On a general matrix valued unbalanced optimal transport and its fully discretization: dynamic formulation and convergence framework

In this work, we present a rather general class of transport distances over the space of positive semidefinite matrix valued Radon measures, called the weighted Wasserstein Bures distance, and consider the convergence property of their fully discretized counterparts. These distances are defined via a generalization of Benamou Brenier formulation of the quadratic optimal transport, based on a new weighted action functional and an abstract matricial continuity equation. It gives rise to a convex optimization problem. We shall give a complete characterization of its minimizer (i.e., the geodesic) and discuss some topological and geometrical properties of these distances. Some recently proposed models: the interpolation distance by Chen et al. [18] and the Kantorovich Bures distance by Brenier et al. [11], as well as the well studied Wasserstein Fisher Rao distance [43, 19, 40], fit in our model. The second part of this work is devoted to the numerical analysis of the fully discretization of the new transport model. We reinterpret the convergence framework proposed very recently by Lavenant [41] for the quadratic optimal transport from the perspective of Lax equivalence theorem and extend it to our general problem. In view of this abstract framework, we suggest a concrete fully discretized scheme inspired by the finite element theory, and show the unconditional convergence under mild assumptions. In particular, these assumptions are removed in the case of Wasserstein Fisher Rao distance due to the existence of a static formulation.Comment: 47 pages, raw draf

Super-resolution in recovering embedded electromagnetic sources in high contrast media

The purpose of this work is to provide a rigorous mathematical analysis of the expected super-resolution phenomenon in the time-reversal imaging of electromagnetic (EM) radiating sources embedded in a high contrast medium. It is known that the resolution limit is essentially determined by the sharpness of the imaginary part of the EM Green's tensor for the associated background. We first establish the close connection between the resolution and the material parameters and the resolvent of the electric integral operator, via the Lippmann-Schwinger representation formula. We then present an insightful characterization of the spectral structure of the integral operator for a general bounded domain and derive the pole-pencil decomposition of its resolvent in the high contrast regime. For the special case of a spherical domain, we provide some quantitative asymptotic behavior of the eigenvalues and eigenfunctions. These mathematical findings shall enable us to provide a concise and rigorous illustration of the super-resolution in the EM source reconstruction in high contrast media. Some numerical examples are also presented to verify our main theoretical results.Comment: 31 pages, 6 figure

Mathematical analysis of electromagnetic plasmonic metasurfaces

We study the anomalous electromagnetic scattering in the homogenization regime, by a subwavelength thin layer of periodically distributed plasmonic nanoparticles on a perfect conducting plane. By using layer potential techniques, we derive the asymptotic expansion of the electromagnetic field away from the thin layer and quantitatively analyze the field enhancement due to the mixed collective plasmonic resonances, which can be characterized by the spectra of periodic Neumann-Poincar\'{e} type operators. Based on the asymptotic behavior of the scattered field in the macroscopic scale, we further demonstrate that the optical effect of this thin layer can be effectively approximated by a Leontovich boundary condition, which is uniformly valid no matter whether the incident frequency is near the resonant range but varies with the magnetic property of the plasmonic nanoparticles. The quantitative approximation clearly shows the blow-up of the field energy and the conversion of polarization when resonance occurs, resulting in a significant change of the reflection property of the conducting plane. These results confirm essential physical changes of electromagnetic metasurface at resonances mathematically, whose occurrence was verified earlier for the acoustic case \cite{ammari2017bubble} and the transverse magnetic case \cite{ammari2016mathematical}.Comment: 34 page

Fano resonances in all-dielectric electromagnetic metasurfaces

We are interested in the resonant electromagnetic (EM) scattering by the all-dielectric metasurfaces made of a two-dimensional lattice of nanoparticles with high refractive indices. It has been shown that a single high-index nanoresonator can couple with the incident wave and exhibit a strong magnetic dipole response, while the recent physical research reveals that when the particles are arranged in a certain configuration, they may have different anomalous scattering effects in the macroscopic scale, compared to the single particle case. In this work, we shall develop a rigorous mathematical framework for analyzing the resonant behaviors of dielectric metasurfaces. We start with the characterization of subwavelength scattering resonances in this periodic setting and their asymptotic expansions. Then we show that the real resonances always exist below the essential spectrum of the Maxwell operator, and that they are the simple poles of the scattering resolvent with the exponentially decaying resonant modes. We also discuss the implications of the symmetry of the metasurface on the subwavelength band functions and the associated eigenfunctions by the group theory. For the symmetric metasurfaces with the normal incidence, we use the variational method to show the existence of the embedded eigenvalues (i.e., the real subwavelength resonance embedded in the continuous essential spectrum). Furthermore, we slightly break the symmetry by an antisymmetric deformation field and the non-normal incidence, and prove that in this case, the metasurface can present the Fano-type reflection and transmission anomalies

Pretraining Language Models with Text-Attributed Heterogeneous Graphs

In many real-world scenarios (e.g., academic networks, social platforms), different types of entities are not only associated with texts but also connected by various relationships, which can be abstracted as Text-Attributed Heterogeneous Graphs (TAHGs). Current pretraining tasks for Language Models (LMs) primarily focus on separately learning the textual information of each entity and overlook the crucial aspect of capturing topological connections among entities in TAHGs. In this paper, we present a new pretraining framework for LMs that explicitly considers the topological and heterogeneous information in TAHGs. Firstly, we define a context graph as neighborhoods of a target node within specific orders and propose a topology-aware pretraining task to predict nodes involved in the context graph by jointly optimizing an LM and an auxiliary heterogeneous graph neural network. Secondly, based on the observation that some nodes are text-rich while others have little text, we devise a text augmentation strategy to enrich textless nodes with their neighbors' texts for handling the imbalance issue. We conduct link prediction and node classification tasks on three datasets from various domains. Experimental results demonstrate the superiority of our approach over existing methods and the rationality of each design. Our code is available at https://github.com/Hope-Rita/THLM.Comment: Accepted by EMNLP 2023 Finding

NADiffuSE: Noise-aware Diffusion-based Model for Speech Enhancement

The goal of speech enhancement (SE) is to eliminate the background interference from the noisy speech signal. Generative models such as diffusion models (DM) have been applied to the task of SE because of better generalization in unseen noisy scenes. Technical routes for the DM-based SE methods can be summarized into three types: task-adapted diffusion process formulation, generator-plus-conditioner (GPC) structures and the multi-stage frameworks. We focus on the first two approaches, which are constructed under the GPC architecture and use the task-adapted diffusion process to better deal with the real noise. However, the performance of these SE models is limited by the following issues: (a) Non-Gaussian noise estimation in the task-adapted diffusion process. (b) Conditional domain bias caused by the weak conditioner design in the GPC structure. (c) Large amount of residual noise caused by unreasonable interpolation operations during inference. To solve the above problems, we propose a noise-aware diffusion-based SE model (NADiffuSE) to boost the SE performance, where the noise representation is extracted from the noisy speech signal and introduced as a global conditional information for estimating the non-Gaussian components. Furthermore, the anchor-based inference algorithm is employed to achieve a compromise between the speech distortion and noise residual. In order to mitigate the performance degradation caused by the conditional domain bias in the GPC framework, we investigate three model variants, all of which can be viewed as multi-stage SE based on the preprocessing networks for Mel spectrograms. Experimental results show that NADiffuSE outperforms other DM-based SE models under the GPC infrastructure. Audio samples are available at: https://square-of-w.github.io/NADiffuSE-demo/