Kernel Coefficients Research Articles

Heat kernel coefficients on Kähler manifolds

Polterovich proved a remarkable closed formula for heat kernel coefficients of the Laplace operator on compact Riemannian manifolds involving powers of Laplacians acting on the distance function. In the case of Kähler manifolds, we prove a combinatorial formula for powers of the complex Laplacian and use it to derive an explicit graph-theoretic formula for the numerics in heat coefficients as a linear combination of metric jets based on Polterovich’s formula.

Optimization of parameters for image denoising algorithm pertaining to generalized Caputo-Fabrizio fractional operator

The aim of the present paper is to optimize the values of different parameters related to the image denoising algorithm involving Caputo Fabrizio fractional integral operator of non-singular type with the Mittag-Leffler function in generalized form. The algorithm aims to find the coefficients of a kernel to remove the noise from images. The optimization of kernel coefficients are done on the basis of different numerical parameters like Mean Square Error (MSE), Peak Signal-to-Noise Ratio (PSNR), Structure Similarity Index measure (SSIM) and Image Enhancement Factor (IEF). The performance of the proposed algorithm is investigated through above-mentioned numeric parameters and visual perception with the other prevailed algorithms. Experimental results demonstrate that the proposed optimized kernel based on generalized fractional operator performs favorably compared to state of the art methods. The uniqueness of the paper is to highlight the optimized values of performance parameters for different values of fractional order. The novelty of the presented work lies in the development of a kernel utilizing coefficients from a fractional integral operator, specifically involving the Mittag-Leffler function in a more generalized form.

Tunable optical matrix convolution of 20-Gbit/s QPSK 2-D data with a kernel using optical wave mixing.

Compared to its electronic counterpart, optically performed matrix convolution can accommodate phase-encoded data at high rates while avoiding optical-to-electronic-to-optical (OEO) conversions. We experimentally demonstrate a reconfigurable matrix convolution of quadrature phase-shift keying (QPSK)-encoded input data. The two-dimensional (2-D) input data is serialized, and its time-shifted replicas are generated. This 2-D data is convolved with a 1-D kernel with coefficients, which are applied by adjusting the relative phase and amplitude of the kernel pumps. Time-shifted data replicas (TSDRs) and kernel pumps are coherently mixed using nonlinear wave mixing in a periodically poled lithium niobate (PPLN) waveguide. To show the tunability and reconfigurability of this approach, we vary the kernel coefficients, kernel sizes (e.g., 2 × 1 or 3 × 1), and input data rates (e.g., 6-20 Gbit/s). The convolution results are verified to be error-free under an applied: (a) 2 × 1 kernel, resulting in a 16-quadrature amplitude modulation (QAM) output with an error vector magnitude (EVM) of ∼5.1-8.5%; and (b) 3 × 1 kernel, resulting in a 64-QAM output with an EVM of ∼4.9-5.5%.

Thermal Casimir effect for a Dirac field on flat space with a nontrivial circular boundary condition

This work investigates the thermal Casimir effect associated with a massive spinor field defined on a four-dimensional flat space with a circularly compactified spatial dimension whose periodicity is oriented along a vector in the xy plane. We employ the generalized zeta function method to establish a finite definition for the vacuum free energy density. This definition conveniently separates into the zero-temperature Casimir energy density and additional terms accounting for temperature corrections. The structure of existing divergences is analyzed from the asymptotic behavior of the spinor heat kernel function and removed in the renormalization by subtracting scheme. The only non-null heat coefficient is the one associated with the Euclidean divergence. We also address the need for a finite renormalization to treat the ambiguity in the zeta function regularization prescription associated with this Euclidean heat kernel coefficient and ensure that the renormalization procedure is unique. The high- and low-temperature asymptotic limits are also explored. In particular, we explicitly show that free energy density lacks a classical limit at high temperatures, and the entropy density agrees with the Nernst heat theorem at low temperatures. Published by the American Physical Society 2024

