Wasserstein Distance Research Articles

Adaptive dynamic programming and distributionally robust optimal control of linear stochastic system using the Wasserstein metric

SummaryIn this paper, we consider the optimal control of unknown stochastic dynamical system for both the finite‐horizon and infinite‐horizon cases. The objective of this paper is to find an optimal controller to minimize the expected value of a function which depends on the random disturbance. Throughout this paper, it is assumed that the mean vector and covariance matrix of the disturbance distribution is unknown. An uncertainty set in the space of mean vector and the covariance matrix is introduced. For the finite‐horizon case, we derive a closed‐form expression of the unique optimal policy and the opponents policy that generates the worst‐case distribution. For the infinite‐horizon case, we simplify the Riccati equation obtained in the finite‐hozion setting to an algebraic Riccati equation, which can guarantee the existence of the solution of the Riccati equation. It is shown that the resulting optimal policies obtained in these two cases can stabilize the expected value of the system state under the worst‐case distribution. Furthermore, the unknown system matrices can also be explicitly computed using the adaptive dynamic programming technique, which can help compute the optimal control policy by solving the algebraic Riccati equation. Finally, a simulation example is presented to demonstrate the effectiveness of our theoretical results.

Data-Driven Distributionally Robust Day-Ahead Dispatch for Active Distribution Networks Based on Improved Conditional Generative Adversarial Network

With the increasing penetration of renewable energy generation (REG), strong uncertainty poses a significant challenge to the rationality of day-ahead dispatch plans in distribution systems. We propose a data-driven, distributionally robust day-ahead dispatch method for active distribution networks (ADNs) based on an improved conditional generative adversarial network (CGAN). First, an improved CWGAN-GP model is constructed based on three-dimensional convolution (Conv3D) and the Wasserstein distance to obtain the day-ahead REG scenarios, considering the correlation features of photovoltaic (PV) and wind turbines (WTs), which reduces the conservatism of scenario generation. Accordingly, the Gaussian mixture model (GMM) clustering method based on moment information was used to construct an ambiguity set. Moreover, a two-stage distributionally robust day-ahead dispatch model for an ADN is established, which realizes the fusion of the data-driven method and distributionally robust optimization (DRO) mechanism model. Finally, the validity of the data-driven DRO (DDRO) method is verified through a case analysis.

Stability estimates for the Vlasov–Poisson system in p$p$‐kinetic Wasserstein distances

AbstractWe extend Loeper's ‐estimate (Theorem 2.9 in J. Math. Pures Appl. (9) 86 (2006), no. 1, 68–79) relating the force fields to the densities for the Vlasov–Poisson system to , with , based on the Helmholtz–Weyl decomposition. This allows us to generalize both the classical Loeper's 2‐Wasserstein stability estimate (Theorem 1.2 in J. Math. Pures Appl. (9) 86 (2006), no. 1, 68–79) and the recent stability estimate by the first author relying on the newly introduced kinetic Wasserstein distance (Theorem 3.1 in Arch Rational Mech. Anal. 244 (2022), no. 1, 27–50) to kinetic Wasserstein distances of order .

Study on multi-timescale operation of hydrogen-containing energy coupled with transportation system

Abstract With the widespread attention on hydrogen energy, the operation optimization of the coupling system of hydrogen energy and transportation has become a critical issue. Therefore, this study develops a hydrogen-containing energy transportation coupling system for the system collaborative operation framework, operation mode and equipment modelling of the system. Furthermore, a medium- to long-term operation optimization model and a short-term operation optimization model were constructed considering the differences in the operation of coupled systems at different timescales and operating costs as the objective function and power balance and system equipment as constraints. In the medium- and long-term operation optimization model, the planning scenario reduction method is used to reduce the wind power generation scenario. In the short-term operation optimization model, a multivariate uncertainty model is constructed to represent the uncertainty in the coupling system. Subsequently, the solution method of the model is proposed. Finally, a coupled system is simulated to verify the effectiveness of the model. (i) When the initial scene set is 600, the typical scene reduction method using Latin hypercube sampling and the Wasserstein distance can reduce operating costs by 7.60% and 9.49% compared with K-means reduction and K-media reduction methods. (ii) The sensitivity coefficients of hydrogen sales price, electricity sales price and maintenance rate to operating costs are –0.031%, –1.009% and 0.0105%, respectively. (iii) Considering multiple uncertainties can help make optimal decisions based on the overall consideration of disadvantage scenarios, thereby reducing system operating costs.

