Search Results (14)

We consider the problem of Bayesian parameter inference in the Merton structural credit risk model, where the posterior is induced by a jump-diffusion likelihood and the marginal evidence is not available in closed form. To approximate this posterior, we construct a variational family based on triangular sum-of-squares (SOS) polynomial flows, in which each component map is monotone by construction: its diagonal derivative is a positive definite quadratic form on a monomial basis, yielding a closed-form log-Jacobian and explicit gradients with respect to all flow parameters. The symmetric positive definite matrices parametrizing the flow are optimized by intrinsic Riemannian gradient ascent on the positive definite cone equipped with the affine-invariant metric, which preserves feasibility at every iterate without projection. We show that the rank-one Jacobian gradients produced by the SOS structure have unit norm in the affine-invariant metric, establishing a direct algebraic coupling between the transport family and the optimization geometry and implying a universal 1-Lipschitz bound for the log-Jacobian along geodesics. On the likelihood side, we derive exact score identities for all five structural parameters of the Merton model—drift, volatility, jump intensity, jump mean, and jump volatility—through both the Poisson log-normal mixture and the Fourier inversion representations. Strictly positive parameters are handled via exponential reparametrization, and the resulting gradients propagate end-to-end through the flow. We establish uniform truncation bounds on compact parameter sets for the infinite mixture and its associated score series, providing rigorous control over the finite approximations used in practice. The base distribution is chosen to be uniform on ${[0,1]}^5$, whose bounded support ensures uniform control of the monomial basis and stabilizes the polynomial calculus. These ingredients are assembled into a fully explicit modified ELBO with implementable gradients, combining Euclidean updates for vector parameters and intrinsic manifold updates for matrix parameters. Full article

Background: The decoding of motor imagery electroencephalography (MI-EEG) is constrained by core issues including low signal-to-noise ratio (SNR) and cross-session as well as cross-subject domain shift, which seriously impedes the practical deployment of brain–computer interfaces (BCIs). Methods: To address these challenges, this paper proposes a novel end-to-end MI-EEG decoding method named BARN-DA. Two innovative modules, Band-Aware Channel Attention (BACA) and Multi-Scale Kernel Perception (MSKP), are designed: one enhances discriminative channel features by modeling channel information fused with frequency band feature representation, and the other captures complex data correlations via multi-scale parallel convolutions to improve the discriminability of the network’s feature extraction. Subsequently, the features are mapped onto the Riemannian manifold. For the source and target domain features residing on this manifold, a Riemannian Maximum Mean Discrepancy (R-MMD) loss is designed based on the log-Euclidean metric. This approach enables the effective embedding of Symmetric Positive Definite (SPD) matrices into the Reproducing Kernel Hilbert Space (RKHS), thereby reducing cross-domain discrepancies. Results: Experimental results on four public datasets demonstrate that the BARN-DA method achieves average cross-session classification accuracies of 84.65% ± 8.97% (BCIC IV 2a), 89.19% ± 7.69% (BCIC IV 2b), and 61.76% ± 12.68% (SHU), as well as average cross-subject classification accuracies of 65.49% ± 11.64% (BCIC IV 2a), 78.78% ± 8.44% (BCIC IV 2b), and 78.14% ± 14.41% (BCIC III 4a). Compared with state-of-the-art methods, BARN-DA obtains higher classification accuracy and stronger cross-session and cross-subject generalization ability. Conclusions: These results confirm that BARN-DA effectively alleviates low SNR and domain shift problems in MI-EEG decoding, providing an efficient technical solution for practical BCI systems. Full article

