Construction Of Kernels Research Articles

The Parametrix Construction of the Heat Kernel on a Graph

The Parametrix Construction of the Heat Kernel on a Graph

Nonabelian descent types

We present the notion of nonabelian descent type, which classifies torsors up to twisting by a Galois cocycle. This relies on the previous construction of kernels and nonabelian Galois 2-cohomology due to Springer and Borovoi. The necessity of descent types arises in the context of the descent theory where no torsors are given a priori, for example, when we wish to study the arithmetic properties such as the Brauer–Manin obstruction to the Hasse principle on homogeneous spaces without rational points. This new definition also unifies the types by Colliot-Thélène–Sansuc, the extended types by Harari–Skorobogatov, and the finite descent types by Harpaz–Wittenberg.

Locally-Scaled Kernels and Confidence Voting

Classification, the task of discerning the class of an unlabeled data point using information from a set of labeled data points, is a well-studied area of machine learning with a variety of approaches. Many of these approaches are closely linked to the selection of metrics or the generalizing of similarities defined by kernels. These metrics or similarity measures often require their parameters to be tuned in order to achieve the highest accuracy for each dataset. For example, an extensive search is required to determine the value of K or the choice of distance metric in K-NN classification. This paper explores a method of kernel construction that when used in classification performs consistently over a variety of datasets and does not require the parameters to be tuned. Inspired by dimensionality reduction techniques (DRT), we construct a kernel-based similarity measure that captures the topological structure of the data. This work compares the accuracy of K-NN classifiers, computed with specific operating parameters that obtain the highest accuracy per dataset, to a single trial of the here-proposed kernel classifier with no specialized parameters on standard benchmark sets. The here-proposed kernel used with simple classifiers has comparable accuracy to the ‘best-case’ K-NN classifiers without requiring the tuning of operating parameters.

Noise‐Tailored Constructions for Spin Wigner Function Kernels

AbstractThe effective use of noisy intermediate‐scale quantum devices requires error mitigation to improve the accuracy of sampled measurement distributions. The more accurately the effects of noise on these distributions can be modeled, the more closely error mitigation will be able to approach theoretical bounds. The characterization of noisy quantum channels and the inference of their effects on general observables are challenging problems, but in many cases a change in representation can greatly simplify the analysis. Here, spin Wigner functions for multiqudit systems are investigated. This study generalizes previous kernel constructions, capturing the effects of several probabilistic unitary noise models in few parameters.

Aerodynamic shape optimization based on proper orthogonal decomposition reparameterization under small training sets

Complex shape aerodynamic optimization frequently encounters the problem of dimensionality curse. Extracting geometric features from the original design space proves effective in this issue by reconstructing shapes with lower-dimension. While deep learning methods possess strong capabilities in dimensionality reduction, they suffer from longer training times and higher computational resource requirements. However, Proper Orthogonal Decomposition (POD) presents a more practical algorithm that can be effectively utilized for complex high-dimensional shapes. In this study, we propose a combination of the POD-based method and Gaussian Process Regression with an automatic kernel construction algorithm (AKC-GPR) to optimize complex shapes in a lower-dimensional reparameterization space using small training sets. This approach aims to explore the potential for reducing optimization cost. By reducing the design parameters from 56 to 18 with an error rate of 1.28 %, we establish surrogate models for optimization using AKC-GPR with progressively decreasing training sets. For samples below 100, the Mean Absolute Percentage Errors (MAPEs) remain below 2 %, while the Mean Relative Percentage Errors (MRPEs) approach 10 %. It emphasizes the proficiency of POD in geometric dimensionality reduction. The optimized shapes exhibit smoother contour, resulting in a 10 % improvement in lift and a relatively modest reduction in drag of less than 3 %. The lift-to-drag ratio experiences an increase of over 14 %. The shapes optimized by various training sets possess minimal differences in geometric shapes and aerodynamic characteristics. This confirms the AKC-GPR algorithm's high-precision fitting capability in effectively handling complex aerodynamic coefficients of multi-parameter shapes during small-sample shape optimization.

