Narrow Results

We develop a general approach to setting up and studying classes of quantum dynamical systems close to and structurally similar to systems having specified properties, in particular detailed balance. This is done in terms of transport plans and Wasserstein distances between systems on possibly different observable algebras.

In this paper, we propose a novel approach to address the challenges of Super-Resolution Generative Adversarial Network (SRGAN) in face image super-resolution reconstruction. We introduce a new improved SRGAN algorithm, named Wasserstein SRGAN (W-SRGAN), which addresses the limitations of the original model by enhancing the loss function, generator, and discriminator. Our approach utilizes the embedded residual structure combined with feature fusion as the new generator, while removing the Sigmoid of the last layer of the discriminator of SRGAN by borrowing the idea of Wasserstein GAN (W-GAN). Additionally, we replace the Kullback–Leibler (KL) divergence of SRGAN with Wasserstein distance. The contributions of our research are twofold. Firstly, we propose a new face super-resolution reconstruction algorithm that outperforms existing methods in terms of visual quality. Secondly, we introduce a new loss function and generator–discriminator architecture that can be applied to other image super-resolution tasks, extending the applicability of GANs in this domain. Experimental results demonstrate that our proposed W-SRGAN outperforms Bicubic, Super-Resolution Convolutional Neural Network (SRCNN), and SRGAN in terms of visual quality on all Celeb A datasets. These results confirm the effectiveness of our proposed algorithm and provide a new solution for face super-resolution reconstruction.

Motor imagery (MI) based brain–computer interfaces help patients with movement disorders to regain the ability to control external devices. Common spatial pattern (CSP) is a popular algorithm for feature extraction in decoding MI tasks. However, due to noise and nonstationarity in electroencephalography (EEG), it is not optimal to combine the corresponding features obtained from the traditional CSP algorithm. In this paper, we designed a novel CSP feature selection framework that combines the filter method and the wrapper method. We first evaluated the importance of every CSP feature by the infinite latent feature selection method. Meanwhile, we calculated Wasserstein distance between feature distributions of the same feature under different tasks. Then, we redefined the importance of every CSP feature based on two indicators mentioned above, which eliminates half of CSP features to create a new CSP feature subspace according to the new importance indicator. At last, we designed the improved binary gravitational search algorithm (IBGSA) by rebuilding its transfer function and applied IBGSA on the new CSP feature subspace to find the optimal feature set. To validate the proposed method, we conducted experiments on three public BCI datasets and performed a numerical analysis of the proposed algorithm for MI classification. The accuracies were comparable to those reported in related studies and the presented model outperformed other methods in literature on the same underlying data.

This paper presents a method for adapting the cost function in the Monge–Kantorovich Problem (MKP) to a classification task. More specifically, we introduce a criterion that allows to learn a cost function which tends to produce large distance values for elements belonging to different classes and small distance values for elements belonging to the same class. Under some additional constraints (one of them being the well-known Monge condition), we show that the optimization of this criterion writes as a linear programming problem. Experimental results on synthetic data show that the output optimal cost function provides good retrieval performances in the presence of two types of perturbations commonly found in histograms. When compared to a set of various commonly used cost functions, our optimal cost function performs as good as the best cost function of the set, which shows that it can adapt well to the task. Promising results are also obtained on real data for two-class image retrieval based on grayscale intensity histograms.

A simple model to handle the flow of people in emergency evacuation situations is considered: at every point x, the velocity U(x) that individuals at x would like to realize is given. Yet, the incompressibility constraint prevents this velocity field to be realized and the actual velocity is the projection of the desired one onto the set of admissible velocities. Instead of looking at a microscopic setting (where individuals are represented by rigid discs), here the macroscopic approach is investigated, where the unknown is a density ρ(t,x). If a gradient structure is given, say U = -∇D where D is, for instance, the distance to the exit door, the problem is presented as a Gradient Flow in the Wasserstein space of probability measures. The functional which gives the Gradient Flow is neither finitely valued (since it takes into account the constraints on the density), nor geodesically convex, which requires for an ad hoc study of the convergence of a discrete scheme.

We consider a two-species system of nonlocal interaction PDEs modeling the swarming dynamics of predators and prey, in which all agents interact through attractive/repulsive forces of gradient type. In order to model the predator–prey interaction, we prescribed proportional potentials (with opposite signs) for the cross-interaction part. The model has a particle-based discrete (ODE) version and a continuum PDE version. We investigate the structure of particle stationary solution and their stability in the ODE system in a systematic form, and then consider simple examples. We then prove that the stable particle steady states are locally stable for the fully nonlinear continuum model, provided a slight reinforcement of the particle condition is required. The latter result holds in one space dimension. We complement all the particle examples with simple numerical simulations, and we provide some two-dimensional examples to highlight the complexity in the large time behavior of the system.

