Dimensional Kernels Research Articles

Matrix-valued Allen–Cahn equation and the Keller–Rubinstein–Sternberg problem

In this paper, we consider the sharp interface limit of a matrix-valued Allen–Cahn equation, which takes the form: $$\begin{aligned} \partial _t{\textbf{A}}=\Delta {\textbf{A}}-\varepsilon ^{-2}({\textbf{A}}{\textbf{A}}^{\textrm{T}}{\textbf{A}}-{\textbf{A}})\quad \text {with}\quad {\textbf{A}}:\Omega \subset {\mathbb {R}}^m\rightarrow {\mathbb {R}}^{n\times n}. \end{aligned}$$ We show that the sharp interface system is a two-phases flow system: the interface evolves according to the motion by mean curvature; in the two bulk phase regions, the solution obeys the heat flow of harmonic maps with values in $$O^+(n)$$ and $$O^-(n)$$ (represent the sets of $$n\times n$$ orthogonal matrices with determinant $$+1$$ and $$-1$$ respectively); on the interface, the phase matrices on two sides satisfy a novel mixed boundary condition. The above result provides a solution to the Keller–Rubinstein–Sternberg’s problem in the O(n) setting. Our proof relies on two key ingredients. First, in order to construct the approximate solutions by matched asymptotic expansions, as the standard approach does not seem to work, we introduce the notion of quasi-minimal connecting orbits. They satisfy the usual leading order equations up to some small higher order terms. In addition, the linearized systems around these quasi-minimal orbits needs to be solvable up to some good remainders. These flexibilities are needed for the possible “degenerations” and higher dimensional kernels for the linearized operators on matrix-valued functions due to intriguing boundary conditions at the sharp interface. The second key point is to establish a spectral uniform lower bound estimate for the linearized operator around approximate solutions. To this end, we introduce additional decompositions to reduce the problem into the coercive estimates of several linearized operators for scalar functions and some singular product estimates which are accomplished by exploring special cancellation structures between eigenfunctions of these linearized operators.

First-Kind Boundary Integral Equations for the Dirac Operator in 3-Dimensional Lipschitz Domains

We develop novel first-kind boundary integral equations for Euclidean Dirac operators in 3D Lipschitz domains comprising square-integrable potentials and involving only weakly singular kernels. Generalized Garding inequalities are derived and we establish that the obtained boundary integral operators are Fredholm of index zero. Their finite dimensional kernels are characterized and we show that their dimension is equal to the number of topological invariants of the domain's boundary, in other words to the sum of its Betti numbers. This is explained by the fundamental discovery that the associated bilinear forms agree with those induced by the 2D surface Dirac operators for H-1/2 surface de Rham Hilbert complexes whose underlying inner-products are the non-local inner products defined through the classical single-layer boundary integral operators for the Laplacian. Decay conditions for well-posedness in natural energy spaces of the Dirac system in unbounded exterior domains are also presented.

The bounds of the odd dimensional Clifford-Fourier kernels

The bounds of the odd dimensional Clifford-Fourier kernels

On principal graphical models with application to gene network

The principal graphical model is introduced by incorporating the ideas from the linear sufficient dimension reduction (SDR) methods such as the sliced inverse regression and sliced average variance estimation to a nonparametric graphical model. A nonparametric graphical model is a widely used method to investigate undirected graphs. However, when the number of nodes is large, a nonparametric graphical model suffers from the ‘curse of dimensionality’ because they contain intrinsic high dimensional kernels. The parametric graphical models such as the Gaussian or copula Gaussian graphical models are also well known for their intuitive structure and interpretability. However, they hinge on strong parametric model assumptions. The principal graphical model applies well-known linear SDR techniques to the nonparametric graphical models to enhance performance in high dimensional networks, avoid model assumptions, and maintain interpretability. We use components of linear SDR as modules and implement them in the (p2) pairs of variables in the network to evaluate conditional independence. In the numerical experiment, our methods have competitive accuracy in both low and high-dimensional settings. Our methods are applied to the DREAM 4 challenge gene network dataset and they work well in high dimensional settings with a limited number of observations.