Fast flow field prediction of three-dimensional hypersonic vehicles using an improved Gaussian process regression algorithm

Conducting large-scale numerical computations to obtain flow field during the hypersonic vehicle engineering design phase can be excessively costly. Although deep learning algorithms enable rapid flow field prediction with high-precision, they require a significant investment in training samples, contradicting the motivation of reducing the cost of acquiring flow field. The combination of feature extraction algorithms and regression algorithms can also achieve high-precision prediction of flow fields, which is more suitable to tackle three-dimensional flow prediction with a small dataset. In this study, we propose a reduced-order model (ROM) for the three-dimensional hypersonic vehicle flow prediction utilizing proper orthogonal decomposition to extract representative features and Gaussian process regression with improved automatic kernel construction (AKC-GPR) to perform a nonlinear mapping of physical features for prediction. The selection of variables is based on sensitivity analysis and modal assurance criterion. The underlying relationship is unveiled between flow field variables and inflow conditions. The ROM exhibits high predictive accuracy, with mean absolute percentage error (MAPE) of total field less than 3.5%, when varying altitudes and Mach numbers. During angle of attack variations, the ROM only effectively reconstructs flow distribution by interpolation with a MAPE of 7.02%. The excellent small-sample fitting capability of our improved AKC-GPR algorithm is demonstrated by comparing with original AKC-GPRs with a maximum reduction in a MAPE of 35.28%. These promising findings suggest that the proposed ROM can serve as an effective approach for rapid and accurate vehicle flow predicting, enabling its application in engineering design analysis.

Metalens for Accelerated Optoelectronic Edge Detection under Ambient Illumination.

Analog systems may allow image processing, such as edge detection, with low computational power. However, most demonstrated analog systems, based on either conventional 4-f imaging systems or nanophotonic structures, rely on coherent laser sources for illumination, which significantly restricts their use in routine imaging tasks with ambient, incoherent illumination. Here, we demonstrated a metalens-assisted imaging system that can allow optoelectronic edge detection under ambient illumination conditions. The metalens was designed to generate polarization-dependent optical transfer functions (OTFs), resulting in a synthetic OTF with an isotropic high-pass frequency response after digital subtraction. We integrated the polarization-multiplexed metalens with a polarization camera and experimentally demonstrated single-shot edge detection of indoor and outdoor scenes, including a flying airplane, under ambient sunlight illumination. The proposed system showcased the potential of using polarization multiplexing for the construction of complex optical convolution kernels toward accelerated machine vision tasks such as object detection and classification under ambient illumination.

Comments on Quasi-Linear Support Vector Machine for Nonlinear Classification

In 2014, the above paper entitled 'Quasi-Linear Support Vector Machine for Nonlinear Classification' was published by Zhou, et al. [1]. They proposed a quasi-linear kernel function for support vector machine (SVM). However, in this letter, we point out that this proposed kernel function is a part of multiple kernel functions generated by well-known multiple kernel learning which is proposed by Bach, et al. [2] in 2004. Since then, there have been a lot of related papers on multiple kernel learning with several applications [3]. This letter verifies that the main kernel function proposed by Zhou, et al. [1] can be derived using multiple kernel learning algorithms [3]. In the kernel construction, Zhou, et al. [1] used Gaussian kernels, but the multiple kernel learning had already discussed the locality of additive Gaussian kernels or other kernels in the framework [4][5]. Especially additive Gaussian or other kernels were discussed in tutorial at major international conference ECCV2012 [6]. The authors did not discuss these matters.