Selective multiple kernel fuzzy clustering with locality preserved ensemble

Multiple kernel fuzzy clustering (MKFC) has demonstrated promising performance in capturing the non-linear relationships within data. However, its effectiveness relies heavily on the appropriate selection of the fuzzification coefficient and kernel functions. To address this challenge, this paper proposes a novel clustering ensemble approach for improving the robustness and accuracy of the MKFC algorithm. The proposed method employs a multi-objective evolutionary optimization approach to heuristically select the optimal fuzzification and kernel coefficients. By employing the selected coefficients, the MKFC algorithm generates a diverse set of accurate candidate clustering results. Subsequently, a locality preserved ensemble mechanism is introduced to derive the final partition matrix, ensuring that all candidate clusterings within the Pareto non-dominated set contribute to the final consensus matrix. Moreover, this mechanism incorporates the knowledge about the locality of the dataset via a graph regularization term, thereby further enhancing the clustering performance. Comprehensive experiments conducted on widely adopted benchmark datasets demonstrate the superiority of the proposed method over the state-of-the-art approaches.

Heat kernel coefficients for massive gravity

Heat kernel coefficients for massive gravity

Vacuum energy, temperature corrections and heat kernel coefficients in (D + 1)-dimensional spacetimes with nontrivial topology

Vacuum energy, temperature corrections and heat kernel coefficients in (<i>D</i> + 1)-dimensional spacetimes with nontrivial topology

Gradient estimates for mixed local and nonlocal parabolic problems with measure data

We obtain a nonlinear Calderón-Zygmund type estimate for the spatial gradient of SOLA (solutions obtained as limits of approximations) to a mixed local and nonlocal parabolic problem involving measure data. Specifically, we consider a non-homogeneous problem with a source term as a Radon measure, where the leading nonlinear local operator has linear growth and the nonlinearity is allowed to be merely measurable. Moreover, the nonlocal operator is of the type fractional Laplacian with only measurable kernel coefficients. Under some smallness assumption on the local operator, we prove that the gradient of SOLA has the same integrability as that of a fractional maximal function of order 1 of the Radon measure. The proof relies on the use of a new type of maximal functions and novel parabolic tail estimates, which to the best of our knowledge, have not been observed in the previous works. Furthermore, to complement the aforementioned gradient estimates, we prove the existence of SOLA to an initial value problem and derive some gradient estimates with respect to the L1-norm.

One-loop effective action in chiral Einstein–Cartan gravity

In chiral Einstein–Cartan gravity, a new gauge fixing procedure is implemented recently, leading to a very economical perturbation expansion of the action. Using this formulation and the relevant gauge fixing, we develop the ghost Lagrangian on an arbitrary Einstein background using the Becchi-Rouet-Stora-Tyutin (BRST) formalism. The novelty is the appearance of a new term quadratic in the tetrad field. We next compute the heat kernel coefficients and understand the divergences arising in the gravitational one-loop effective action. In our computation, the arising heat kernel coefficients depend only on the self-dual part of the Weyl curvature. We make a comparison between our results and what has been obtained for metric general relativity.

Local momentum space: scalar field and gravity

We use the local momentum space technique to obtain an expansion of the Feynman propagators for scalar field and graviton up to first order in the background curvature. The expressions for the propagators are cross-checked with the past literature as well as with the expressions for the traced heat kernel coefficients. The propagators so obtained are used to compute one-loop divergences in the Vilkovisky-Dewitt’s (VD’s) effective action for a scalar field non-minimally coupled with gravity for an arbitrary spacetime metric background. The VD effective action is then compared with the standard effective action in the limit κ = 0, where in terms of the Planck mass. The comparison yields the important result that taking the limit κ = 0 after computing the VD effective action is not equivalent to computing the VD effective action for the same theory in the absence of gravity.