Image Enhancement Based on Dual-Branch Generative Adversarial Network Combining Spatial and Frequency Domain Information for Imbalanced Fault Diagnosis of Rolling Bearing

To address the problems of existing 2D image-based imbalanced fault diagnosis methods for rolling bearings, which generate images with inadequate texture details and color degradation, this paper proposes a novel image enhancement model based on a dual-branch generative adversarial network (GAN) combining spatial and frequency domain information for an imbalanced fault diagnosis of rolling bearing. Firstly, the original vibration signals are converted into 2D time–frequency (TF) images by a continuous wavelet transform, and a dual-branch GAN model with a symmetric structure is constructed. One branch utilizes an auxiliary classification GAN (ACGAN) to process the spatial information of the TF images, while the other employs a GAN with a frequency generator and a frequency discriminator to handle the frequency information of the input images after a fast Fourier transform. Then, a shuffle attention (SA) module based on an attention mechanism is integrated into the proposed model to improve the network’s expression ability and reduce the computational burden. Simultaneously, mean square error (MSE) is integrated into the loss functions of both generators to enhance the consistency of frequency information for the generated images. Additionally, a Wasserstein distance and gradient penalty are also incorporated into the losses of the two discriminators to prevent gradient vanishing and mode collapse. Under the supervision of the frequency WGAN-GP branch, an ACWGAN-GP can generate high-quality fault samples to balance the dataset. Finally, the balanced dataset is utilized to train the auxiliary classifier to achieve fault diagnosis. The effectiveness of the proposed method is validated by two rolling bearing datasets. When the imbalanced ratios of the four datasets are 0.5, 0.2, 0.1, and 0.05, respectively, their average classification accuracy reaches 99.35% on the CWRU bearing dataset. Meanwhile, the average classification accuracy reaches 96.62% on the MFS bearing dataset.

YOLOv7-SN: Underwater Target Detection Algorithm Based on Improved YOLOv7

Exploring the ocean’s resources requires finding underwater objects, which is a challenging task due to blurry images and small, densely packed targets. To improve the accuracy of underwater target detection, we propose an enhanced version of the YOLOv7 network called YOLOv7-SN. Our goal is to optimize the effectiveness and accuracy of underwater target detection by introducing a series of innovations. We incorporate the channel attention module SE into the network’s key part to improve the extraction of relevant features for underwater targets. We also introduce the RFE module with dilated convolution behind the backbone network to capture multi-scale information. Additionally, we use the Wasserstein distance as a new metric to replace the traditional loss function and address the challenge of small target detection. Finally, we employ probe heads carrying implicit knowledge to further enhance the model’s accuracy. These methods aim to optimize the efficacy of underwater target detection and improve its ability to deal with the complexity and challenges of underwater environments. We conducted experiments on the URPC2020, and RUIE datasets. The results show that the mean accuracy (mAP) is improved by 5.9% and 3.9%, respectively, compared to the baseline model.

Altered topological structure of the brain white matter in maltreated children through topological data analysis.

Childhood maltreatment may adversely affect brain development and consequently influence behavioral, emotional, and psychological patterns during adulthood. In this study, we propose an analytical pipeline for modeling the altered topological structure of brain white matter in maltreated and typically developing children. We perform topological data analysis (TDA) to assess the alteration in the global topology of the brain white matter structural covariance network among children. We use persistent homology, an algebraic technique in TDA, to analyze topological features in the brain covariance networks constructed from structural magnetic resonance imaging and diffusion tensor imaging. We develop a novel framework for statistical inference based on the Wasserstein distance to assess the significance of the observed topological differences. Using these methods in comparing maltreated children with a typically developing control group, we find that maltreatment may increase homogeneity in white matter structures and thus induce higher correlations in the structural covariance; this is reflected in the topological profile. Our findings strongly suggest that TDA can be a valuable framework to model altered topological structures of the brain. The MATLAB codes and processed data used in this study can be found at https://github.com/laplcebeltrami/maltreated.

YOLO‐UOD: An underwater small object detector via improved efficient layer aggregation network

AbstractAccurate detection of underwater objects is a key indicator technology to effectively enhance the field of marine development and application, and is of great importance to various fields including marine military defense and seafood aquaculture. Efficient and rapid detection of underwater targets is a crucial technological challenge in this field. To meet the challenges posed by these issues, this study applies the convolutional omni‐efficient layer aggregation network (CO‐ELAN) module to the detector backbone to improve the ability of the network structure to acquire underwater objects from image information. The module improves the feature representation of gradient branching through a multi‐dimensional dynamic convolution and attention mechanism. In terms of loss calculation, the optimized normalized Wasserstein distance approach is used to predict the box distribution probabilistic modelling method to determine comparable distances to the ground box and obtain better samples of small target labels. Here, an underwater image enhancement algorithm based on white balance and underwater blur fusion is used to obtain clear images that enable improved detector performance. After the verification experiment on the URPC2018 dataset, it is found that the detector has better underwater detection ability compared with other detectors in the complex underwater environment. The proposed method achieves a 2.4% improvement over the YOLOv7 baseline model, while reducing computation costs by 5%.