Improved automatic kernel construction for Gaussian process regression in small sample learning for predicting lift body aerodynamic performance

A Gaussian process regression (GPR) model based on an improved automatic kernel construction (AKC) algorithm using beam search is proposed to establish a surrogate model between lift body shape parameters and aerodynamic coefficients with various training sets sizes. The precision of our proposed surrogate model is assessed through tenfold cross-validation. The improved AKC-GPR algorithm, polynomial regression, and support vector regression (SVR) are employed to construct the regression model. The interpolation and extrapolation capabilities of the model, as generated by the improved AKC-GPR algorithm, are examined using six shapes beyond the sample set. The results show that the three models perform similarly with a large training set. However, when the training set size is less than 40% sample dataset, the model constructed by the improved AKC-GPR algorithm has better fitting and prediction capabilities than the other models. Specifically, the max relative error of the improved model is one-fourth of that of SVR and one-half of that of polynomial regression with the training set size of 8% of the sample dataset. Furthermore, the lift-to-drag ratio relative error of interpolation is only 3%, and extrapolation error is 6%. In terms of the fitting and prediction abilities for small samples, the lift-to-drag ratio model outperforms the drag coefficient model, while the lift coefficient model performs the poorest. These findings suggest that the proposed AKC-GPR algorithm can be an effective approach for building a surrogate model in the field of aerodynamics.

Construction of Multivariate Polynomial Approximation Kernels via Semidefinite Programming

In this paper we construct a hierarchy of multivariate polynomial approximation kernels for uniformly continuous functions on the hypercube via semidefinite programming. We give details on the implementation of the semidefinite programs defining the kernels. Finally, we show how symmetry reduction may be performed to increase numerical tractability.

Dimension‐independent Markov chain Monte Carlo on the sphere

AbstractWe consider Bayesian analysis on high‐dimensional spheres with angular central Gaussian priors. These priors model antipodally symmetric directional data, are easily defined in Hilbert spaces and occur, for instance, in Bayesian density estimation and binary level set inversion. In this paper we derive efficient Markov chain Monte Carlo methods for approximate sampling of posteriors with respect to these priors. Our approaches rely on lifting the sampling problem to the ambient Hilbert space and exploit existing dimension‐independent samplers in linear spaces. By a push‐forward Markov kernel construction we then obtain Markov chains on the sphere which inherit reversibility and spectral gap properties from samplers in linear spaces. Moreover, our proposed algorithms show dimension‐independent efficiency in numerical experiments.

Software Realization of Progress Control Platform for Country House Tourism Development from the Perspective of Integrated Communication Network Operation

This paper aims to build a system platform that meets the current needs of homestay management by introducing a set of progress control platform of an advanced country homestay tourism development system and using the Android operating system as an application object by using smart phones as operating tools. The problem of Internet application in the homestay industry. This research is technically based on Android system, and introduces Java and H5 for system development. Java language can realize the basic framework of App, and H5 can realize the construction of kernel. In terms of software service, the service attitude of customer service and the consulting service provided by the owner affect consumers Buying intentions, good homestay buyers evaluations in terms of booking intentions will speed up consumers’ purchasing decisions.

Geographically convolutional neural network weighted regression: a method for modeling spatially non-stationary relationships based on a global spatial proximity grid

Geographically weighted regression (GWR) is a classical method of modeling spatially non-stationary relationships. The geographically neural network weighted regression (GNNWR) model solves the problem of the inaccurate construction of spatial weight kernels using a spatially weighted neural network. However, when the spatial distribution of observations is uneven, the spatial proximity expression in the input of GWR and GNNWR models does not fully represent the impact of the whole research space on the estimating point. Therefore, we established a global spatial proximity grid (GSPG) to express the spatial proximity of each estimating point and proposed a spatially weighted convolutional neural network (SWCNN) to extract the relationship between the GSPG and spatial weights. Finally, we proposed a geographically convolutional neural network weighted regression (GCNNWR) model combining SWCNN and ordinary linear regression (OLR) model to estimate spatial non-stationarity. We used two case studies of simulated data and real environment data to demonstrate the advancements of the GCNNWR model. The GCNNWR model achieved higher estimation accuracy and greater predictive power than the OLR, GWR, multi-scale GWR (MGWR), and GNNWR models. Moreover, the GCNNWR model maintained its better stability and accuracy in estimating spatially non-stationary relationships when the distribution of observations was uneven.

