A quasi-conforming embedded reproducing kernel particle method for heterogeneous materials
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Schlinkman, Ryan T.↗
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Verification and Large Deformation Analysis Using the Reproducing Kernel Particle Method.
Abstract not provided.
Beckwith, Frank↗
The Conforming Reproducing Kernel (CRK) Method: Recent Developments and Applications.
Abstract not provided.
Koester, Jacob↗
The Conforming Reproducing Kernel (CRK) Method: Recent Developments and Applications.
Abstract not provided.
Koester, Jacob↗
Treatment of near-incompressibility in meshfree and immersed-particle methods
We propose here new projection methods for treating near-incompressibility in small and large deformation elasticity and plasticity within the framework of particle and meshfree methods. Using the $\bar{B}$ and $\bar{F}$ techniques as our point of departure, we develop projection methods for the conforming reproducing kernel method and the immersed-particle or material point-like methods. The methods are based on the projection of the dilatational part of the appropriate measure of deformation onto lower-dimensional approximation spaces, according to the traditional $\bar{B}$ and $\bar{F}$ approaches, but tailored to meshfree and particle methods. The presented numerical examples exhibit reduced stress oscillations and are free of volumetric locking and hourglassing phenomena.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Predicting carbon dioxide and energy fluxes across global FLUXNET sites with regression algorithms
Spatio-temporal fields of land–atmosphere fluxes derived from data-driven models can complement simulations by process-based land surface models. While a number of strategies for empirical models with eddy-covariance flux data have been applied, a systematic intercomparison of these methods has been missing so far. In this study, we performed a cross-validation experiment for predicting carbon dioxide, latent heat, sensible heat and net radiation fluxes across different ecosystem types with 11 machine learning (ML) methods from four different classes (kernel methods, neural networks, tree methods, and regression splines). We applied two complementary setups: (1) 8-day average fluxes based on remotely sensed data and (2) daily mean fluxes based on meteorological data and a mean seasonal cycle of remotely sensed variables. The patterns of predictions from different ML and experimental setups were highly consistent. There were systematic differences in performance among the fluxes, with the following ascending order: net ecosystem exchange ( R 2 < 0.5), ecosystem respiration ( R 2 > 0.6), gross primary production ( R 2 > 0.7), latent heat ( R 2 > 0.7), sensible heat ( R 2 > 0.7), and net radiation ( R 2 > 0.8). The ML methods predicted the across-site variability and the mean seasonal cycle of the observed fluxes very well ( R 2 > 0.7), while the 8-day deviations from the mean seasonal cycle were not well predicted ( R 2 < 0.5). Fluxes were better predicted at forested and temperate climate sites than at sites in extreme climates or less represented by training data (e.g., the tropics). Finally, the evaluated large ensemble of ML-based models will be the basis of new global flux products.
54 ENVIRONMENTAL SCIENCES↗
Application of Quantum Machine Learning to High Energy Physics Analysis at LHC Using Quantum Computer Simulators and Quantum Computer Hardware
Machine learning enjoys widespread success in High Energy Physics (HEP) analyses at LHC. However the ambitious HL-LHC program will require much more computing resources in the next two decades. Quantum computing may offer speed-up for HEP physics analyses at HL-LHC, and can be a new computational paradigm for big data analyses in High Energy Physics.We have successfully employed three methods (1) Variational Quantum Classifier (VQC) method, (2) Quantum Support Vector Machine Kernel (QSVM-kernel) method and (3) Quantum Neural Network (QNN) method for two LHC flagship analyses: ttH (Higgs production in association with two top quarks) and H->mumu (Higgs decay to two muons, the second generation fermions). We shall address the progressive improvements in performance from method (1) to method (3).We will present our experiences and results of a study on LHC High Energy Physics data analyses with IBM Quantum Simulator and Quantum Hardware (using IBM Qiskit framework), Google Quantum Simulator (using Google Cirq framework), and Amazon Quantum Simulator (using Amazon Braket cloud service). The work is in the context of a Qubit platform (a gate-model quantum computer). Taking into account the present limitation of hardware access, different quantum machine learning methods are studied on simulators and the results are compared with classical machine learning methods (BDT, classical Support Vector Machine and classical Neural Network). Furthermore, we do apply quantum machine learning on IBM quantum hardware to compare performance between quantum simulator and quantum hardware. The work is performed by an international and interdisciplinary collaboration with the Department of Physics and Department of Computer Sciences of University of Wisconsin, CERN Quantum Technology Initiative, IBM Research Zurich, IBM T.J. Watson Research Center, Fermilab Quantum Institute, BNL Computational Science Initiative, State University of New York at Stony Brook, and Quantum Computing and AI Research of Amazon Web Services. This work pioneers a close collaboration of academic institutions with industrial corporations in the High Energy Physics analyses effort. Though the size of event samples in future HL-LHC physics and the limited number of qubits pose some challenges to the Quantum Machine learning studies for High Energy Physics, more advanced quantum computers with larger number of qubits, reduced noise and improved running time (as envisioned by IBM and Google) may outperform classical machine learning in both classification power and in speed.Although the era of efficient quantum computing may still be years away, we have made promising progress and obtained preliminary results in applying quantum machine learning to High Energy Physics. A PROOF OF PRINCIPLE.