Plateau–orogenic belts host a substantial share of global gold resources, yet quantitative prospectivity mapping is challenged by complex mineralization and strongly heterogeneous, multi-scale datasets. Using the Mayoumu area (Tibet) as a representative orogenic gold district, we develop an integrated multi-source workflow that fuses remote-sensing alteration information with regional geochemical and structural constraints within an ensemble-learning framework. Alteration anomalies were mapped from GF-5 hyperspectral imagery using mixture-tuned matched filtering (MTMF) and from Sentinel-2 multispectral imagery using the iCrosta method to extend alteration signals across scales. Geochemical anomalies were extracted from 1:200,000 stream-sediment data through isometric log-ratio (ILR) transformation and robust principal component analysis (RPCA). At the same time, ore-controlling structures were quantified using Euclidean-distance-to-fault layers. Three Boosting-based ensemble models—gradient boosting decision tree (GBDT), extreme gradient boosting (XGBoost), and light gradient boosting machine (LightGBM)—were trained to predict mineral prospectivity. Performance was evaluated using confusion matrix metrics and ROC–AUC, and key predictors were interpreted using SHAP. All three models achieved AUC values > 0.90, with LightGBM performing best (AUC = 0.94) and delineating high-prospectivity zones that coincide with known occurrences and highlight additional targets. The proposed workflow provides a practical, transferable reference for gold prospectivity mapping in complex orogenic belts worldwide. Full article

Sleep stage classification is crucial for diagnosing Obstructive Sleep Apnea (OSA). OSA patients’ sleep electroencephalography (EEG) signals often exhibit frequent oscillations due to abnormal apnea. Additionally, EEG signals are weak and nonlinear; it is more suitable to analyze EEG signals in the nonlinear space. Hence, we proposed a novel cross-subject EEG-based Sleep Stage Classification (EEGSSC) method for OSA patients in Riemannian manifold space. Firstly, each sleep EEG instance was converted into a sequence of symmetric positive definite matrices by calculating the multichannel covariance. Next, a domain similarity detection technique is introduced to select similar patients in the manifold space. Centroid alignment is then applied to minimize differences in marginal probability distributions between patients by aligning the Riemannian means of their covariance matrices. To extract the comprehensive features of the sleep EEG signals on the manifold, we not only used a transported square-root vector field to capture dynamic features but also computed static features by the log-Euclidean Riemannian metric. A multi-layer perceptron classifier is then used for classification. The proposed method has been tested on ISRUC and Dreem datasets, and the results demonstrate that EEGSSC can serve as an effective tool for automated sleep stage classification in OSA patients. Full article

This study focuses on a hydrocarbon reservoir located in southeastern Mexico. The analysis uses well log data derived from petrophysical evaluations of storage capacity. The structural complexity of the reservoir and observed heterogeneity in Cretaceous units motivate a fractal-based characterization of spatial fluctuations. The objective is to assess the fractal scaling of storage capacity fluctuations using the dynamic Family–Vicsek framework. Critical exponents $\alpha$ (roughness), $\beta$ (growth), and z (dynamic) are obtained through structure function metrics. Data collapse techniques and local Hurst exponent distributions are used to explore long-range memory and spatial heterogeneity across wells. This study aims to classify storage capacity fluctuation records based on Euclidean or fractal geometries. This analysis allows a novel characterization of storage trends in the reservoir. The analysis reveals persistent scaling behavior, indicating long-range correlations in the storage capacity fluctuations. Multiscale patterns and variations in local Hurst exponents highlight the presence of multifractality and regional heterogeneity. Specifically, the spatial distribution of local Hurst exponents obtained in this study enables the inference of statistical properties in synthetic wells, providing key input for the structural and functional characterization of the reservoir’s geological model. This approach aims to identify preferential subsurface flow pathways for hydrocarbons and gas. Full article