Improved deep mixed kernel randomized network for wind speed prediction

Abstract Forecasting wind speed is an extremely complicated and challenging problem due to its chaotic nature and its dependence on several atmospheric conditions. Although there are several intelligent techniques in the literature for wind speed prediction, their accuracies are not yet very reliable. Therefore, in this paper, a new hybrid intelligent technique named the deep mixed kernel random vector functional-link network auto-encoder (AE) is proposed for wind speed prediction. The proposed method eliminates manual tuning of hidden nodes with random weights and biases, providing prediction model generalization and representation learning. This reduces reconstruction error due to the exact inversion of the kernel matrix, unlike the pseudo-inverse in a random vector functional-link network, and shortens the execution time. Furthermore, the presence of a direct link from the input to the output reduces the complexity of the prediction model and improves the prediction accuracy. The kernel parameters and coefficients of the mixed kernel system are optimized using a new chaotic sine–cosine Levy flight optimization technique. The lowest errors in terms of mean absolute error (0.4139), mean absolute percentage error (4.0081), root mean square error (0.4843), standard deviation error (1.1431) and index of agreement (0.9733) prove the efficiency of the proposed model in comparison with other deep learning models such as deep AEs, deep kernel extreme learning machine AEs, deep kernel random vector functional-link network AEs, benchmark models such as least square support vector machine, autoregressive integrated moving average, extreme learning machines and their hybrid models along with different state-of-the-art methods.

Sparse self-attention transformer for image inpainting

Learning-based image inpainting methods have made remarkable progress in recent years. Nevertheless, these methods still suffer from issues such as blurring, artifacts, and inconsistent contents. The use of vanilla convolution kernels, which have limited perceptual fields and spatially invariant kernel coefficients, is one of the main causes for these problems. In contrast, the multi-headed attention in the transformer can effectively model non-local relations among input features by generating adaptive attention scores. Therefore, this paper explores the feasibility of employing the transformer model for the image inpainting task. However, the multi-headed attention transformer blocks pose a significant challenge due to their overwhelming computational cost. To address this issue, we propose a novel U-Net style transformer-based network for the inpainting task, called the sparse self-attention transformer (Spa-former). The Spa-former retains the long-range modeling capacity of transformer blocks while reducing the computational burden. It incorporates a new channel attention approximation algorithm that reduces attention calculation to linear complexity. Additionally, it replaces the canonical softmax function with the ReLU function to generate a sparse attention map that effectively excludes irrelevant features. As a result, the Spa-former achieves effective long-range feature modeling with fewer parameters and lower computational resources. Our empirical results on challenging benchmarks demonstrate the superior performance of our proposed Spa-former over state-of-the-art approaches.

An adaptive modelling approach using a novel modified AOA/SVR for prediction of drilling-induced delamination in CFRP/Ti stacks

Delamination is the most critical damage in drilling the CFRP/Ti stacks under the impact of drilling parameters and tool structure, which makes the traditional theoretical or empirical models not have enough accuracy and be time-consuming due to the multi variables, while the machine learning model would suffer the unsuitable hyper-parameters and have a bad accuracy and generalization ability. This paper proposed an adaptive modelling approach to predict the delamination while drilling the CFRP/Ti stacks. This approach adapted the original arithmetic optimization algorithm (AOA) by adding a random disturbance phase to update the penalty coefficient C and the kernel coefficient γ of the support vector regression (SVR) automatically. In the meanwhile, the approach made use of the energy of the 5 stages in drilling the CFRP/Ti stacks and predicted the delamination damage both at the entrance and exit. The modified AOA optimized the training mean squared error(MSE) in predicting the entrance and exit delamination by 10.27 % and 33.63 %, while the accuracy of the proposed model can reach 96.7 % and 97.17 % respectively. The model got validated, and had a comprehensive ability containing the accuracy and generalization ability.