AeroDetectNet: A Lightweight, High-Precision Network for Enhanced Detection of Small Objects in Aerial Remote Sensing Imagery.

Abstract Object detection in remote sensing imagery exhibits difficulties due to complex backgrounds, diverse object scales, and intricate spatial context relationships. Motivated by the problems mentioned above, this paper introduces AeroDetectNet, a novel lightweight and high-precision object detection network custom-designed for aerial remote sensing scenarios, building upon the YOLOv7-tiny algorithm. It enhances performance through four key improvements: the Normalized Wasserstein Distance for consistent object size sensitivity, the Involution module for reduced background noise, a self-designed RCS-Biformer module for better spatial context interpretation, and a self-designed WF-CoT SPPCSP feature pyramid for improved feature map weighting and context capture. Ablation studies conducted on a hybrid dataset composed of three open-source remote sensing datasets (including NWPU VHR-10 remote sensing images, RSOD remote sensing images, and VisDrone UAV images) have demonstrated the effectiveness of four improvements specifically for small-size object detection. Visualizations through Grad-CAM further demonstrate AeroDetectNet's capacity to extract and focus on key object features. Upon individual testing across three open-source datasets, AeroDetectNet has successfully demonstrated its ability to identify objects in images with a smaller pixel area. Through experimental comparisons with other related studies, the AeroDetectNet achieved a competitive mAP while maintaining fewer model parameters, highlighting its highly accurate and lightweight properties.

DL-YOLOX: Real-time object detection via adjustable dilated enhancement for autonomous driving scene

In the domain of autonomous driving, object detection presents several complex challenges, particularly concerning the accurate identification of small and salient objects. This paper introduces DL-YOLOX (Dilated Enhancement YOLOX), which flexibly uses dilated convolution to enhance features to achieve the purpose of improving small objects and silent objects. As we all know, a large receptive field covers a larger area and has greater contextual information, which is more advantageous for detecting large targets. A small receptive field helps capture local details and has better detection capabilities for detecting small targets. To bolster the representation of objects across various scales, we propose the integration of Dilated Adaptive Feature Fusion (DAFF) which has the ability to adaptively fuse features with different receptive fields. This innovative fusion mechanism allows for a more comprehensive understanding of objects, enabling improved detection accuracy even for objects of varying sizes. In addition, we tackle the issue of small object loss during feature propagation by introducing Stack Dilated Module (SDM), a powerful module that mitigates this phenomenon and contributes to better detection performance. Moreover, we endeavor to enhance small object detection further by replacing the conventional Intersection over Union (IoU) metric with Normalized Gaussian Wasserstein Distance (NWD), a novel distance metric that proves to be more effective in accurately gauging small object detection, thus elevating the precision of our algorithm. To thoroughly evaluate the robustness and generalization capabilities of our proposed method, we conduct extensive experiments on two benchmark datasets, namely MS COCO 2017 and BDD100K. The results from our evaluation not only affirm the significant improvements achieved in multi-scale object detection but also highlight the real-time capability of our approach. The impressive performance across these datasets demonstrates the promising potential of DL-YOLOX in revolutionizing object detection techniques in the context of autonomous driving.

Augmented projection Wasserstein distances: Multi-dimensional projection with neural surface

The Wasserstein distance is a fundamental tool for comparing probability distributions and has found broad applications in various fields, including image generation using generative adversarial networks. Despite its useful properties, the performance of the Wasserstein distance decreases when data is high-dimensional, known as the curse of dimensionality. To mitigate this issue, an extension of the Wasserstein distance has been developed, such as the sliced Wasserstein distance using one-dimensional projection. However, such an extension loses information on the original data, due to the linear projection onto the one-dimensional space. In this paper, we propose novel distances named augmented projection Wasserstein distances (APWDs) to address these issues, which utilize multi-dimensional projection with a nonlinear surface by a neural network. The APWDs employ a two-step procedure; it first maps data onto a nonlinear surface by a neural network, then linearly projects the mapped data into a multidimensional space. We also give an algorithm to select a subspace for the multi-dimensional projection. The APWDs are computationally effective while preserving nonlinear information of data. We theoretically confirm that the APWDs mitigate the curse of dimensionality from data. Our experiments demonstrate the APWDs’ outstanding performance and robustness to noise, particularly in the context of nonlinear high-dimensional data.

