The Experts below are selected from a list of 279 Experts worldwide ranked by ideXlab platform
Jun Ohkubo - One of the best experts on this subject based on the ideXlab platform.
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Approximation scheme for master equations: variational approach to Multivariate Case
The Journal of chemical physics, 2008Co-Authors: Jun OhkuboAbstract:We study an approximation scheme based on a second quantization method for a chemical master equation. Small systems, such as cells, could not be studied by the traditional rate equation approach because fluctuation effects are very large in such small systems. Although a Fokker–Planck equation obtained by the system size expansion includes the fluctuation effects, it needs large computational costs for complicated chemical reaction systems. In addition, discrete characteristics of the original master equation are neglected in the system size expansion scheme. It has been shown that the use of the second quantization description and a variational method achieves tremendous reduction in the dimensionality of the master equation approximately, without loss of the discrete characteristics. Here, we propose a new scheme for the choice of variational functions, which is applicable to Multivariate Cases. It is revealed that the new scheme gives better numerical results than old ones and the computational cost i...
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approximation scheme for master equations variational approach to Multivariate Case
Journal of Chemical Physics, 2008Co-Authors: Jun OhkuboAbstract:We study an approximation scheme based on a second quantization method for a chemical master equation. Small systems, such as cells, could not be studied by the traditional rate equation approach because fluctuation effects are very large in such small systems. Although a Fokker–Planck equation obtained by the system size expansion includes the fluctuation effects, it needs large computational costs for complicated chemical reaction systems. In addition, discrete characteristics of the original master equation are neglected in the system size expansion scheme. It has been shown that the use of the second quantization description and a variational method achieves tremendous reduction in the dimensionality of the master equation approximately, without loss of the discrete characteristics. Here, we propose a new scheme for the choice of variational functions, which is applicable to Multivariate Cases. It is revealed that the new scheme gives better numerical results than old ones and the computational cost increases only slightly.
T. Pham-gia - One of the best experts on this subject based on the ideXlab platform.
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System Stress-Strength Reliability: The Multivariate Case
IEEE Transactions on Reliability, 2007Co-Authors: N. Turkkan, T. Pham-giaAbstract:Present day complex systems with dependence between their components require more advanced models to evaluate their reliability. We compute the reliability of a system consisting of two subsystems S 1, and S2 connected in series, where the reliability of each subsystem is of general stress-strength type, defined by R1 = P(A TX > BTY). A & B are column-constant vectors, and strength X & stress Y are multigamma random vectors, i.e. (X, Y) ~ MG (alpha, beta), where alpha and beta are k-dimensional constant vectors. A Bayesian approach is adopted for R2 = P(B TW > 0), where W is multinormal, i.e. W ~ MN(mu, T), with the mean vector mu, and the precision matrix T having a joint s-normal-Wishart prior distribution. Final computations are carried out by simulation, an approach which plays a major role in this article. The results obtained show that the approach adopted can deal effectively with the dependence between components of X & Y
Nathan Srebro - One of the best experts on this subject based on the ideXlab platform.
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a function space view of bounded norm infinite width relu nets the Multivariate Case
International Conference on Learning Representations, 2020Co-Authors: Greg Ongie, Rebecca Willett, Daniel Soudry, Nathan SrebroAbstract:A key element of understanding the efficacy of overparameterized neural networks is characterizing how they represent functions as the number of weights in the network approaches infinity. In this paper, we characterize the norm required to realize any function as a single hidden-layer ReLU network with an unbounded number of units (infinite width), but where the Euclidean norm of the weights is bounded, including precisely characterizing which functions can be realized with finite norm. This was settled for univariate functions in Savarese et al. (2019), where it was shown that the required norm is determined by the L1-norm of the second derivative of the function. We extend the characterization to multi-variate functions (i.e., multiple input units), relating the required norm to the L1-norm of the Radon transform of a higher-order Laplacian of the function. This characterization allows us to show that all functions in a Sobolev space, can be represented with bounded norm, to calculate the required norm for several specific functions, and to obtain a depth separation result. These results have important implications for understanding generalization performance and the distinction between neural networks and more traditional kernel learning.
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a function space view of bounded norm infinite width relu nets the Multivariate Case
arXiv: Learning, 2019Co-Authors: Greg Ongie, Rebecca Willett, Daniel Soudry, Nathan SrebroAbstract:A key element of understanding the efficacy of overparameterized neural networks is characterizing how they represent functions as the number of weights in the network approaches infinity. In this paper, we characterize the norm required to realize a function $f:\mathbb{R}^d\rightarrow\mathbb{R}$ as a single hidden-layer ReLU network with an unbounded number of units (infinite width), but where the Euclidean norm of the weights is bounded, including precisely characterizing which functions can be realized with finite norm. This was settled for univariate univariate functions in Savarese et al. (2019), where it was shown that the required norm is determined by the L1-norm of the second derivative of the function. We extend the characterization to Multivariate functions (i.e., networks with d input units), relating the required norm to the L1-norm of the Radon transform of a (d+1)/2-power Laplacian of the function. This characterization allows us to show that all functions in Sobolev spaces $W^{s,1}(\mathbb{R})$, $s\geq d+1$, can be represented with bounded norm, to calculate the required norm for several specific functions, and to obtain a depth separation result. These results have important implications for understanding generalization performance and the distinction between neural networks and more traditional kernel learning.
N. Turkkan - One of the best experts on this subject based on the ideXlab platform.
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System Stress-Strength Reliability: The Multivariate Case
IEEE Transactions on Reliability, 2007Co-Authors: N. Turkkan, T. Pham-giaAbstract:Present day complex systems with dependence between their components require more advanced models to evaluate their reliability. We compute the reliability of a system consisting of two subsystems S 1, and S2 connected in series, where the reliability of each subsystem is of general stress-strength type, defined by R1 = P(A TX > BTY). A & B are column-constant vectors, and strength X & stress Y are multigamma random vectors, i.e. (X, Y) ~ MG (alpha, beta), where alpha and beta are k-dimensional constant vectors. A Bayesian approach is adopted for R2 = P(B TW > 0), where W is multinormal, i.e. W ~ MN(mu, T), with the mean vector mu, and the precision matrix T having a joint s-normal-Wishart prior distribution. Final computations are carried out by simulation, an approach which plays a major role in this article. The results obtained show that the approach adopted can deal effectively with the dependence between components of X & Y
Ian M. Thompson - One of the best experts on this subject based on the ideXlab platform.
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Multivariate Case-control study of survival following transurethral resection of prostate.
Urology, 1992Co-Authors: Michael E. Hooten, Kenn Finstuen, Ian M. ThompsonAbstract:Abstract Although prevalence of preooperative mortality following transurethral resection of the prostate (TURP) has steadily declined as reported in several review studies, it has been suggested that with extended follow-up after TURP, mortality exceeds that of an age-matched population. The sample selected for this study was drawn from a data base of 2,005 men who entered a urologic health-screening program. The sample included a group of 25 patients who underwent TURP and a group of 50 age-matched control patients with symptoms of prostatism. Patients were followed for six years, a total of 450 observed person-years. Multivariate analyses based on a general linear model approach for the dependent variable of survival were used to assess gains in predictive efficiency due to combinations of TURP versus control, time, and patient variables. F-ratio hypothesis tests of coefficients of multiple determination for the models indicated that TURP did account for a significant amount of the variability of survival but only after five years of follow-up. However, a far larger proportion of the variance in survival was explained by other patient variables of age, preoperative risk, comorbidity factors, and postoperative urinary disease after all effects due to TURP, follow-up years, and operation cohort years were held constant in the survival prediction equation.
