Low-Rank Optimal Transport for Robust Domain Adaptation

When encountering the distribution shift between the source(training) and target(test) domains, domain adaptation attempts to adjust the classifiers to be capable of dealing with different domains. Previous domain adaptation research has achieved a lot of success both in theory and practice under the assumption that all the examples in the source domain are welllabeled and of high quality. However, the methods consistently lose robustness in noisy settings where data from the source domain have corrupted labels or features which is common in reality. Therefore, robust domain adaptation has been introduced to deal with such problems. In this paper, we attempt to solve two interrelated problems with robust domain adaptation:distribution shift across domains and sample noises of the source domain. To disentangle these challenges, an optimal transport approach with low-rank constraints is applied to guide the domain adaptation model training process to avoid noisy information influence. For the domain shift problem, the optimal transport mechanism can learn the joint data representations between the source and target domains using a measurement of discrepancy and preserve the discriminative information. The rank constraint on the transport matrix can help recover the corrupted subspace structures and eliminate the noise to some extent when dealing with corrupted source data. The solution to this relaxed and regularized optimal transport framework is a convex optimization problem that can be solved using the Augmented Lagrange Multiplier method, whose convergence can be mathematically proved. The effectiveness of the proposed method is evaluated through extensive experiments on both synthetic and real-world datasets.

WEAK SOLUTIONS OF MONGE-AMPRE TYPE EQUATIONS IN OPTIMAL TRANSPORTATION

This paper concerns the weak solutions of some Monge-Amp^re type equa- tions in the optimal transportation theory. The relationship between the Aleksandrov solutions and the viscosity solutions of the Monge-Ampere type equations is discussed. A uniform estimate for solution of the Dirichlet problem with homogeneous boundary value is obtained.

ENTROPICAL OPTIMAL TRANSPORT,SCHRODINGER'S SYSTEM AND ALGORITHMS

In this exposition paper we present the optimal transport problem of Monge-Ampère-Kantorovitch(MAK in short)and its approximative entropical regularization.Contrary to the MAK optimal transport problem,the solution of the entropical optimal transport problem is always unique,and is characterized by the Schrödinger system.The relationship between the Schrödinger system,the associated Bernstein process and the optimal transport was developed by Léonard[32,33](and by Mikami[39]earlier via an h-process).We present Sinkhorn’s algorithm for solving the Schrödinger system and the recent results on its convergence rate.We study the gradient descent algorithm based on the dual optimal question and prove its exponential convergence,whose rate might be independent of the regularization constant.This exposition is motivated by recent applications of optimal transport to different domains such as machine learning,image processing,econometrics,astrophysics etc..

Domain adaptive semantic segmentation by optimal transport

Scene segmentation is widely used in autonomous driving for environmental perception.Semantic scene segmentation has gained considerable attention owing to its rich semantic information.It assigns labels to the pixels in an image,thereby enabling automatic image labeling.Current approaches are based mainly on convolutional neural networks(CNN),however,they rely on numerous labels.Therefore,the use of a small amount of labeled data to achieve semantic segmentation has become increasingly important.In this study,we developed a domain adaptation framework based on optimal transport(OT)and an attention mechanism to address this issue.Specifically,we first generated the output space via a CNN owing to its superior of feature representation.Second,we utilized OT to achieve a more robust alignment of the source and target domains in the output space,where the OT plan defined a well attention mechanism to improve the adaptation of the model.In particular,the OT reduced the number of network parameters and made the network more interpretable.Third,to better describe the multiscale properties of the features,we constructed a multiscale segmentation network to perform domain adaptation.Finally,to verify the performance of the proposed method,we conducted an experiment to compare the proposed method with three benchmark and four SOTA methods using three scene datasets.The mean intersection-over-union(mIOU)was significantly improved,and visualization results under multiple domain adaptation scenarios also show that the proposed method performed better than semantic segmentation methods.

Autoencoder-based conditional optimal transport generative adversarial network for medical image generation

Medical image generation has recently garnered significant interest among researchers.However,the primary generative models,such as Generative Adversarial Networks(GANs),often encounter challenges during training,including mode collapse.To address these issues,we proposed the AECOT-GAN model(Autoencoder-based Conditional Optimal Transport Generative Adversarial Network)for the generation of medical images belonging to specific categories.The training process of our model comprises three fundamental components.The training process of our model encompasses three fundamental components.First,we employ an autoencoder model to obtain a low-dimensional manifold representation of real images.Second,we apply extended semi-discrete optimal transport to map Gaussian noise distribution to the latent space distribution and obtain corresponding labels effectively.This procedure leads to the generation of new latent codes with known labels.Finally,we integrate a GAN to train the decoder further to generate medical images.To evaluate the performance of the AE-COT-GAN model,we conducted experiments on two medical image datasets,namely DermaMNIST and BloodMNIST.The model’s performance was compared with state-of-the-art generative models.Results show that the AE-COT-GAN model had excellent performance in generating medical images.Moreover,it effectively addressed the common issues associated with traditional GANs.