Matrix Wasserstein distance generative adversarial network with gradient penalty for fast low-carbon economic dispatch of novel power systems

For novel power systems, access to sustainable energy sources will undoubtedly increase the uncertainty of power systems significantly. Consequently, to guarantee efficient, secure, smooth, and expedited functioning, this work proposes a matrix Wasserstein distance generative adversarial network with gradient penalty (MWGAN-GP), which considers the minimization of generation cost and carbon emission simultaneously for low-carbon economic dispatch (ED) of novel power systems. The proposed MWGAN-GP generates a series of matrix data by a trained generator, equalizes load values and the sum of unit outputs at each period by relaxation operation, and rapidly generates numerous ED solutions. The speed of ED based on MWGAN-GP is greatly faster than the speed of ED based on a traditional genetic algorithm (GA). Furthermore, the introduction of the relaxation operation decreases the average cost of ED based on MWGAN-GP by 7.94 % compared to GA according to the case with 4000 sets of load data.

Persistent Homology based Wasserstein distance for graph networks

The technique of measuring similarity between topological spaces using Wasserstein distance
 between persistence diagrams is extended to graph networks in this paper. A relationship
 between the Wasserstein distance of the Cartesian product of topological spaces and the
 Wasserstein distance of individual spaces is found to ease the comparative study of the
 Cartesian product of topological spaces. The Cartesian product and the strong product of
 weighted graphs are defined, and the relationship between the Wasserstein distance between
 graph products and the Wasserstein distance between individual graphs is determined. For
 this, clique complex filtration and the Vietoris- Rips filtration are used.

Wasserstein upper bounds of [formula omitted]-norms for multivariate densities in Besov spaces

While the total variation between two probability measures cannot be bounded by Wasserstein metrics in general, it is possible if they possess smooth densities. More generally, we demonstrate that Lp-distances between densities in Besov spaces can be bounded by powers of the Wasserstein metrics.

Electroencephalographic Signal Data Augmentation Based on Improved Generative Adversarial Network.

EEG signals combined with deep learning play an important role in the study of human-computer interaction. However, the limited dataset makes it challenging to study EEG signals using deep learning methods. Inspired by the GAN network in image generation, this paper presents an improved generative adversarial network model L-C-WGAN-GP to generate artificial EEG data to augment training sets and improve the application of BCI in various fields. The generator consists of a long short-term memory (LSTM) network and the discriminator consists of a convolutional neural network (CNN) which uses the gradient penalty-based Wasserstein distance as the loss function in model training. The model can learn the statistical features of EEG signals and generate EEG data that approximate real samples. In addition, the performance of the compressed sensing reconstruction model can be improved by using augmented datasets. Experiments show that, compared with the existing advanced data amplification techniques, the proposed model produces EEG signals closer to the real EEG signals as measured by RMSE, FD and WTD indicators. In addition, in the compressed reconstruction of EEG signals, adding the new data reduces the loss by about 15% compared with the original data, which greatly improves the reconstruction accuracy of the EEG signals' compressed sensing.

Application of CNN-LSTM Model for Vehicle Acceleration Prediction Using Car-following Behavior Data

Accurate vehicle acceleration prediction is useful for developing reliable Advanced Driving Assistance Systems (ADAS) and improving road safety. The existence of driver heterogeneity magnifies the variations in acceleration data, leading to consequential impacts on the precision of vehicle acceleration prediction. However, few studies have fully considered the driver heterogeneity when predicting vehicle acceleration. To model the characteristics of individual drivers, this study first identifies the driving behavior semantics which is defined as the underlying patterns of driving behaviors. The analysis results from the coupled hidden Markov model (CHMM) are used to evaluate the driving behavior differences between different drivers by Wasserstein distance. Then the convolutional neural network (CNN) and long short-term memory (LSTM) network are applied to predict vehicle acceleration. To validate the accuracy of the proposed prediction framework, vehicle acceleration data in car-following conditions is extracted from the safety pilot model deployment (SPMD) dataset. The segmentation results indicate that the CHMM possesses a robust capacity for modeling driving behavior. The prediction results demonstrate that the proposed framework, which incorporates driver clustering before prediction, significantly improves the accuracy of predictions. And the CNN-LSTM outperforms the LSTM in predicting vehicle acceleration during car-following scenarios. The findings from this study can enhance the development of personalized functionalities within ADAS to promote its deployment, thereby improving its acceptance and safety.

