Applications of the ensemble Kalman filter to high-dimensional problems are feasible only with small ensembles. This necessitates a kind of regularization of the analysis (observation update) problem. We propose a regularization technique based on a new non-stationary, non-parametric spatial model on the sphere. The model termed the Locally Stationary Convolution Model is a constrained version of the general Gaussian process convolution model. The constraints on the location-dependent convolution kernel include local isotropy, positive definiteness as a function of distance, and smoothness as a function of location. The model allows for a rigorous definition of the local spectrum, which is required to be a smooth function of spatial wavenumber. We propose and test an ensemble filter in which prior covariances are postulated to obey the Locally Stationary Convolution Model. The model is estimated online in a two-stage procedure. First, ensemble perturbations are bandpass filtered in several wavenumber bands to extract aggregated local spatial spectra. Second, a neural network recovers the local spectra from sample variances of the filtered fields. In simulation experiments, the new filter was capable of outperforming several existing techniques. With small to moderate ensemble sizes, the improvement was substantial.
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Deep neural networks have shown remarkable performance when trained on independent and identically distributed data from a fixed set of classes. However, in real-world scenarios, it can be desirable to train models on a continuous stream of data where multiple classification tasks are presented sequentially. This scenario, known as Continual Learning (CL) poses challenges to standard learning algorithms which struggle to maintain knowledge of old tasks while learning new ones. This stability-plasticity dilemma remains central to CL and multiple metrics have been proposed to adequately measure stability and plasticity separately. However, none considers the increasing difficulty of the classification task, which inherently results in performance loss for any model. In that sense, we analyze some limitations of current metrics and identify the presence of setup-induced forgetting. Therefore, we propose new metrics that account for the task's increasing difficulty. Through experiments on benchmark datasets, we demonstrate that our proposed metrics can provide new insights into the stability-plasticity trade-off achieved by models in the continual learning environment.
A significant limitation of one-class classification anomaly detection methods is their reliance on the assumption that unlabeled training data only contains normal instances. To overcome this impractical assumption, we propose two novel classification-based anomaly detection methods. Firstly, we introduce a semi-supervised shallow anomaly detection method based on an unbiased risk estimator. Secondly, we present a semi-supervised deep anomaly detection method utilizing a nonnegative (biased) risk estimator. We establish estimation error bounds and excess risk bounds for both risk minimizers. Additionally, we propose techniques to select appropriate regularization parameters that ensure the nonnegativity of the empirical risk in the shallow model under specific loss functions. Our extensive experiments provide strong evidence of the effectiveness of the risk-based anomaly detection methods.
In this paper, we propose a human trajectory prediction model that combines a Long Short-Term Memory (LSTM) network with an attention mechanism. To do that, we use attention scores to determine which parts of the input data the model should focus on when making predictions. Attention scores are calculated for each input feature, with a higher score indicating the greater significance of that feature in predicting the output. Initially, these scores are determined for the target human position, velocity, and their neighboring individual's positions and velocities. By using attention scores, our model can prioritize the most relevant information in the input data and make more accurate predictions. We extract attention scores from our attention mechanism and integrate them into the trajectory prediction module to predict human future trajectories. To achieve this, we introduce a new neural layer that processes attention scores after extracting them and concatenates them with positional information. We evaluate our approach on the publicly available ETH and UCY datasets and measure its performance using the final displacement error (FDE) and average displacement error (ADE) metrics. We show that our modified algorithm performs better than the Social LSTM in predicting the future trajectory of pedestrians in crowded spaces. Specifically, our model achieves an improvement of 6.2% in ADE and 6.3% in FDE compared to the Social LSTM results in the literature.
We introduce new control-volume finite-element discretization schemes suitable for solving the Stokes problem. Within a common framework, we present different approaches for constructing such schemes. The first and most established strategy employs a non-overlapping partitioning into control volumes. The second represents a new idea by splitting into two sets of control volumes, the first set yielding a partition of the domain and the second containing the remaining overlapping control volumes required for stability. The third represents a hybrid approach where finite volumes are combined with finite elements based on a hierarchical splitting of the ansatz space. All approaches are based on typical finite element function spaces but yield locally mass and momentum conservative discretization schemes that can be interpreted as finite volume schemes. We apply all strategies to the inf-sub stable MINI finite-element pair. Various test cases, including convergence tests and the numerical observation of the boundedness of the number of preconditioned Krylov solver iterations, as well as more complex scenarios of flow around obstacles or through a three-dimensional vessel bifurcation, demonstrate the stability and robustness of the schemes.
Textual geographic information is indispensable and heavily relied upon in practical applications. The absence of clear distribution poses challenges in effectively harnessing geographic information, thereby driving our quest for exploration. We contend that geographic information is influenced by human behavior, cognition, expression, and thought processes, and given our intuitive understanding of natural systems, we hypothesize its conformity to the Gamma distribution. Through rigorous experiments on a diverse range of 24 datasets encompassing different languages and types, we have substantiated this hypothesis, unearthing the underlying regularities governing the dimensions of quantity, length, and distance in geographic information. Furthermore, theoretical analyses and comparisons with Gaussian distributions and Zipf's law have refuted the contingency of these laws. Significantly, we have estimated the upper bounds of human utilization of geographic information, pointing towards the existence of uncharted territories. Also, we provide guidance in geographic information extraction. Hope we peer its true countenance uncovering the veil of geographic information.