Graph Kernels Encoding Features of All Subgraphs by Quantum Superposition

Graph kernels are often used in bioinformatics and network applications to measure the similarity between graphs; therefore, they may be used to construct efficient graph classifiers. Many graph kernels have been developed thus far, but to the best of our knowledge there is no existing graph kernel that uses some features explicitly extracted from all subgraphs to measure similarity. We propose a novel graph kernel that applies a quantum computer to measure the similarity obtained from all subgraphs by fully exploiting the power of quantum superposition to encode every subgraph into a feature of particular form. For the construction of the quantum kernel, we develop an efficient protocol that clears the index information of the subgraphs encoded in the quantum state. We also prove that the quantum computer requires less query complexity to construct the feature vector than the classical sampler used to approximate the same vector. A detailed numerical simulation of a bioinformatics problem is presented to demonstrate that, in many cases, the proposed quantum kernel achieves better classification accuracy than existing graph kernels.

An alternative for Laplace Birnbaum-Saunders distribution

Abstract In this paper, we first propose a new general method to introduce various lifetime distributions by choosing an appropriate kernel distribution. They have some characteristics in common with the well-known Birnbaum-Saunders distribution. Then, we choose the triangular distribution as a kernel model and construct the new distribution. This distribution has its support on the positive real axis and consists of two-pieces. We show that the newly defined distribution is in fact a generalized Birnbaum-Saunders distribution. It is mathematically tractable for studying its theoretical properties in detail. Different methods of estimation of parameters are proposed. The existence and uniqueness problem of the maximum likelihood estimation method is discussed. The performances of the estimators are evaluated through simulation studies. A real data fitting which compares it with the ordinary Birnbaum-Saunders, Laplace Birnbaum-Saunders and other some generalized Birnbaum-Saunders distributions is also given.

The power particle-in-cell method

This paper introduces a new weighting scheme for particle-grid transfers that generates hybrid Lagrangian/Eulerian fluid simulations with uniform particle distributions and precise volume control. At its core, our approach reformulates the construction of Power Particles [de Goes et al. 2015] by computing volume-constrained density kernels. We employ these optimized kernels as particle domains within the Generalized Interpolation Material Point method (GIMP) in order to incorporate Power Particles into the Particle-In-Cell framework, hence the name the Power Particle-In-Cell method. We address the construction of volume-constrained density kernels as a regularized optimal transportation problem and describe an iterative solver based on localized Gaussian convolutions that leads to a significant performance speedup compared to [de Goes et al. 2015]. We also present novel extensions for handling free surfaces and solid obstacles that bypass the need for cell clipping and ghost particles. We demonstrate the advantages of our transfer weights by improving hybrid schemes for fluid simulation such as the Fluid Implicit Particle (FLIP) method and the Affine Particle-In-Cell (APIC) method with volume preservation and robustness to varying particle-per-cell ratio, while retaining low numerical dissipation, conserving linear and angular momenta, and avoiding particle reseeding or post-process relaxations.

Resolvent and new activation functions for linear programming kernel sparse learning

The resolvent operator and the corresponding Green’s function occupy a central position in the realms of differential and integral equations, operator theory, and in particular the modern physics. However, in the field of machine learning, when confronted with the complex and highly challenging learning tasks from the real world, the prowess of Green’s function of resolvent is rarely explored and exploited. This paper aims at innovating the conventional translation-invariant kernels and rotation-invariant kernels, through theoretical investigation into a new view of constructing kernel functions by means of the resolvent operator and its Green’s function. From the practical perspective, the newly developed kernel functions are applied for robust signal recovery from noise corrupted data in the scenario of linear programming support vector learning. In particular, the monotonic and non-monotonic activation functions are used for kernel design to improve the representation capability. In this manner, a new dimension is given for kernel-based robust sparse learning from the following two aspects: firstly, a new theoretical framework by bridging the gap between the mathematical subtleties of resolvent operator and Green’s function theory and kernel construction; secondly, a concretization for the fusion between activation functions design in neural networks and nonlinear kernels design. Finally, the experimental study demonstrates the potential and superiority of the newly developed kernel functions in robust signal recovery and multiscale sparse modeling, as one step towards removing the apparent boundaries between the realms of modern signal processing and computational intelligence.