In the present paper, we deal with an optimal control problem related to a model in population dynamics; more precisely, the goal is to modify the behavior of a given density of individuals via another population of agents interacting with the first. The cost functional to be minimized to determine the dynamics of the second population takes into account the desired target or configuration to be reached as well as the quantity of control agents. Several applications may fall into this framework, as for instance driving a mass of pedestrian in (or out of) a certain location; influencing the stock market by acting on a small quantity of key investors; controlling a swarm of unmanned aerial vehicles by means of few piloted drones.

We develop a method to compute effectively the Young measures associated to sequences of numerical solutions of the compressible Euler system. Our approach is based on the concept of ${𝒦}$-convergence adapted to sequences of parameterized measures. The convergence is strong in space and time (a.e. pointwise or in certain $L^q$ spaces) whereas the measures converge narrowly or in the Wasserstein distance to the corresponding limit.

In this work, we demonstrate that a functional modeling the self-aggregation of stochastically distributed lipid molecules can be obtained as the $\mathrm{\Gamma }$-limit of a family of discrete energies driven by a sequence of independent and identically distributed random variables. These random variables are intended to describe the asymptotic behavior of lipid molecules that satisfy an incompressibility condition. The discrete energy keeps into account the interactions between particles. We resort to transportation maps to compare functionals defined on discrete and continuous domains, and we prove that, under suitable conditions on the scaling of these maps as the number of random variables increases, the limit functional features an interfacial term with a Wasserstein-type penalization.

Speech emotion analysis plays an important role in English teaching by analyzing the reading state of students. Teachers can dynamically adjust the teaching content according to the emotional feedback of students and improve the teaching quality of the school. Due to unstable student emotions and background noise, the accuracy of speech emotion recognition is constrained. Although multimodal data can alleviate the deficiency of a single modality, collecting and annotating multimodal samples requires a significant amount of resources. To resolve this issue, this paper proposes a novel multimodal sentiment analysis framework based on domain adaptive learning mechanisms to assist English teaching. We construct a novel multi-task variation autoencoder framework in which we simultaneously complete reconstruction and classification tasks. To improve speech emotion recognition performance, we introduce domain adaptive learning based on the Wasserstein distance between two variational hidden layers from the video domain (source domain) and speech domain (target domain). To validate the effectiveness of our proposed model, we conducted extensive comparative experiments on two public datasets and a self-built English oral dataset. All experimental results indicate that domain adaptation learning mechanisms can effectively improve the recognition performance of the target domain. On the self-built dataset for English teaching, the proposed model achieves higher performance compared to other deep models.

We consider the problem of finding a consistent upper price bound for exotic options whose payoff depends on the stock price at two different predetermined time points (e.g. Asian option), given a finite number of observed call prices for these maturities. A model-free approach is used, only taking into account that the (discounted) stock price process is a martingale under the no-arbitrage condition. In case the payoff is directionally convex we obtain the worst case marginal pricing measures. The speed of convergence of the upper price bound is determined when the number of observed stock prices increases. We illustrate our findings with some numerical computations.

Strategies for selecting base probability measures that both respect market valuations of traded cash flows and simultaneously monitor departures from physical measures are suggested, proposed and implemented in an illustrative option hedging context. Mathematical objectives for financial decision making that are financial are advocated and developed using the theory of nonlinear expectations. Financial objectives should equate the additional value of a dollar to a dollar. More exactly, the value of a claim paying out an additional dollar, should exceed the original claim value by a dollar.

We prove that distribution-dependent (also called McKean–Vlasov) stochastic delay equations of the form

In this paper, we establish some estimates related to the Gaussian densities and to Hermite polynomials in order to obtain an almost sure estimate for each term of the Itô-Wiener expansion of the self-intersection local times of the Brownian motion. In dimension $d\ge 4$ the self-intersection local times of the Brownian motion can be considered as a family of measures on the classical Wiener space. We provide some asymptotics relative to these measures. Finally, we try to estimate the quadratic Wasserstein distance between these measures and the Wiener measure.

In this paper, we study the existence and continuous dependence on coefficients of mild solutions for first-order McKean–Vlasov integrodifferential equations with delay driven by a cylindrical Wiener process using resolvent operator theory and Wasserstein distance. Under the situation that the nonlinear term depends on the probability distribution of the state, the existence and uniqueness of solutions are established. An example illustrating the general results is included.