Simple multiple kernel k-means with kernel weight regularization

Multiple kernel clustering (MKC) aims to determine the optimal kernel from several pre-computed basic kernels. Most of MKC algorithms follow a common assumption that the optimal kernel is linearly combined by basic kernels. Based on a min–max framework, a newly proposed MKC method termed simple multiple kernel k-means can acquire a high-quality unified kernel. Although it has achieved promising clustering performance, we have observed that it cannot benefit from any regularization or prior knowledge, thus the learned kernel weight coefficients may be seriously sparse or over-selected. To tackle this issue, we have developed a new algorithm termed simple multiple kernel k-means with kernel weight regularization (SMKKM-KWR), where we introduce average coefficients to avoid too sparse kernel weights. Specially, we add the average kernel coefficients as a regularization term to prevent the learned weight coefficients being far from the average values. After that, an efficient optimization strategy is proposed to solve the new resultant problem. By this way, the unified partition can learn clustering structure from fusion information of all the kernel views, with the goal to reach better clustering performance. Extensive experiments on nine benchmark datasets and four large-scale datasets have demonstrated the effectiveness and efficiency of the proposed algorithm.

A Novel Method of Forecasting Chaotic and Random Wind Speed Regimes Based on Machine Learning with the Evolution and Prediction of Volterra Kernels

This study aims to focus on using the Volterra series and machine learning for forecasting random and chaotic wind speed regimes, since calm weather is mostly noticed at the local site, making dataset selection difficult. A novel method is proposed to predict Volterra kernels up to the third order, using a forward–back propagation neural network with 12-month measurements at Fujairah site (United Arab Emirates). Both daily and monthly wind speed datasets are investigated for forecasting. The three dominant hourly and daily kernels are extracted for each day and each month. Predicted future Volterra kernels are estimated from past values using both statistical analysis and individual neuro networks for each of the Volterra kernel coefficients. Using the evolved Volterra kernels, the hourly and daily wind speeds are forecasted with similar patterns of the measured values. Due to the random nature of wind speed at the local site, a two-layer with four neurons per layer neuro network is used to locate the most variable and intense speed during 8 h in the day. Forecasted wind speed is determined with errors arising from different sources, such as the utilization of only third-order Volterra kernels and the difficulty of machine training of the employed shallow network. Nevertheless, this work depicts a useful algorithm to forecast chaotic and random wind speed regimes. Computational time is a trade of the complexity of Volterra mathematical analysis.

PET Image Reconstruction with Kernel and Kernel Space Composite Regularizer.

Led by the kernelized expectation maximization (KEM) method, the kernelized maximum-likelihood (ML) expectation maximization (EM) methods have recently gained prominence in PET image reconstruction, outperforming many previous state-of-the-art methods. But they are not immune to the problems of non-kernelized MLEM methods in potentially large reconstruction variance and high sensitivity to iteration numbers, and the difficulty in preserving image details and suppressing image variance simultaneously. To solve these problems, this paper derives, using the ideas of data manifold and graph regularization, a novel regularized KEM (RKEM) method with a kernel space composite regularizer for PET image reconstruction. The composite regularizer consists of a convex kernel space graph regularizer that smooths the kernel coefficients, a concave kernel space energy regularizer that enhances the coefficients' energy, and a composition constant that is analytically set to guarantee the convexity of composite regularizer. The composite regularizer renders easy use of PET-only image priors to overcome KEM's difficulty caused by the mismatch of MR prior and underlying PET images. Using this kernel space composite regularizer and the technique of optimization transfer, a globally convergent iterative algorithm is derived for RKEM reconstruction. Tests and comparisons on the simulated and in vivo data are presented to validate and evaluate the proposed algorithm, and demonstrate its better performance and advantages over KEM and other conventional methods.

New covariant Feynman rules for effective field theories

We provide a new and completely general formalism to compute the effective field theory matching contributions from integrating out massive fields in a manifestly gauge covariant way, at any desired loop order. The formalism is based on old ideas such as the background field method and the heat kernel, however we add some crucial new ingredients that greatly improve the simplicity and general applicability of the approach. We formulate our method in terms of Feynman rules, the resulting effective action is expressed in terms of local heat kernel coefficients. We also provide as supplementary material a mathematica code that facilitates the computation of these coefficients.

