the kinematic quantities are globally continuous. The method is general in the sense that the thermodynamic and kinetic parameters are described by an arbitrary degree of polynomials. In practice, the difference between the degrees of the kinematic parameters and the thermodynamic ones {is set} to $1$. The integration in time is done using the forward Euler method but can be extended straightforwardly to higher-order methods. In order to guarantee that the limit solution will be a weak solution of the problem, we introduce a general correction method in the spirit of the Lagrangian staggered method described in \cite{Svetlana,MR4059382, MR3023731}, and we prove a Lax Wendroff theorem. The proof is valid for multidimensional versions of the scheme, even though most of the numerical illustrations in this work, on classical benchmark problems, are one-dimensional because we have easy access to the exact solution for comparison. We conclude by explaining that the method is general and can be used in different settings, for example, Finite Volume, or discontinuous Galerkin method, not just the specific one presented in this paper.
Decision-making algorithms are being used in important decisions, such as who should be enrolled in health care programs and be hired. Even though these systems are currently deployed in high-stakes scenarios, many of them cannot explain their decisions. This limitation has prompted the Explainable Artificial Intelligence (XAI) initiative, which aims to make algorithms explainable to comply with legal requirements, promote trust, and maintain accountability. This paper questions whether and to what extent explainability can help solve the responsibility issues posed by autonomous AI systems. We suggest that XAI systems that provide post-hoc explanations could be seen as blameworthy agents, obscuring the responsibility of developers in the decision-making process. Furthermore, we argue that XAI could result in incorrect attributions of responsibility to vulnerable stakeholders, such as those who are subjected to algorithmic decisions (i.e., patients), due to a misguided perception that they have control over explainable algorithms. This conflict between explainability and accountability can be exacerbated if designers choose to use algorithms and patients as moral and legal scapegoats. We conclude with a set of recommendations for how to approach this tension in the socio-technical process of algorithmic decision-making and a defense of hard regulation to prevent designers from escaping responsibility.
Seeking the equivalent entities among multi-source Knowledge Graphs (KGs) is the pivotal step to KGs integration, also known as \emph{entity alignment} (EA). However, most existing EA methods are inefficient and poor in scalability. A recent summary points out that some of them even require several days to deal with a dataset containing 200,000 nodes (DWY100K). We believe over-complex graph encoder and inefficient negative sampling strategy are the two main reasons. In this paper, we propose a novel KG encoder -- Dual Attention Matching Network (Dual-AMN), which not only models both intra-graph and cross-graph information smartly, but also greatly reduces computational complexity. Furthermore, we propose the Normalized Hard Sample Mining Loss to smoothly select hard negative samples with reduced loss shift. The experimental results on widely used public datasets indicate that our method achieves both high accuracy and high efficiency. On DWY100K, the whole running process of our method could be finished in 1,100 seconds, at least 10* faster than previous work. The performances of our method also outperform previous works across all datasets, where Hits@1 and MRR have been improved from 6% to 13%.
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