Aiming at the nonlinear radiometric differences between multi-source sensor images and coherent spot noise and other factors that lead to alignment difficulties, the registration method of gradient weakly sensitive multi-source sensor images is proposed, which does not need to extract the image gradient in the whole process and has rotational invariance. In the feature point detection stage, the maximum moment map is obtained by using the phase consistency transform to replace the gradient edge map for chunked Harris feature point detection, thus increasing the number of repeated feature points in the heterogeneous image. To have rotational invariance of the subsequent descriptors, a method to determine the main phase angle is proposed. The phase angle of the region near the feature point is counted, and the parabolic interpolation method is used to estimate the more accurate main phase angle under the determined interval. In the feature description stage, the Log-Gabor convolution sequence is used to construct the index map with the maximum phase amplitude, the heterogeneous image is converted to an isomorphic image, and the isomorphic image of the region around the feature point is rotated by using the main phase angle, which is in turn used to construct the feature vector with the feature point as the center by the quadratic interpolation method. In the feature matching stage, feature matching is performed by using the sum of squares of Euclidean distances as a similarity metric. Finally, after qualitative and quantitative experiments of six groups of five pairs of different multi-source sensor image alignment correct matching rates, root mean square errors, and the number of correctly matched points statistics, this algorithm is verified to have the advantage of robust accuracy compared with the current algorithms. Full article

This study utilized a max-stable process (MSP) model with a dependence structure defined via a non-Euclidean distance metric, with the goal of modelling extreme flood data on a river network. The dataset was composed of mean daily discharge observations from 22 United States Geological Survey streamflow gaging stations for river basins in Missouri and Arkansas. The analysis included the application of the elastic-net penalty to automatically build spatially varying trend surfaces to model the marginal distributions. The dependence model accounted for the river distance between hydrologically connected gaging sites and the hydrologic distance, defined as the Euclidean distance between the centers of site’s associated drainage areas, for all stations. Modelling the marginal distributions and spatial dependence among the extremes are two key components for spatially modelling extremes. Among the 16 covariates evaluated for marginal fitting, 7 were selected to spatially model the generalized extreme value (GEV) location parameter (for each gaging station’s contributing drainage basin, its outlet elevation, centroid x coordinate, centroid elevation, area, average basin width, elevation range, and median land surface slope). The three covariates selected for the GEV scale parameter included the area, average basin width, and median land surface slope. The GEV shape parameter was assumed to be constant throughout the entire study area. Comparisons of estimates obtained from the spatial covariate model with their corresponding “at-site” estimates resulted in computed values of 0.95, 0.95, 0.94 and 0.85, 0.84, 0.90 for the coefficient of determination, Nash–Sutcliffe efficiency, and Kling–Gupta efficiency for the GEV location and scale parameters, respectively. Brown–Resnick MSP models were fit to independent multivariate events extracted from a set of common discharge data, transformed to unit Fréchet margins while considering different permutations of the non-Euclidean dependence model. Each of the fitted model’s log-likelihood values indicated improved fits when using hydrologic distance rather than Euclidean distance. They also demonstrated that accounting for flow-connected dependence and anisotropy further improved model fit. In this study, the results from both parts were illustrative; however, further research with larger datasets and more heterogeneous systems is recommended. Full article

Background: Recording the calibration data of a brain–computer interface is a laborious process and is an unpleasant experience for the subjects. Domain adaptation is an effective technology to remedy the shortage of target data by leveraging rich labeled data from the sources. However, most prior methods have needed to extract the features of the EEG signal first, which triggers another challenge in BCI classification, due to small sample sets or a lack of labels for the target. Methods: In this paper, we propose a novel domain adaptation framework, referred to as kernel-based Riemannian manifold domain adaptation (KMDA). KMDA circumvents the tedious feature extraction process by analyzing the covariance matrices of electroencephalogram (EEG) signals. Covariance matrices define a symmetric positive definite space (SPD) that can be described by Riemannian metrics. In KMDA, the covariance matrices are aligned in the Riemannian manifold, and then are mapped to a high dimensional space by a log-Euclidean metric Gaussian kernel, where subspace learning is performed by minimizing the conditional distribution distance between the sources and the target while preserving the target discriminative information. We also present an approach to convert the EEG trials into 2D frames (E-frames) to further lower the dimension of covariance descriptors. Results: Experiments on three EEG datasets demonstrated that KMDA outperforms several state-of-the-art domain adaptation methods in classification accuracy, with an average Kappa of 0.56 for BCI competition IV dataset IIa, 0.75 for BCI competition IV dataset IIIa, and an average accuracy of 81.56% for BCI competition III dataset IVa. Additionally, the overall accuracy was further improved by 5.28% with the E-frames. KMDA showed potential in addressing subject dependence and shortening the calibration time of motor imagery-based brain–computer interfaces. Full article