RFPose-OT:RF-based 3D human pose estimation via optimal transport theory

This paper introduces a novel framework,i.e.,RFPose-OT,to enable three-dimensional(3D)human pose estimation from radio frequency(RF)signals.Different from existing methods that predict human poses from RF signals at the signal level directly,we consider the structure difference between the RF signals and the human poses,propose a transformation of the RF signals to the pose domain at the feature level based on the optimal transport(OT)theory,and generate human poses from the transformed features.To evaluate RFPose-OT,we build a radio system and a multi-view camera system to acquire the RF signal data and the ground-truth human poses.The experimental results in a basic indoor environment,an occlusion indoor environment,and an outdoor environment demonstrate that RFPose-OT can predict 3D human poses with higher precision than state-of-the-art methods.

Analysis of minimum specific energy consumption and optimal transport concentration of slurry pipeline transport systems

Slurry pipeline transport is widely used in several industrial processes.Calculating the specific power consumption(SPC)and determining the best working conditions are important for the design and operation of transportation systems.Based on the Shanghai Jiao Tong University high-concentration multi-sized slurry pressure drop(SJTU-HMSPD)pipeline-resistance-calculation model,the SJTU-SPC model for calculating the power required to transport a unit volume of solid materials over a unit pipeline length is established for a slurry transport system.The said system demonstrates a uniformity coefficient in the 1.26–7.98 range,median particle size of 0.075–4 mm,particle volume concentration of 10–60%,and pipeline diameter of 0.203–0.8 m.The results obtained were successfully verified against existing experimental data.The influence of parameters,such as particle-gradation uniformity coefficient,median particle size,pipe diameter,and particle volume concentration,on the SPC were analysed.The results revealed that the greater is the uniformity coefficient,the smaller is the minimum specific energy consumption and the larger the optimal transport concentration for a constant,median particle size slurry.As observed,the optimal transport concentration for broad-graded sand equalled approximately 48%.These results supplement the conclusions of existing research,indicating that the optimal transport concentration is approximately 30%and provides theoretical support for high concentration transportation of broad graded slurry.

Robust and accurate optimal transportation map by self-adaptive

Optimal transportation plays a fundamental role in many fi elds in engineering and medicine,including surface parameterization in graphics,registration in computer vision,and generative models in deep learning.For quadratic distance cost,optimal transportation map is the gradient of the Brenier potential,which can be obtained by solving the Monge-Ampère equation.Furthermore,it is induced to a geometric convex optimization problem.The Monge-Ampère equation is highly non-linear,and during the solving process,the intermediate solutions have to be strictly convex.Specifi cally,the accuracy of the discrete solution heavily depends on the sampling pattern of the target measure.In this work,we propose a self-adaptive sampling algorithm which greatly reduces the sampling bias and improves the accuracy and robustness of the discrete solutions.Experimental results demonstrate the efficiency and efficacy of our method.

Optimal transport maps on infinite dimensional spaces

We will give a survey on results concerning Girsanov transforma- tions, transportation cost inequalities, convexity of entropy, and optimal transport maps on some infinite dimensional spaces. Some open Problems will be arisen.

Optimal Transportation for Generalized Lagrangian

This paper deals with the optimal transportation for generalized Lagrangian L = L（x, u, t）, and considers the following cost function： c（x, y） = inf x（0）=x x（1）=y u∈U∫0^1 L（x（s）, u（x（s）, s）, s）ds, where U is a control set, and x satisfies the ordinary equation x（s） = f（x（s）, u（x（s）, s））.It is proved that under the condition that the initial measure μ0 is absolutely continuous w.r.t. the Lebesgue measure, the Monge problem has a solution, and the optimal transport map just walks along the characteristic curves of the corresponding Hamilton-Jacobi equation：Vt（t, x） ＋ sup u∈U = 0,V（0, x） = Φ0（x）.

Optimal paths planning in dynamic transportation networks with random link travel times

A theoretical study was conducted on finding optimal paths in transportation networks where link travel times were stochastic and time-dependent(STD). The methodology of relative robust optimization was applied as measures for comparing time-varying, random path travel times for a priori optimization. In accordance with the situation in real world, a stochastic consistent condition was provided for the STD networks and under this condition, a mathematical proof was given that the STD robust optimal path problem can be simplified into a minimum problem in specific time-dependent networks. A label setting algorithm was designed and tested to find travelers' robust optimal path in a sampled STD network with computation complexity of O(n2+n·m). The validity of the robust approach and the designed algorithm were confirmed in the computational tests. Compared with conventional probability approach, the proposed approach is simple and efficient, and also has a good application prospect in navigation system.

