Wasserstein Distance Research Articles

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.

Wasserstein distance-based full waveform inversion method for density reconstruction

Reliable density information is an essential factor in lithologic interpretation and reservoir evaluation. However, density reconstruction is hampered by a lack of long-wavelength information and the crosstalk between different parameters. The powerful nonlinearity in density inversion constrains the multi-parameter full waveform inversion application. This paper introduces the Wasserstein distance into the full waveform inversion for velocity and density based on the optimal transport theory. We construct an objective function with better convexity to avoid the cycle-skipping caused by the poor initial model. Furthermore, due to the sensitivity of the quadratic Wasserstein distance to low-frequency, we add the low-wavenumber component absent in the conventional density inversion. We acquire a long-wavelength density background using optimal transport inversion and then apply it as an initial model for least-squares inversion. Numerical examples show that our optimal transport inversion effectively builds a reasonable density background model for FWI. Density inversion can be improved by incorporating optimal transport and least-squares theory, leading to a reliable quantitative description for petroleum exploration and development.

Wasserstein distance regularized graph neural networks

Distribution shift widely exists in graph representation learning and often reduces model performance. This work investigates how to improve the performance of a graph neural network (GNN) in a single graph by controlling distribution shift between embedding spaces. Specifically, we provide an upper error-bound estimation, which quantitatively analyzes how distribution shift affects GNNs' performance in a single graph. Considering that there is no natural domain division in a single graph, we propose PW-GNN to simultaneously learn discriminative embedding and reduce distribution shift. PW-GNN measures distribution discrepancy using the distance between test embeddings and prototypes, and transfers minimizing distribution shift to minimizing the power of Wasserstein distance, which is introduced into GNNs as a regularizer. A series of theoretical analyses are carried out to demonstrate the effectiveness of PW-GNN. Besides, a low-complexity training algorithm is designed by exploring entropy-regularized strategy and block coordinate descent method. Extensive numerical experiments are conducted on different datasets with both biased and unbiased splits. We empirically test our model equipped with four backbone models. Results show that PW-GNN outperforms state-of-the-art baselines and mitigates up to 8% of negative effects off distribution shift on backbones.

A light-weight defect detection model for capacitor appearance based on the Yolov5

As one of the most important electronic components, capacitors are very important for appearance inspection in the production process. However, the current production process mainly relies on manual inspection, which not only reduces product quality and production efficiency but also increases production costs. The automated detection is limited by the computational resources of the equipment, which is difficult to apply in practice. Therefore, in this paper, we propose a lightweight method for capacitor appearance inspection. We use the YOLOv5 (You Only Look Once Version 5) framework, MobileNet as the backbone network, and GSConv (Ghost convolution) and GSCSP module as the neck depth compression network model to reduce the computational cost. In addition, we incorporate the CBAM (Convolutional Block Attention Module) into the backbone network to improve the network's ability to extract features. For the problem of difficult detection of small targets, we use CIoU (Complete Intersection Union) and NWD (Normalised Gaussian Wasserstein Distance) to design a new loss function. By testing our method on capacitor appearance defect data, compared to the baseline, the model computational cost FLOPs was reduced by 130 %, the model size was reduced by 94%, the accuracy reached 92.5%, and the mAP (mean average precision) reached 92.3%, while the number of frames detected per second was up to 58 frames. The experimental results show that our method is capable of real-time detection of capacitor appearance defects, providing strong theoretical support for practical applications.

Causal Discovery on Discrete Data via Weighted Normalized Wasserstein Distance.

The task of causal discovery from observational data (X,Y) is defined as the task of deciding whether X causes Y , or Y causes X or if there is no causal relationship between X and Y . Causal discovery from observational data is an important problem in many areas of science. In this study, we propose a method to address this problem when the cause-and-effect relationship is represented by a discrete additive noise model (ANM). First, assuming that X causes Y , we estimate the conditional distributions of the noise given X using regression. Similarly, assuming that Y causes X , we also estimate the conditional distributions of noise given Y . Based on the structural characteristics of the discrete ANM, we find that the dissimilarity of the conditional distributions of noise in the causal direction is smaller than that in the anticausal direction. Then, we propose a weighted normalized Wasserstein distance to measure the dissimilarity of the conditional distributions of noise. Finally, we propose a decision rule for casual discovery by comparing two computed weighted normalized Wasserstein distances. An empirical investigation demonstrates that our method performs well on synthetic data and outperforms state-of-the-art methods on real data.

Novel Distance Measures of Multigranular Unbalanced Hesitant Fuzzy Linguistic Term Sets Based on Semantics Intervals.