For multivariate data with noise variables, tandem clustering is a well-known technique that aims to improve cluster identification by first reducing the dimension. However, the usual approach using principal component analysis (PCA) has been criticized for focusing only on inertia so that the first components do not necessarily retain the structure of interest for clustering. To overcome this drawback, a new tandem clustering approach based on invariant coordinate selection (ICS) is proposed. By jointly diagonalizing two scatter matrices, ICS is designed to find structure in the data while returning affine invariant components. Some theoretical results have already been derived and guarantee that under some elliptical mixture models, the group structure can be highlighted on a subset of the first and/or last components. Nevertheless, ICS has received little attention in a clustering context. Two challenges are the choice of the pair of scatter matrices and the selection of the components to retain. For clustering purposes, it is demonstrated that the best scatter pairs consist of one scatter matrix that captures the within-cluster structure and another that captures the global structure. For the former, local shape or pairwise scatters are of great interest, as is the minimum covariance determinant (MCD) estimator based on a carefully selected subset size that is smaller than usual. The performance of ICS as a dimension reduction method is evaluated in terms of preserving the cluster structure present in data. In an extensive simulation study and in empirical applications with benchmark data sets, different combinations of scatter matrices as well as component selection criteria are compared in situations with and without outliers. Overall, the new approach of tandem clustering with ICS shows promising results and clearly outperforms the approach with PCA.
We present a framework for approximate Bayesian inference when only a limited number of noisy log-likelihood evaluations can be obtained due to computational constraints, which is becoming increasingly common for applications of complex models. We model the log-likelihood function using a Gaussian process (GP) and the main methodological innovation is to apply this model to emulate the progression that an exact Metropolis-Hastings (MH) sampler would take if it was applicable. Informative log-likelihood evaluation locations are selected using a sequential experimental design strategy until the MH accept/reject decision is done accurately enough according to the GP model. The resulting approximate sampler is conceptually simple and sample-efficient. It is also more robust to violations of GP modelling assumptions compared with earlier, related "Bayesian optimisation-like" methods tailored for Bayesian inference. We discuss some theoretical aspects and various interpretations of the resulting approximate MH sampler, and demonstrate its benefits in the context of Bayesian and generalised Bayesian likelihood-free inference for simulator-based statistical models.
Normalizing flow is a class of deep generative models for efficient sampling and density estimation. In practice, the flow often appears as a chain of invertible neural network blocks; to facilitate training, existing works have regularized flow trajectories and designed special network architectures. The current paper develops a neural ODE flow network inspired by the Jordan-Kinderleherer-Otto (JKO) scheme, which allows efficient block-wise training of the residual blocks without sampling SDE trajectories or inner loops of score matching or variational learning. As the JKO scheme unfolds the dynamic of gradient flow, the proposed model naturally stacks residual network blocks one by one, reducing the memory load and difficulty in performing end-to-end deep flow network training. We also develop adaptive time reparameterization of the flow network with a progressive refinement of the trajectory in probability space, which improves the model training efficiency and accuracy in practice. Using numerical experiments with synthetic and real data, we show that the proposed JKO-iFlow model achieves similar or better performance in generating new samples compared with the existing flow and diffusion models at a significantly reduced computational and memory cost.
Model order reduction provides low-complexity high-fidelity surrogate models that allow rapid and accurate solutions of parametric differential equations. The development of reduced order models for parametric nonlinear Hamiltonian systems is still challenged by several factors: (i) the geometric structure encoding the physical properties of the dynamics; (ii) the slowly decaying Kolmogorov $n$-width of conservative dynamics; (iii) the gradient structure of the nonlinear flow velocity; (iv) high variations in the numerical rank of the state as a function of time and parameters. We propose to address these aspects via a structure-preserving adaptive approach that combines symplectic dynamical low-rank approximation with adaptive gradient-preserving hyper-reduction and parameters sampling. Additionally, we propose to vary in time the dimensions of both the reduced basis space and the hyper-reduction space by monitoring the quality of the reduced solution via an error indicator related to the projection error of the Hamiltonian vector field. The resulting adaptive hyper-reduced models preserve the geometric structure of the Hamiltonian flow, do not rely on prior information on the dynamics, and can be solved at a cost that is linear in the dimension of the full order model and linear in the number of test parameters. Numerical experiments demonstrate the improved performances of the resulting fully adaptive models compared to the original and reduced order models.
Explaining artificial intelligence (AI) predictions is increasingly important and even imperative in many high-stakes applications where humans are the ultimate decision-makers. In this work, we propose two novel architectures of self-interpretable image classifiers that first explain, and then predict (as opposed to post-hoc explanations) by harnessing the visual correspondences between a query image and exemplars. Our models consistently improve (by 1 to 4 points) on out-of-distribution (OOD) datasets while performing marginally worse (by 1 to 2 points) on in-distribution tests than ResNet-50 and a $k$-nearest neighbor classifier (kNN). Via a large-scale, human study on ImageNet and CUB, our correspondence-based explanations are found to be more useful to users than kNN explanations. Our explanations help users more accurately reject AI's wrong decisions than all other tested methods. Interestingly, for the first time, we show that it is possible to achieve complementary human-AI team accuracy (i.e., that is higher than either AI-alone or human-alone), in ImageNet and CUB image classification tasks.
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