A note on the Radiant formula and its relations to the sliced Wasserstein distance

AbstractIn this note, we show that the squared Wasserstein distance can be expressed as the average over the sphere of one dimensional Wasserstein distances. We name this new expression for the Wasserstein Distance Radiant Formula. Using this formula, we are able to highlight new connections between the Wasserstein distances and the Sliced Wasserstein distance, an alternative Wasserstein-like distance that is cheaper to compute.

Damage Detection in Bridge Structures through Compressed Sensing of Crowdsourced Smartphone Data

Traditional bridge health monitoring methods that necessitate sensor installation are not only costly but also time-consuming. In contrast, utilizing smartphone data collected from vehicles as they traverse bridges offers an efficient and cost-effective alternative. This paper introduces a cutting-edge damage detection framework for indirect monitoring of bridge structures, leveraging a substantial volume of acceleration data collected from smartphones in vehicles passing over the bridge. Our innovative approach addresses the challenge of collecting and transmitting high-frequency data while preserving smartphone battery life and data plans through the integration of compressed sensing (CS) into the crowdsensing-based monitoring framework. CS employs random sampling and signal recovery from a significantly reduced number of samples compared to the requirements of the Nyquist–Shannon sampling theorem. In the proposed framework, acceleration signals from vehicles are initially acquired using smartphone sensors, undergo compression, and are then transmitted for signal reconstruction. Subsequently, feature extraction and dimensionality reduction are performed using Mel-frequency cepstral coefficients and principal component analysis. Damage indexes are computed based on the dissimilarity between probability distribution functions utilizing the Wasserstein distance metric. The efficacy of the proposed methodology in bridge monitoring has been substantiated through the utilization of numerical models and a lab-scale bridge. Furthermore, the feasibility of implementing the framework in a real-world application has been investigated, leveraging the smartphone data from 102 vehicle trips on the Golden Gate Bridge. The results demonstrate that damage detection using the reconstructed signals obtained through compressed sensing achieves comparable performance to that obtained with the original data sampled at the Nyquist measurement sampling rate. However, it is observed that to retain severity information within the signals for accurate damage severity identification, the compression level should be limited to 20%. These findings affirm that compressed sensing significantly reduces the data collection requirements for crowdsensing-based monitoring applications, without compromising the accuracy of damage detection while preserving essential damage-sensitive information within the dataset.

An extreme worst-case risk measure by expectile

Abstract Expectiles have received increasing attention as a risk measure in risk management because of their coherency and elicitability at the level $\alpha\geq1/2$ . With a view to practical risk assessments, this paper delves into the worst-case expectile, where only partial information on the underlying distribution is available and there is no closed-form representation. We explore the asymptotic behavior of the worst-case expectile on two specified ambiguity sets: one is through the Wasserstein distance from a reference distribution and transforms this problem into a convex optimization problem via the well-known Kusuoka representation, and the other is induced by higher moment constraints. We obtain precise results in some special cases; nevertheless, there are no unified closed-form solutions. We aim to fully characterize the extreme behaviors; that is, we pursue an approximate solution as the level $\alpha $ tends to 1, which is aesthetically pleasing. As an application of our technique, we investigate the ambiguity set induced by higher moment conditions. Finally, we compare our worst-case expectile approach with a more conservative method based on stochastic order, which is referred to as ‘model aggregation’.

The stability and statistic of domain decomposition algorithm with mini-batch learning for optimal transport

The Wasserstein distance in the optimal transport (OT) model is being widely used in machine learning applications due to its mathematical and geometric properties. Since the entropic regularization OT is a convex problem, and for its large-scale problems, many algorithms have been proposed. Existing methods mainly adopt iterative random projection techniques to approximate the Wasserstein distance in the OT model. In this paper, we employ the entropic domain decomposition algorithm with mini-batch learning to calculate the entropic regularization OT problem and investigate the stability of this algorithm. The algorithm performs on different cells of the domain instead of using the projection technique and it is based on a mini-batch loss function. The corresponding solution in each iteration is derived from the dual problem of entropic regularized OT on each decomposed cell. Once the solutions to the dual problem on all decomposed cells are obtained, the corresponding transport plan in each iteration can be obtained. We show that the transport plan computed by its dual variables is admissible, and we also give the sub-gradient approximation result. In addition, we exploit the statistical concentration result of the subsampled mini-batch Wasserstein distance on decomposed cells. Finally, the numerical experiments validate that the proposed algorithm can save computational time further.