A non-iterative method for the difference of means is presented to calculate the log-Euclidean distance between a symmetric positive-definite matrix and the mean matrix on the Lie group of symmetric positive-definite matrices. Although affine-invariant Riemannian metrics have a perfect theoretical framework and avoid the drawbacks of the Euclidean inner product, their complex formulas also lead to sophisticated and time-consuming algorithms. To make up for this limitation, log-Euclidean metrics with simpler formulas and faster calculations are employed in this manuscript. Our new approach is to transform a symmetric positive-definite matrix into a symmetric matrix via logarithmic maps, and then to transform the results back to the Lie group through exponential maps. Moreover, the present method does not need to compute the mean matrix and retains the usual Euclidean operations in the domain of matrix logarithms. In addition, for some randomly generated positive-definite matrices, the method is compared using experiments with that induced by the classical affine-invariant Riemannian metric. Finally, our proposed method is applied to denoise the point clouds with high density noise via the K-means clustering algorithm. Full article

Symmetric positive definite (SPD) data have become a hot topic in machine learning. Instead of a linear Euclidean space, SPD data generally lie on a nonlinear Riemannian manifold. To get over the problems caused by the high data dimensionality, dimensionality reduction (DR) is a key subject for SPD data, where bilinear transformation plays a vital role. Because linear operations are not supported in nonlinear spaces such as Riemannian manifolds, directly performing Euclidean DR methods on SPD matrices is inadequate and difficult in complex models and optimization. An SPD data DR method based on Riemannian manifold tangent spaces and global isometry (RMTSISOM-SPDDR) is proposed in this research. The main contributions are listed: (1) Any Riemannian manifold tangent space is a Hilbert space isomorphic to a Euclidean space. Particularly for SPD manifolds, tangent spaces consist of symmetric matrices, which can greatly preserve the form and attributes of original SPD data. For this reason, RMTSISOM-SPDDR transfers the bilinear transformation from manifolds to tangent spaces. (2) By log transformation, original SPD data are mapped to the tangent space at the identity matrix under the affine invariant Riemannian metric (AIRM). In this way, the geodesic distance between original data and the identity matrix is equal to the Euclidean distance between corresponding tangent vector and the origin. (3) The bilinear transformation is further determined by the isometric criterion guaranteeing the geodesic distance on high-dimensional SPD manifold as close as possible to the Euclidean distance in the tangent space of low-dimensional SPD manifold. Then, we use it for the DR of original SPD data. Experiments on five commonly used datasets show that RMTSISOM-SPDDR is superior to five advanced SPD data DR algorithms. Full article

A diffusion tensor models the covariance of the Brownian motion of water at a voxel and is required to be symmetric and positive semi-definite. Therefore, image processing approaches, designed for linear entities, are not effective for diffusion tensor data manipulation, and the existence of artefacts in diffusion tensor imaging acquisition makes diffusion tensor data segmentation even more challenging. In this study, we develop a spatial fuzzy c-means clustering method for diffusion tensor data that effectively segments diffusion tensor images by accounting for the noise, partial voluming, magnetic field inhomogeneity, and other imaging artefacts. To retain the symmetry and positive semi-definiteness of diffusion tensors, the log and root Euclidean metrics are used to estimate the mean diffusion tensor for each cluster. The method exploits spatial contextual information and provides uncertainty information in segmentation decisions by calculating the membership values for assigning a diffusion tensor at one voxel to different clusters. A regularisation model that allows the user to integrate their prior knowledge into the segmentation scheme or to highlight and segment local structures is also proposed. Experiments on simulated images and real brain datasets from healthy and Spinocerebellar ataxia 2 subjects showed that the new method was more effective than conventional segmentation methods. Full article