Unbounded Toeplitz Operators with Rational Symbols

Unbounded (and bounded) Toeplitz operators (TO) with rational symbols are analysed in detail showing that they are densely defined closed and have finite dimensional kernels and deficiency spaces. The latter spaces as well as the domains, ranges, spectral and Fredholm points are determined. In particular, in the symmetric case, i.e., for a real rational symbol the deficiency spaces and indices are explicitly available. The concluding section gives a brief overview on the research on unbounded TO in order to locate the present contribution. Regarding properties of unbounded TO in general, it furnishes some new results recalling the close relationship to Wiener-Hopf operators and, in case of semiboundedness, to singular operators of Hilbert transformation type. Specific symbols considered in the literature admit further analysis. Some conclusions are drawn for semibounded integrable and real square-integrable symbols. There is an approach to semibounded TO, which starts from closable semibounded forms related to a Toeplitz matrix. The Friedrichs extension of the TO associated with such a form is studied. Finally, analytic TO and Toeplitz-like operators are briefly examined, which in general differ from the TO treated here.

MS-Net: A lightweight separable ConvNet for multi-dimensional image processing

As the core technology of deep learning, convolutional neural networks have been widely applied in a variety of computer vision tasks and have achieved state-of-the-art performance. However, it’s difficult and inefficient for them to deal with high dimensional image signals due to the dramatic increase of training parameters. In this paper, we present a lightweight and efficient MS-Net for the multi-dimensional(MD) image processing, which provides a promising way to handle MD images, especially for devices with limited computational capacity. It takes advantage of a series of one dimensional convolution kernels and introduces a separable structure in the ConvNet throughout the learning process to handle MD image signals. Meanwhile, multiple group convolutions with kernel size 1 × 1 are used to extract channel information. Then the information of each dimension and channel is fused by a fusion module to extract the complete image features. Thus the proposed MS-Net significantly reduces the training complexity, parameters and memory cost. The proposed MS-Net is evaluated on both 2D and 3D benchmarks CIFAR-10, CIFAR-100 and KTH. Extensive experimental results show that the MS-Net achieves competitive performance with greatly reduced computational and memory cost compared with the state-of-the-art ConvNet models.

Unbounded Wiener\u2013Hopf Operators and Isomorphic Singular Integral Operators

Some basics of a theory of unbounded Wiener–Hopf operators (WH) are developed. The alternative is shown that the domain of a WH is either zero or dense. The symbols for non-trivial WH are determined explicitly by an integrability property. WH are characterized by shift invariance. We study in detail WH with rational symbols showing that they are densely defined, closed and have finite dimensional kernels and deficiency spaces. The latter spaces as well as the domains, ranges, spectral and Fredholm points are explicitly determined. Another topic concerns semibounded WH. There is a canonical representation of a semibounded WH using a product of a closable operator and its adjoint. The Friedrichs extension is obtained replacing the operator by its closure. The polar decomposition gives rise to a Hilbert space isomorphism relating a semibounded WH to a singular integral operator of Hilbert transformation type. This remarkable relationship, which allows to transfer results and methods reciprocally, is new also in the thoroughly studied case of bounded WH.

Fluctuations of the arctic curve in the tilings of the Aztec diamond on restricted domains

We consider uniform random domino tilings of the restricted Aztec diamond which is obtained by cutting off an upper triangular part of the Aztec diamond by a horizontal line. The restriction line asymptotically touches the arctic circle that is the limit shape of the north polar region in the unrestricted model. We prove that the rescaled boundary of the north polar region in the restricted domain converges to the Airy$_2$ process conditioned to stay below a parabola with explicit continuous statistics and the finite dimensional distribution kernels. The limit is the hard-edge tacnode process which was first discovered in the framework of non-intersecting Brownian bridges. The proof relies on a random walk representation of the correlation kernel of the non-intersecting line ensemble which corresponds to a random tiling.

Solvability for Multi-Point BVP of Nonlinear Fractional Differential Equations at Resonance with Three Dimensional Kernels

This work deals with the BVP multi-point existence of solutions of a nonlinear fractional differential equations at resonance, where the kernel’s dimension of the fractional differential operator is equal to three. Our results are based on Mawhin’s theory of coincidence. As application, we give an example to illustrate our results.

Lower dimensional kernels for video discriminators

This work presents an analysis of the discriminators used in Generative Adversarial Networks (GANs) for Video. We show that unconstrained video discriminator architectures induce a loss surface with high curvature which make optimization difficult. We also show that this curvature becomes more extreme as the maximal kernel dimension of video discriminators increases. With these observations in hand, we propose a methodology for the design of a family of efficient Lower-Dimensional Video Discriminators for GANs (LDVD-GANs). The proposed methodology improves the performance and efficiency of video GAN models it is applied to and demonstrates good performance on complex and diverse datasets such as UCF-101. In particular, we show that LDVDs can double the performance of Temporal-GANs and provide for state-of-the-art performance on a single GPU using the proposed methodology.