A Multilevel Method for the Solution of Time Dependent Optimal Transport

In this paper we present a new computationally efficientnumerical scheme for the minimizing flow for the computation of the optimal L 2 mass transport map-ping using the fluid approach.We review the method and discuss its numerical properties.We then derive a new scaleable,efficient discretization and a solution technique for the problem and show that the problem is equivalent to a mixed form formulation of a nonlinear fluid flow in porous media.We demonstrate the effec-tiveness of our approach using a number of numerical experiments.

Watertight surface reconstruction method for CAD models based on optimal transport

Feature-preserving mesh reconstruction from point clouds is challenging.Implicit methods tend to fit smooth surfaces and cannot be used to reconstruct sharp features.Explicit reconstruction methods are sensitive to noise and only interpolate sharp features when points are distributed on feature lines.We propose a watertight surface reconstruction method based on optimal transport that can accurately reconstruct sharp features often present in CAD models.We formalize the surface reconstruction problem by minimizing the optimal transport cost between the point cloud and the reconstructed surface.The algorithm consists of initialization and refinement steps.In the initialization step,the convex hull of the point cloud is deformed under the guidance of a transport plan to obtain an initial approximate surface.Next,the mesh surface was optimized using operations including vertex relocation and edge collapses/fips to obtain feature-preserving results.Experiments demonstrate that our method can preserve sharp features while being robust to noise and missing data.

Self-Organized Optimization of Transport on Complex Networks

We propose a self-organized optimization mechanism to improve the transport capacity of complex gradient networks. We find that, regardless of network topology, the congestion pressure can be strongly reduced by the self-organized optimization mechanism. Furthermore, the random scale-free topology is more efficient to reduce congestion compared with the random Poisson topology under the optimization mechanism. The reason is that the optimization mechanism introduces the correlations between the gradient field and the local topology of the substrate network. Due to the correlations, the cutoff degree of the gradient network is strongly reduced and the number of the nodes exerting their maximal transport capacity consumedly increases. Our work presents evidence supporting the idea that scale-free networks can efficiently improve their transport capacity by self- organized mechanism under gradient-driven transport mode.

Spatial batch optimal design based on self-learning Gaussian process models for LPCVD processes

Low pressure chemical vapor deposition(LPCVD) is one of the most important processes during semiconductor manufacturing.However,the spatial distribution of internal temperature and extremely few samples makes it hard to build a good-quality model of this batch process.Besides,due to the properties of this process,the reliability of the model must be taken into consideration when optimizing the MVs.In this work,an optimal design strategy based on the self-learning Gaussian process model(GPM) is proposed to control this kind of spatial batch process.The GPM is utilized as the internal model to predict the thicknesses of thin films on all spatial-distributed wafers using the limited data.Unlike the conventional model based design,the uncertainties of predictions provided by GPM are taken into consideration to guide the optimal design of manipulated variables so that the designing can be more prudent Besides,the GPM is also actively enhanced using as little data as possible based on the predictive uncertainties.The effectiveness of the proposed strategy is successfully demonstrated in an LPCVD process.

The Role of Short Sea Shipping Concept in Black SeaRegion as a Connection between Eurasian TransportSystems

Despite of fluctuations in world trade as a result of economic cycles,the evolution of the political processes remains the trend of sustained growth of trade flows.This ends up in a rise in both the demand for logistics services and the requirements for them.In this sense,the critically important is the strategic development of the transport systems as a support for the improvement of competitive logistics.An important aspect is the promotion of multimodal transport,which in search of the best transport solutions will reduce the use of relatively expensive and environmentally unfriendly road transportation.This will be at the expense of the efficient combination of different modes in which the concept of short sea shipping(SSS)occupies a central place.Although this concept is widely applied in many places in the Black Sea,it still has significant potential.It was prompted by stagnation in economic relations as a result of political and economic crises in the region since the late twentieth and early twenty-first century.To evaluate the potential of the concept in the development of transport is done research on intermodal logistics network in the logistics corridor Central Asia-Central Europe.To optimize intermodal transport links a comparative analysis of the various transport alternatives on the route Tehran-Budapest is done.On this basis it is made optimization assessment on three main criteria cost,delivery time and environmental protection and basic recommendations on strategic planning development of the Bulgarian transport infrastructure are given.An essential aspect is the encouragement of multimodal transportation,which in looking for the best transport solutions can cut back the utilization of comparatively costly and environmentally harmful road transportation.This would be at the expense of the adequate combination of different modes of transportation in which the concept of SSS has a fundamental area.Despite this concept is widely applied in various regions,in the Black Sea it still has an important future due to stagnation in economic relations as a result of political confrontations and economic crises within the region since the late twentieth and early twenty-first century.To assess the capability of the concept in the development of transport is done research on intermodal logistics network in the logistics corridor Central Asia-Central Europe.To improve intermodal transport links a comparative analysis of the various transport options on the routes Astana-Budapest and Tehran-Budapest are made.On this basis it is proposed an optimization assessment on three main criteria cost,delivery time and environmental protection,and fundamental suggestions on strategic development of the Bulgarian transport infrastructure are proposed.