We investigated the energy of N points on an infinite compact metric space $(A,d)$ of a diameter less than 1 that interact through the potential $(1/d^s){(log1/d)}^t,$ where $s,t\ge 0$ and d is the metric distance. With ${\mathcal{E}}_{{log}^t}^s(A,N)$ denoting the minimal energy for such N-point configurations, we studied certain continuity and differentiability properties of ${\mathcal{E}}_{{log}^t}^s(A,N)$ in the variable $s.$ Then, we showed that in the limits, as $s\to \infty$ and as $s\to s_0>0,$N-point configurations that minimize the $s,{log}^t$-energy tends to an N-point best-packing configuration and an N-point configuration that minimizes the $s_0,{log}^t$-energy, respectively. Furthermore, we considered when A are circles in the Euclidean space ${\mathbb{R}}^2.$ In particular, we proved the minimality of N distinct equally spaced points on circles in ${\mathbb{R}}^2$ for some certain s and t. The study on circles shows a possibility for the utilization of N points generated through such new potential to uniformly discretize on objects with very high symmetry. Full article

Remote sensing image scene classification, which consists of labeling remote sensing images with a set of categories based on their content, has received remarkable attention for many applications such as land use mapping. Standard approaches are based on the multi-layer representation of first-order convolutional neural network (CNN) features. However, second-order CNNs have recently been shown to outperform traditional first-order CNNs for many computer vision tasks. Hence, the aim of this paper is to show the use of second-order statistics of CNN features for remote sensing scene classification. This takes the form of covariance matrices computed locally or globally on the output of a CNN. However, these datapoints do not lie in an Euclidean space but a Riemannian manifold. To manipulate them, Euclidean tools are not adapted. Other metrics should be considered such as the log-Euclidean one. This consists of projecting the set of covariance matrices on a tangent space defined at a reference point. In this tangent plane, which is a vector space, conventional machine learning algorithms can be considered, such as the Fisher vector encoding or SVM classifier. Based on this log-Euclidean framework, we propose a novel transfer learning approach composed of two hybrid architectures based on covariance pooling of CNN features, the first is local and the second is global. They rely on the extraction of features from models pre-trained on the ImageNet dataset processed with some machine learning algorithms. The first hybrid architecture consists of an ensemble learning approach with the log-Euclidean Fisher vector encoding of region covariance matrices computed locally on the first layers of a CNN. The second one concerns an ensemble learning approach based on the covariance pooling of CNN features extracted globally from the deepest layers. These two ensemble learning approaches are then combined together based on the strategy of the most diverse ensembles. For validation and comparison purposes, the proposed approach is tested on various challenging remote sensing datasets. Experimental results exhibit a significant gain of approximately $2\%$ in overall accuracy for the proposed approach compared to a similar state-of-the-art method based on covariance pooling of CNN features (on the UC Merced dataset). Full article

This paper presents an overview of coding methods used to encode a set of covariance matrices. Starting from a Gaussian mixture model (GMM) adapted to the Log-Euclidean (LE) or affine invariant Riemannian metric, we propose a Fisher Vector (FV) descriptor adapted to each of these metrics: the Log-Euclidean Fisher Vectors (LE FV) and the Riemannian Fisher Vectors (RFV). Some experiments on texture and head pose image classification are conducted to compare these two metrics and to illustrate the potential of these FV-based descriptors compared to state-of-the-art BoW and VLAD-based descriptors. A focus is also applied to illustrate the advantage of using the Fisher information matrix during the derivation of the FV. In addition, finally, some experiments are conducted in order to provide fairer comparison between the different coding strategies. This includes some comparisons between anisotropic and isotropic models, and a estimation performance analysis of the GMM dispersion parameter for covariance matrices of large dimension. Full article