Human Activity Recognition

The topic of Human activity recognition (HAR) is a prominent research area topic in the field of computer vision and image processing area. It has empowered state-of-art application in multiple sectors, surveillance, digital entertainment and medical healthcare. It is interesting to observe and intriguing to predict such kind of movements. Several sensor-based approaches have also been introduced to study and predict human activities such accelerometer, gyroscope, etc., it has its own advantages and disadvantages.[10] In this paper, an intelligent human activity recognition system is developed. Convolutional neural network (CNN) with spatiotemporal three dimensional (3D) kernels are trained using Kinetics data set which has 400 classes that depicts activities of humans in their everyday life and work and consist of 400 and more videos for each class. The 3D CNN model used in this model is RESNET-34. The videos were temporally cut down and last around tenth of a second. The trained model show satisfactory performance in all stages of training, testing. Finally the results show promising activity recognition of over 400 human actions.

On the problem of existence in principal value of a Calderón–Zygmund operator on a space of non‐homogeneous type

In this paper, we study the relationship between two fundamental regularity properties of an s-dimensional Calderón–Zygmund operator (CZO) acting on a Borel measure μ in R d , with s ∈ ( 0 , d ) . In the classical case when s = d and μ is equal to the Lebesgue measure, Calderón and Zygmund showed that if a CZO is bounded in L 2 , then the principal value integral exists almost everywhere. However, there are by now several examples showing that this implication may fail for lower dimensional kernels and measures, even when the CZO has a homogeneous kernel consisting of spherical harmonics. We introduce sharp geometric conditions on μ, in terms of certain scaled transportation distances, which ensure that an extension of the Calderón–Zygmund theorem holds. These conditions are necessary and sufficient in the cases of the Riesz transform and the Huovinen transform. Our techniques build upon prior work by Mattila and Verdera, and incorporate the machinery of symmetric measures, introduced to the area by Mattila.

Volatility Model Calibration With Convolutional Neural Networks

We use a supervised deep convolution neural network to replicate the calibration of the Heston model to equity volatility surfaces. For this purpose we treat the implied volatility surface together with some auxiliary data, namely the strikes and moneyness of the corresponding options and the equity forwards, as a 3-dimensional input tensor for the neural network, in analogy to a colour channel image representation like the RGB. To extract the main features of the input data we are using inception layers with (1;1), (1;3) and (2;1) dimensional kernels. The specific choice is motivated by the no-arbitrage conditions on the call price surface. In terms of a local surface modelling the (1;3) filters with different weights can model the position, slope and curvature in the moneyness direction while the (2;1) filters can model Position and slope in the maturity direction. The neural network has been implemented using the open source library tensorflow.

Fast SAR target recognition based on random convolution features and ensemble extreme learning machines

Deep convolution neural network has demonstrated excellent performance in target detection and recognition tasks. However, few training samples and optimization design of deep models are two main problems to be solved when applied to SAR target recognition. This paper proposes an algorithm for SAR target recognition by combination of two dimensional random convolution features and ensemble extreme learning machines. Firstly, two dimensional random convolution kernels with different widths are generated, and convolution and pooling operations are performed in input image to extract random convolution feature vectors. Secondly, random samplings based on ensemble learning theory are done for extracted feature vectors to improve generalization performance of classifier and reduce training time, and base classifiers are then trained by extreme learning machines (ELM). Finally, majority vote method is adopted to combine the classification results of base classifiers. SAR target recognition experiments based on MSTAR database were performed, and experimental results demonstrate that, training time has dropped by ten times due to fast training capability of ELM, and the proposed algorithm achieves comparable classification performance with deep-learning-based methods which use data augmentation and multiple convolution layers. The proposed algorithm has the advantages of easy implementation and fewer adjustable parameters, and improves classifier's generalization performance through adoption of ensemble learning ideas.

Clifford–Fourier transform on hyperbolic space

In this paper, we introduce a new generalization of the Helgason–Fourier transform using the angular Dirac operator on both the hyperboloid and unit ball models. The explicit integral kernels of even dimension are derived. Furthermore, we establish the formal generating function of the even dimensional kernels. In the computations, fractional integration plays a key unifying role. Copyright © 2016 John Wiley & Sons, Ltd.