In the field of qualitative decision making with hesitant fuzzy linguistic term sets (HFLTSs), distance measure (DM) is a significant concept that reflects the degree of difference between HFLTSs. Various DMs among HFLTSs have been proposed, which enhance the applicability of HFLTSs in multicriteria decision making (MCDM) under qualitative hesitation information. However, existing strategies not only have their own pros and cons but also fail to measure linguistic assessments from multiple linguistic term sets with different distributions (i.e., multigranular unbalanced linguistic information). In this article, we first propose a new strict DM between two hesitant fuzzy linguistic elements (HFLEs) based on the Wasserstein distance of their semantics intervals. Then, two novel kinds of DMs (Euclidean and Chebyshev forms) for HFLTSs are proposed and their strictness is proved. Weighted and ordered weighted versions of HFLTSs based on Euclidean form are derived. Afterward, illustrative examples, simulations, and related analyses are given to demonstrate the rationality of the proposals. Finally, two cases are applied to verify the efficiency and practicality of the novel measures in real life.

The rival coffee shop problem

In this paper, we will address a modification of the following optimization problem: given a positive integer $N$ and a compact Riemannian manifold X, the goal is to place a point $x_N\in X$ in such a way that the sequence $\{x_1,\dots,x_N\}\subset X$ is distributed as uniformly as possible, considering that $\{x_1,\dots,x_{N-1}\}\subset X$ already is. This can be thought as a way of placing coffee shops in a certain area one at a time in order to cover it optimally. So, following this modelization we will denote this problem as the coffee shop problem. This notion of optimal settlement is formalized in the context of optimal transport and Wasserstein distance. As a novel aspect, we introduce a new element to the problem: the presence of a rival brand, which competes against us by opening its own coffee shops. As our main tool, we use a variation of the Wasserstein distance, that allows us to work with finite signed measures and fits our problem. We present different results depending on how fast the rival is able to grow. With the Signed Wasserstein distance, we are able to obtain similar inequalities to the ones produced by the canonical Wasserstein one.

S-YOLOv5: A Lightweight Model for Detecting Objects Thrown from Tall Buildings in Communities

Objects thrown from tall buildings in communities are characterized by their small size, inconspicuous features, and high speed. Existing algorithms for detecting such objects face challenges, including excessive parameters, overly complex models that are difficult to implement, and insufficient detection accuracy. This study proposes a lightweight detection model for objects thrown from tall buildings in communities, named S-YOLOv5, to address these issues. The model is based on the YOLOv5 algorithm, and a lightweight convolutional neural network, Enhanced ShuffleNet (ESNet), is chosen as the backbone network to extract image features. On this basis, the initial stage of the backbone network is enhanced and the simplified attention module (SimAM) attention mechanism is added to utilize the rich position information and contour information in the shallow feature map to improve the detection of small targets. For feature fusion, the sparsely connected Path Aggregation Network (SCPANet) module is designed to use sparsely connected convolution (SCConv) instead of the regular convolution of the Path Aggregation Network (PANet) to fuse features efficiently. In addition, the model uses the normalized Wasserstein distance (NWD) loss function to reduce the sensitivity of positional bias. The accuracy of the model is further improved. Test results from the self-built objects thrown from tall buildings dataset show that S-YOLOv5 can detect objects thrown from tall buildings quickly and accurately, with an accuracy of 90.2% and a detection rate of 34.1 Fps/s. Compared with the original YOLOv5 model, the parameters are reduced by 87.3%, and the accuracy and rate are improved by 0.8% and 63%, respectively.

Detecting Weak Distribution Shifts via Displacement Interpolation

Detecting weak, systematic distribution shifts and quantitatively modeling individual, heterogeneous responses to policies or incentives have found increasing empirical applications in social and economic sciences. Given two probability distributions P (null) and Q (alternative), we study the problem of detecting weak distribution shift deviating from the null P toward the alternative Q, where the level of deviation vanishes as a function of n, the sample size. We propose a model for weak distribution shifts via displacement interpolation between P and Q, drawing from the optimal transport theory. We study a hypothesis testing procedure based on the Wasserstein distance, derive sharp conditions under which detection is possible, and provide the exact characterization of the asymptotic Type I and Type II errors at the detection boundary using empirical processes. We demonstrate how the proposed testing procedure works in modeling and detecting weak distribution shifts in real data sets using two empirical examples: distribution shifts in consumer spending after COVID-19, and heterogeneity in the published p-values of statistical tests in journals across different disciplines.