46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
Systems and methods for customizing kernel machines with deep neural networks
A method including receiving an input data set. The input data set can include one of a feature domain set or a kernel matrix. The method also can include constructing dense embeddings using: (i) Nyström approximations on the input data set when the input data set comprises the kernel matrix, and (ii) clustered Nyström approximations on the input data set when the input data set comprises the feature domain set. The method additionally can include performing representation learning on each of the dense embeddings using a multi-layer fully-connected network for each of the dense embeddings to generate latent representations corresponding to each of the dense embeddings. The method further can include applying a fusion layer to the latent representations corresponding to the dense embeddings to generate a combined representation. The method additionally can include performing classification on the combined representation. Other embodiments of related systems and methods are also disclosed.
Song, Huan↗
Data assimilation in operator algebras
We develop an algebraic framework for sequential data assimilation of partially observed dynamical systems. In this framework, Bayesian data assimilation is embedded in a nonabelian operator algebra, which provides a representation of observables by multiplication operators and probability densities by density operators (quantum states). In the algebraic approach, the forecast step of data assimilation is represented by a quantum operation induced by the Koopman operator of the dynamical system. Moreover, the analysis step is described by a quantum effect, which generalizes the Bayesian observational update rule. Projecting this formulation to finite-dimensional matrix algebras leads to computational schemes that are i) automatically positivity-preserving and ii) amenable to consistent data-driven approximation using kernel methods for machine learning. Moreover, these methods are natural candidates for implementation on quantum computers. Applications to the Lorenz 96 multiscale system and the El Niño Southern Oscillation in a climate model show promising results in terms of forecast skill and uncertainty quantification.
97 MATHEMATICS AND COMPUTING↗
Optimizing Kernel Machines Using Deep Learning
Building highly non-linear and non-parametric models is central to several state-of-the-art machine learning systems. Kernel methods form an important class of techniques that induce a reproducing kernel Hilbert space (RKHS) for inferring non-linear models through the construction of similarity functions from data. These methods are particularly preferred in cases where the training data sizes are limited and when prior knowledge of the data similarities is available. Despite their usefulness, they are limited by the computational complexity and their inability to support end-to-end learning with a task specific objective. On the other hand, deep neural networks have become the de facto solution for end-to-end inference in several learning paradigms. In this article, we explore the idea of using deep architectures to perform kernel machine optimization, for both computational efficiency and end-to-end inferencing. To this end, we develop the DKMO (Deep Kernel Machine Optimization) framework, that creates an ensemble of dense embeddings using Nystrõm kernel approximations and utilizes deep learning to generate task-specific representations through the fusion of the embeddings. Intuitively, the filters of the network are trained to fuse information from an ensemble of linear subspaces in the RKHS. Furthermore, we introduce the kernel dropout regularization to enable improved training convergence. Finally, we extend this framework to the multiple kernel case, by coupling a global fusion layer with pre-trained deep kernel machines for each of the constituent kernels. Using case studies with limited training data, and lack of explicit feature sources, we demonstrate the effectiveness of our framework over conventional model inferencing techniques.
97 MATHEMATICS AND COMPUTING↗
A Study of Clustering Techniques and Hierarchical Matrix Formats for Kernel Ridge Regression
We present memory-efficient and scalable algorithms for kernel methods used in machine learning. Using hierarchical matrix approximations for the kernel matrix the memory requirements, the number of floating point operations, and the execution time are drastically reduced compared to standard dense linear algebra routines. We consider both the general H matrix hierarchical format as well as Hierarchically Semi-Separable (HSS) matrices. Furthermore, we investigate the impact of several preprocessing and clustering techniques on the hierarchical matrix compression. Effective clustering of the input leads to a ten-fold increase in efficiency of the compression. The algorithms are implemented using the STRUMPACK solver library. These results confirm that - with correct tuning of the hyperparameters - classification using kernel ridge regression with the compressed matrix does not lose prediction accuracy compared to the exact - not compressed - kernel matrix and that our approach can be extended to O(1M) datasets, for which computation with the full kernel matrix becomes prohibitively expensive. We present numerical experiments in a distributed memory environment up to 1,024 processors of the NERSC's Cori supercomputer using well-known datasets to the machine learning community that range from dimension 8 up to 784.