A new construction of the Clifford-Fourier kernel

In this paper, we develop a new method based on the Laplace transform to study the Clifford-Fourier transform. First, the kernel of the Clifford-Fourier transform in the Laplace domain is obtained. When the dimension is even, the inverse Laplace transform may be computed and we obtain the explicit expression for the kernel as a finite sum of Bessel functions. We equally obtain the plane wave decomposition and find new integral representations for the kernel in all dimensions. Finally we define and compute the formal generating function for the even dimensional kernels.

Microscopic Images Restoration of Chinese Herbal Medicine Using the TV Model

Chinese herbal medicine (CHM) samples defiled, extraction of microscopic images, collection site light not equal and images collected uneven illumination point which can make the acquisition of CHM microscopic images sub- ject to different unfavorable factors. Those often make microscopic images of CHM become blurred or introduce noise. In this paper, some restoration methods for the microscopic images of CHM using the total variation (TV) model are put forward. Firstly, the microscopic images of CHM are blurred utilizing the two dimensional different Gaussian convolution kernels and different Gauss white noise are added, and then the damaged images are restored by using the improved TV model. The TV restoration results are compared with the results using Wiener filtering, median filtering method etc. The theoretical analysis and experimental results show that the novel method is effectively able to restore image, and maintain the image detail information.

Connectivity Modeling System: A probabilistic modeling tool for the multi-scale tracking of biotic and abiotic variability in the ocean

Pelagic organisms' movement and motion of buoyant particles are driven by processes operating across multiple, spatial and temporal scales. We developed a probabilistic, multi-scale model, the Connectivity Modeling System (CMS), to gain a mechanistic understanding of dispersion and migration processes in the ocean. The model couples offline a new nested-grid technique to a stochastic Lagrangian framework where individual variability is introduced by drawing particles' attributes at random from specified probability distributions of traits. This allows 1) to track seamlessly a large number of both actively swimming and inertial particles over multiple, independent ocean model domains and 2) to generate ensemble forecasts or hindcasts of the particles' three dimensional trajectories, dispersal kernels, and transition probability matrices used for connectivity estimates. In addition, CMS provides Lagrangian descriptions of oceanic phenomena (advection, dispersion, retention) and can be used in a broad range of oceanographic applications, from the fate of pollutants to the pathways of water masses in the global ocean. Here we describe the CMS modular system where particle behavior can be augmented with specific features, and a parallel module implementation simplifies data management and CPU intensive computations associated with solving for the tracking of millions of active particles. Some novel features include on-the-fly data access of operational hydrodynamic models, individual particle variability and inertial motion, and multi-nesting capabilities to optimize resolution. We demonstrate the performance of the interpolation algorithm by testing accuracy in tracing the flow stream lines in both time and space and the efficacy of probabilistic modeling in evaluating the bio-physical coupling against empirical data. Finally, following recommended practices for the development of community models, we provide an open source code with a series of coupled standalone, optional modules detailed in a user's guide.

On variational eigenvalue approximation of semidefinite operators

Eigenvalue problems for semidefinite operators with infinite dimensional kernels appear for instance in electromagnetics. Variational discretizations with edge elements have long been analyzed in terms of a discrete compactness property. As an alternative, we show here how the abstract theory can be developed in terms of a geometric property called the vanishing gap condition. This condition is shown to be equivalent to eigenvalue convergence and intermediate between two different discrete variants of Friedrichs estimates. Next we turn to a more practical means of checking these properties. We introduce a notion of compatible operator and show how the previous conditions are equivalent to the existence of such operators with various convergence properties. In particular the vanishing gap condition is shown to be equivalent to the existence of compatible operators satisfying an Aubin-Nitsche estimate. Finally we give examples demonstrating that the implications not shown to be equivalences, indeed are not.

Reconstructing Brain White Matter Pathways with Diffusion Tensor MRI Using Kernel-Based Diffusion Simulations

A novel algorithm for approximating anatomical brain connectivity in vivo is presented using diffusion tensor magnetic resonance imaging (DT-MRI). This technique relies on simulating diffusion process within a series of overlapping three dimensional diffusion kernels that cover only a small portion of the human brain volume. The shape of the anisotropic diffusion represented by diffusion fronts is used to estimate the directional organization of the underlying white matter fiber tracts. The proposed algorithm is tested on both simulated and real DT-MRI data. The demonstration shows that the synthetic tracts are accurately replicated, while various examples of white matter fiber pathways can be reconstructed as well, with assigned connectivity indices showing uncertainty. Several features of the algorithm are elucidated by the tracking experiments, including its capability of handling fiber branching and crossing, and robustness to noise. Impact of thresholding settings and the kernel size on performance of the algorithm is also analyzed.