Constructing a Memory Kernel of the Returning Probability to Efficiently Describe Molecular Binding Processes

Returning probability theory enables us to systematically analyze host-guest binding kinetics in terms of the thermodynamic and kinetic properties of the intermediate state existing in the binding processes with the molecular dynamics (MD) simulation. In order to employ this theory, we need to compute the returning probability, a time-correlation function for describing the kinetics of the intermediate state. Here, we present a methodology to realize the efficient calculation of the probability with the memory kernel construction.

Maximum margin Riemannian manifold-based hyperdisk for fault diagnosis of roller bearing with multi-channel fusion covariance matrix

For rotating machinery, the sudden failure of roller bearing would lead to the downtime of the whole system and even catastrophic accidents. Therefore, multiple accelerometers are usually arranged to comprehensively evaluate the health of roller bearing, enhancing the stability and reliability of monitoring results. This paper proposes a novel fault diagnosis framework by utilizing a multi-channel fusion covariance matrix (MFCM) and Riemannian manifold-based hyperdisk. First, 22 statistical features are acquired from each channel data. Then, MFCM is calculated as the fault feature representation of roller bearing to achieve multi-channel feature fusion, where the element of MFCM represents the correlation information between different channels. Finally, since MFCM is a symmetric positive definite (SPD) matrix, lying on a Riemannian manifold, we design a maximum margin Riemannian manifold-based hyperdisk (MMRMHD) classifier to conduct fault classification, where Log-Euclidean metric (LEM) is introduced to calibrate the distribution of MFCMs. Moreover, to further improve the classification ability of nonlinear SPD data, we map MFCMs into a high-dimensional Hilbert space with the LEM-based kernel function and construct a novel kernelized MMRMHD model. The experimental results on two bearing datasets with multi-channel vibration signals demonstrate the effectiveness and superiority of the proposed fault diagnosis framework.

On-the-Fly Full Hessian Kernel Calculations Based upon Seismic-Wave Simulations

AbstractFull waveform inversion or adjoint tomography has routinely been performed to image the internal structure of the Earth at high resolution. This is typically done using the Fréchet kernels and the approximate Hessian or the approximate inverse Hessian because of the high-computational cost of computing and storing the full Hessian. Alternatively, the full Hessian kernels can be used to improve inversion resolutions and convergence rates, as well as possibly to mitigate interparameter trade-offs. The storage requirements of the full Hessian kernel calculations can be reduced by compression methods, but often at a price of accuracy depending on the compression factor. Here, we present open-source codes to compute both Fréchet and full Hessian kernels on the fly in the computer random access memory (RAM) through simultaneously solving four wave equations, which we call Quad Spectral-Element Method (QuadSEM). By recomputing two forward fields at the same time that two adjoint fields are calculated during the adjoint simulation, QuadSEM constructs the full Hessian kernels using the exact forward and adjoint fields. In addition, we also implement an alternative approach based on the classical wavefield storage method (WSM), which stores forward wavefields every kth (k≥1) timestep during the forward simulation and reads required fields back into memory during the adjoint simulation for kernel construction. Both Fréchet and full Hessian kernels can be computed simultaneously through the QuadSEM or the WSM code, only doubling the computational cost compared with the computation of Fréchet kernels alone. Compared with WSM, QuadSEM can reduce the disk space and input/output cost by three orders of magnitude in the presented examples that use 15,000 timesteps. Numerical examples are presented to demonstrate the functionality of the methods, and the computer codes are provided with this contribution.