Optimal transport with nonlinear mobilities: A deterministic particle approximation result

Abstract We study the discretisation of generalised Wasserstein distances with nonlinear mobilities on the real line via suitable discrete metrics on the cone of N ordered particles, a setting which naturally appears in the framework of deterministic particle approximation of partial differential equations. In particular, we provide a Γ-convergence result for the associated discrete metrics as N → ∞ {N\to\infty} to the continuous one and discuss applications to the approximation of one-dimensional conservation laws (of gradient flow type) via the so-called generalised minimising movements, proving a convergence result of the schemes at any given discrete time step τ > 0 {\tau>0} . This the first work of a series aimed at sheding new lights on the interplay between generalised gradient-flow structures, conservation laws, and Wasserstein distances with nonlinear mobilities.

Learning and generalization of task-parameterized skills through few human demonstrations

In the field of robotics, the demand for adaptable skills capable of effectively handling diverse situations has surpassed the reliance on repetitive tasks. To enhance the generalization of motion policies, task-parameterized learning has emerged as a valuable approach, as it encodes pertinent contextual information in task parameters, facilitating the flexible execution of tasks. This process requires the collection of multiple demonstrations in various situations. Generate a set of diverse situations covering all possible cases is a complex task. Due to this, the search for training with fewer demonstrations becomes highly desirable. In this article, we present a novel algorithm focused on generating synthetic information to facilitate the generalization of parameterized tasks. This algorithm makes use of Kinesthetic Fast Marching Learning, a Learning from Demonstration (LfD) algorithm that allows obtaining optimal movement paths based on velocity fields. The algorithm enables autonomous data generation, producing demonstrations that are equal to or more optimal than those generated by users themselves. Evaluation is done through a metric based on Wasserstein distance that takes into account the probabilistic data of the generated paths. The algorithm has been evaluated through tests in simulated environments, comparing its efficiency against two widely used algorithms (where it has shown greater efficiency in generalization and the generation of more optimal paths) and with real-world tests in a task-oriented environment (sweeping) carried out by the ADAM robot.

A Small Target Tea Leaf Disease Detection Model Combined with Transfer Learning

Tea cultivation holds significant economic value, yet the leaves of tea plants are frequently susceptible to various pest and disease infestations. Consequently, there is a critical need for research focused on precisely and efficiently detecting these threats to tea crops. The investigation of a model capable of effectively identifying pests and diseases in tea plants is often hindered by challenges, such as limited datasets of pest and disease samples and the small size of detection targets. To address these issues, this study has chosen TLB, a common pest and disease in tea plants, as the primary research subject. The approach involves the application of transfer learning in conjunction with data augmentation as a fundamental methodology. This technique entails transferring knowledge acquired from a comprehensive source data domain to the model, aiming to mitigate the constraints of limited sample sizes. Additionally, to tackle the challenge of detecting small targets, this study incorporates the decoupling detection head TSCODE and integrates the Triplet Attention mechanism into the E-ELAN structure within the backbone to enhance the model’s focus on the TLB’s small targets and optimize detection accuracy. Furthermore, the model’s loss function is optimized based on the Wasserstein distance measure to mitigate issues related to sensitivity in localizing small targets. Experimental results demonstrate that, in comparison to the conventional YOLOv7 tiny model, the proposed model exhibits superior performance on the TLB small sample dataset, with precision increasing by 6.5% to 92.2%, recall by 4.5% to 86.6%, and average precision by 5.8% to 91.5%. This research offers an effective solution for identifying tea pests and diseases, presenting a novel approach to developing a model for detecting such threats in tea cultivation.

A detection network for small defects of steel surface based on YOLOv7

Defect detection is crucial in steel production to prevent safety risks and prolong lifespan. However, current methods still face challenges in accurately detecting small and vague targets. In response to this problem, we propose a model named SRN-YOLO for steel surface detection based on YOLOv7. Firstly, a split residual convolution network (SResNet) is designed to capture gradient feature information. Then, a re-fusion feature pyramid network (RFPN) is built to minimize the loss of features. Furthermore, a location regression loss function named NWD-CIOU combined the Normalized Wasserstein Distance (NWD) and Complete Intersection Over Union (CIOU) is employed to increase the number of positive samples of small targets. The experiment results show that SRN-YOLO has excellent detection performance for steel surface defects. Specifically, the recognition accuracies of the model on NEU-DET and GC10-DET reach 81.2 mAP and 71.6 mAP, respectively, which are improved by 5.8% and 3.8% compared to YOLOv7.