Study on Rural Solid Waste Collection and Transportation Mode and Route Optimization Strategy in the Cold Regions of Northern China

Rural vitalization is a major strategy for reform and development of agriculture and rural areas in China,the key task of which is improving rural living environment.Imperfect rural solid waste(RSW)collection and transportation system exacerbates the pollution of RSW to rural living environment,while it has not been established and improved in the cold region of Northern China due to climate and economy.Through the analysis of the current situation of RSW source separation,collection,transportation and disposal in China,an RSW collection and transportation system suitable for the northern cold region was developed.Considering the low winter temperature in the northern cold region,different requirements for RSW collection,transportation and terminal disposal,scattered source points and single terminal disposal nodes in rural areas,the study focused on determining the number and location of transfer stations,established a model for transfer stations selection and RSW collection and transportation routes optimization for RSW collection and transportation system,and proposed the elite retention particle swarm optimization–genetic algorithm(ERPSO–GA).The rural area of Baiquan County was taken as a representative case,the collection and transportation scheme of which was given,and the feasibility of the scheme was clarified by simulation experiment.

A Dynamical System-Based Framework for Dimension Reduction

We propose a novel framework for learning a low-dimensional representation of data based on nonlinear dynamical systems,which we call the dynamical dimension reduction(DDR).In the DDR model,each point is evolved via a nonlinear flow towards a lower-dimensional subspace;the projection onto the subspace gives the low-dimensional embedding.Training the model involves identifying the nonlinear flow and the subspace.Following the equation discovery method,we represent the vector field that defines the flow using a linear combination of dictionary elements,where each element is a pre-specified linear/nonlinear candidate function.A regularization term for the average total kinetic energy is also introduced and motivated by the optimal transport theory.We prove that the resulting optimization problem is well-posed and establish several properties of the DDR method.We also show how the DDR method can be trained using a gradient-based optimization method,where the gradients are computed using the adjoint method from the optimal control theory.The DDR method is implemented and compared on synthetic and example data sets to other dimension reduction methods,including the PCA,t-SNE,and Umap.

A Geometric Understanding of Deep Learning

This work introduces an optimal transportation(OT)view of generative adversarial networks(GANs).Natural datasets have intrinsic patterns,which can be summarized as the manifold distribution principle:the distribution of a class of data is close to a low-dimensional manifold.GANs mainly accomplish two tasks:manifold learning and probability distribution transformation.The latter can be carried out using the classical OT method.From the OT perspective,the generator computes the OT map,while the discriminator computes the Wasserstein distance between the generated data distribution and the real data distribution;both can be reduced to a convex geometric optimization process.Furthermore,OT theory discovers the intrinsic collaborative-instead of competitive-relation between the generator and the discriminator,and the fundamental reason for mode collapse.We also propose a novel generative model,which uses an autoencoder(AE)for manifold learning and OT map for probability distribution transformation.This AE–OT model improves the theoretical rigor and transparency,as well as the computational stability and efficiency;in particular,it eliminates the mode collapse.The experimental results validate our hypothesis,and demonstrate the advantages of our proposed model.

Deep density estimation via invertible block-triangular mapping

In this work,we develop an invertible transport map,called KRnet,for density estimation by coupling the Knothe–Rosenblatt(KR)rearrangement and the flow-based generative model,which generalizes the real-valued non-volume preserving(real NVP)model(arX-iv:1605.08803v3).The triangular structure of the KR rearrangement breaks the symmetry of the real NVP in terms of the exchange of information between dimensions,which not only accelerates the training process but also improves the accuracy significantly.We have also introduced several new layers into the generative model to improve both robustness and effectiveness,including a reformulated affine coupling layer,a rotation layer and a component-wise nonlinear invertible layer.The KRnet can be used for both density estimation and sample generation especially when the dimensionality is relatively high.Numerical experiments have been presented to demonstrate the performance of KRnet.