We consider the geometric ergodicity of the Stochastic Gradient Langevin Dynamics (SGLD) algorithm under nonconvexity settings. Via the technique of reflection coupling, we prove the Wasserstein contraction of SGLD when the target distribution is log-concave only outside some compact sets. The time discretization and the minibatch in SGLD introduce several difficulties when applying the reflection coupling, which are addressed by a series of careful estimates of conditional expectations. As a direct corollary, the SGLD with constant step size has an invariant distribution and we are able to obtain its geometric ergodicity in terms of $W_1$ distance. The generalization to non-gradient drifts is also included.

This work studies the convergence and finite sample approximations of entropic regularized Wasserstein distances in the Hilbert space setting. Our first main result is that for Gaussian measures on an infinite-dimensional Hilbert space, convergence in the 2-Sinkhorn divergence is strictly weaker than convergence in the exact 2-Wasserstein distance. Specifically, a sequence of centered Gaussian measures converges in the 2-Sinkhorn divergence if the corresponding covariance operators converge in the Hilbert–Schmidt norm. This is in contrast to the previous known result that a sequence of centered Gaussian measures converges in the exact 2-Wasserstein distance if and only if the covariance operators converge in the trace class norm. In the reproducing kernel Hilbert space (RKHS) setting, the kernel Gaussian–Sinkhorn divergence, which is the Sinkhorn divergence between Gaussian measures defined on an RKHS, defines a semi-metric on the set of Borel probability measures on a Polish space, given a characteristic kernel on that space. With the Hilbert–Schmidt norm convergence, we obtain dimension-independent convergence rates for finite sample approximations of the kernel Gaussian–Sinkhorn divergence, of the same order as the Maximum Mean Discrepancy. These convergence rates apply in particular to Sinkhorn divergence between Gaussian measures on Euclidean and infinite-dimensional Hilbert spaces. The sample complexity for the 2-Wasserstein distance between Gaussian measures on Euclidean space, while dimension-dependent, is exponentially faster than the worst case scenario in the literature.

Zero-shot sketch-based image retrieval (ZSSBIR) aims at retrieving natural images given free hand-drawn sketches that may not appear during training. Previous approaches used semantic aligned sketch-image pairs or utilized memory expensive fusion layer for projecting the visual information to a low-dimensional subspace, which ignores the significant heterogeneous cross-domain discrepancy between highly abstract sketch and relevant image. This may yield poor performance in the training phase. To tackle this issue and overcome this drawback, we propose a Wasserstein distance-based cross-modal semantic network (WAD-CMSN) for ZSSBIR. Specifically, it first projects the visual information of each branch (sketch, image) to a common low-dimensional semantic subspace via Wasserstein distance in an adversarial training manner. Furthermore, a novel identity matching loss is employed to select useful features, which can not only capture complete semantic knowledge, but also alleviate the over-fitting phenomenon caused by the WAD-CMSN model. Experimental results on the challenging Sketchy (Extended) and TU-Berlin (Extended) datasets indicate the effectiveness of the proposed WAD-CMSN model over several competitors.

We consider non-decreasing entropy solutions to 1-d scalar conservation laws and show that the spatial derivatives of such solutions satisfy a contraction property with respect to the Wasserstein distance of any order. This result extends the L¹-contraction property shown by Kružkov.

This paper is devoted to the study of diagonal hyperbolic systems in one space dimension, with cumulative distribution functions or, more generally, nonconstant monotonic bounded functions as initial data. Under a uniform strict hyperbolicity assumption on the characteristic fields, we construct a multi-type version of the sticky particle dynamics and we obtain the existence of global weak solutions via a compactness argument. We then derive a ${\mathrm{\text{L}}}^p$ stability estimate on the particle system which is uniform in the number of particles. This allows us to construct nonlinear semigroups solving the system in the sense of Bianchini and Bressan [Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. of Math. (2) 161(1) (2005) 223–342]. We also obtain that these semigroup solutions satisfy a stability estimate in Wasserstein distances of all order, which extends the classical ${\mathrm{\text{L}}}^1$ estimate and generalizes to diagonal systems a result by Bolley, Brenier and Loeper [Contractive metrics for scalar conservation laws, J. Hyperbolic Differ. Equ.2(1) (2005) 91–107] in the scalar case. Our results are established without any smallness assumption on the variation of the data, and we only require the characteristic fields to be Lipschitz continuous and the system to be uniformly strictly hyperbolic.