97 MATHEMATICS AND COMPUTING↗
Validation tests of an improved kernel density estimation method for identifying disease clusters
The spatial filter method, which belongs to the class of kernel density estimation methods, has been used to make morbidity and mortality maps in several recent studies. We propose improvements in the method that include a spatial basis of support designed to give a constant standard error for the standardized mortality/morbidity rate; a stair-case weight method for weighting observations to reduce estimation bias; and a method for selecting parameters to control three measures of performance of the method: sensitivity, specificity and false discovery rate. We test the performance of the method using Monte Carlo simulations of hypothetical disease clusters over a test area of four counties in Iowa. The simulations include different types of spatial disease patterns and high resolution population distribution data. Results confirm that the new features of the spatial filter method do substantially improve its performance in realistic situations comparable to those where the method is likely to be used.
59 BASIC BIOLOGICAL SCIENCES↗
Approximate l-fold cross-validation with Least Squares SVM and Kernel Ridge Regression
Kernel methods have difficulties scaling to large modern data sets. The scalability issues are based on computational and memory requirements for working with a large matrix. These requirements have been addressed over the years by using low-rank kernel approximations or by improving the solvers scalability. However, Least Squares Support VectorMachines (LS-SVM), a popular SVM variant, and Kernel Ridge Regression still have several scalability issues. In particular, the O(n^3) computational complexity for solving a single model, and the overall computational complexity associated with tuning hyperparameters are still major problems. We address these problems by introducing an O(n log n) approximate l-fold cross-validation method that uses a multi-level circulant matrix to approximate the kernel. In addition, we prove our algorithm s computational complexity and present empirical runtimes on data sets with approximately 1 million data points. We also validate our approximate method s effectiveness at selecting hyperparameters on real world and standard benchmark data sets. Lastly, we provide experimental results on using a multi-level circulant kernel approximation to solve LS-SVM problems with hyperparameters selected using our method.
Edwards, Richard E↗
Approximate l-Fold Cross-Validation with Least Squares SVM and Kernel Ridge Regression
Kernel methods have difficulties scaling to large modern data sets. The scalability issues are based on computational and memory requirements for working with a large matrix. These requirements have been addressed over the years by using low-rank kernel approximations or by improving the solvers' scalability. However, Least Squares Support Vector Machines (LS-SVM), a popular SVM variant, and Kernel Ridge Regression still have several scalability issues. In particular, the O(n^3) computational complexity for solving a single model, and the overall computational complexity associated with tuning hyper parameters are still major problems. We address these problems by introducing an O(nlog n) approximate l-fold cross-validation method that uses a multi-level circulant matrix to approximate the kernel. In addition, we prove our algorithm's computational complexity and present empirical runtimes on data sets with approximately one million data points. We also validate our approximate method's effectiveness at selecting hyper parameters on real world and standard benchmark data sets. Lastly, we provide experimental results on using a multi level circulant kernel approximation to solve LS-SVM problems with hyper parameters selected using our method.
Computer Science↗
Numerical evidence against advantage with quantum fidelity kernels on classical data
Quantum machine learning techniques are commonly considered one of the most promising candidates for demonstrating practical quantum advantage. In particular, quantum kernel methods have been demonstrated to be able to learn certain classically intractable functions efficiently if the kernel is well aligned with the target function. In the more general case, quantum kernels are known to suffer from exponential “flattening” of the spectrum as the number of qubits grows, preventing generalization and necessitating the control of the inductive bias by hyperparameters. We show that the general-purpose hyperparameter-tuning techniques proposed to improve the generalization of quantum kernels lead to the kernel becoming well approximated by a classical kernel, removing the possibility of quantum advantage. We provide extensive numerical evidence for this phenomenon utilizing multiple previously studied quantum feature maps and both synthetic and real data. Our results show that unless novel techniques are developed to control the inductive bias of quantum kernels, they are unlikely to provide a quantum advantage on classical data that lacks special structure.
Quantum Information, Science & Technology↗