SimpleMKKM: Simple Multiple Kernel K-Means.

We propose a simple yet effective multiple kernel clustering algorithm, termed simple multiple kernel k-means (SimpleMKKM). It extends the widely used supervised kernel alignment criterion to multi-kernel clustering. Our criterion is given by an intractable minimization-maximization problem in the kernel coefficient and clustering partition matrix. To optimize it, we equivalently rewrite the minimization-maximization formulation as a minimization of an optimal value function, prove its differenentiablity, and design a reduced gradient descent algorithm to decrease it. Furthermore, we prove that the resultant solution of SimpleMKKM is the global optimum. We theoretically analyze the performance of SimpleMKKM in terms of its clustering generalization error. After that, we develop extensive experiments to investigate the proposed SimpleMKKM from the perspectives of clustering accuracy, advantage on the formulation and optimization, variation of the learned consensus clustering matrix with iterations, clustering performance with varied number of samples and base kernels, analysis of the learned kernel weight, the running time and the global convergence. The experimental study demonstrates the effectiveness of the proposed SimpleMKKM by considerably and consistently outperforming state of the art multiple kernel clustering alternatives. In addition, the ablation study shows that the improved clustering performance is contributed by both the novel formulation and new optimization. Our work provides a more effective approach to integrate multi-view data for clustering, and this could trigger novel research on multiple kernel clustering. The source code and data for SimpleMKKM are available at https://github.com/xinwangliu/SimpleMKKMcodes/.

Spatial analysis of the phenomenon of poverty in Nasiriyah using geographic information systems technology

The research aims to diagnose the factors of poverty in the city of Nasiriyah, and its spatial analysis by adopting spatial analyzes in the environment of the geographic information systems program by finding patterns of spatial distribution of the poor population and the factors and density of their distribution such as neighborhood, nucleus density, kernel coefficient, and hot spots) with identifying the factors causing this phenomenon, As well as studying their conditions and the resulting effects. (665) questionnaires were distributed to several dwellings in the city of Nasiriyah, which differed in physical and physical distribution the intentional sample method was used to select a number of poor people according to their distribution in residential neighborhoods, and measures were adopted to treat them statistically according to the characteristics, size and quality of the data obtained. The research reached a set of indicators represented by the clear contrast of the phenomenon of poverty in the urban sphere of the city of Nasiriyah and its housing units, as the population and housing factors interact with each other to formulate the most important factors influencing the phenomenon of poverty through the close interdependence between them, as most of the neighborhoods of the city of Nasiriyah suffer from urban problems, and among them suffer from Poverty to varying degrees.

Neural KEM: A Kernel Method With Deep Coefficient Prior for PET Image Reconstruction.

Image reconstruction of low-count positron emission tomography (PET) data is challenging. Kernel methods address the challenge by incorporating image prior information in the forward model of iterative PET image reconstruction. The kernelized expectation-maximization (KEM) algorithm has been developed and demonstrated to be effective and easy to implement. A common approach for a further improvement of the kernel method would be adding an explicit regularization, which however leads to a complex optimization problem. In this paper, we propose an implicit regularization for the kernel method by using a deep coefficient prior, which represents the kernel coefficient image in the PET forward model using a convolutional neural-network. To solve the maximum-likelihood neural network-based reconstruction problem, we apply the principle of optimization transfer to derive a neural KEM algorithm. Each iteration of the algorithm consists of two separate steps: a KEM step for image update from the projection data and a deep-learning step in the image domain for updating the kernel coefficient image using the neural network. This optimization algorithm is guaranteed to monotonically increase the data likelihood. The results from computer simulations and real patient data have demonstrated that the neural KEM can outperform existing KEM and deep image prior methods.