The Experts below are selected from a list of 2358 Experts worldwide ranked by ideXlab platform
Miki Aoyagi - One of the best experts on this subject based on the ideXlab platform.
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learning coefficient of Vandermonde Matrix type singularities in model selection
2019Co-Authors: Miki AoyagiAbstract:In recent years, selecting appropriate learning models has become more important with the increased need to analyze learning systems, and many model selection methods have been developed. The learning coefficient in Bayesian estimation, which serves to measure the learning efficiency in singular learning models, has an important role in several information criteria. The learning coefficient in regular models is known as the dimension of the parameter space over two, while that in singular models is smaller and varies in learning models. The learning coefficient is known mathematically as the log canonical threshold. In this paper, we provide a new rational blowing-up method for obtaining these coefficients. In the application to Vandermonde Matrix-type singularities, we show the efficiency of such methods.
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learning coefficients and reproducing true probability functions in learning systems
2017Co-Authors: Miki AoyagiAbstract:Recently, the widely applicable information criterion (WAIC) model selection method has been considered for reproducing and estimating a probability function from data in a learning system. The learning coefficient in Bayesian estimation serves to measure the learning efficiency in singular learning models, and has an important role in the WAIC method. Mathematically, the learning coefficient is the log canonical threshold of the relative entropy. In this paper, we consider the Vandermonde Matrix-type singularity learning coefficients in statistical learning theory.
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learning coefficient of generalization error in bayesian estimation and Vandermonde Matrix type singularity
2012Co-Authors: Miki Aoyagi, Kenji NagataAbstract:The term algebraic statistics arises from the study of probabilistic models and techniques for statistical inference using methods from algebra and geometry (Sturmfels, 2009). The purpose of our study is to consider the generalization error and stochastic complexity in learning theory by using the log-canonical threshold in algebraic geometry. Such thresholds correspond to the main term of the generalization error in Bayesian estimation, which is called a learning coefficient (Watanabe, 2001a, 2001b). The learning coefficient serves to measure the learning efficiencies in hierarchical learning models. In this letter, we consider learning coefficients for Vandermonde Matrix-type singularities, by using a new approach: focusing on the generators of the ideal, which defines singularities. We give tight new bound values of learning coefficients for the Vandermonde Matrix-type singularities and the explicit values with certain conditions. By applying our results, we can show the learning coefficients of three-layered neural networks and normal mixture models.
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a bayesian learning coefficient of generalization error and Vandermonde Matrix type singularities
2010Co-Authors: Miki AoyagiAbstract:The coefficient of the main term of the generalization error in Bayesian estimation is called a Bayesian learning coefficient. In this article, we first introduce Vandermonde Matrix type singularities and show certain orthogonality conditions of them. Recently, it has been recognized that Vandermonde Matrix type singularities are related to Bayesian learning coefficients for several hierarchical learning models. By applying the orthogonality conditions of them, we show that their log canonical threshold also corresponds to the Bayesian learning coefficient for normal mixture models, and we obtain the explicit computational results in dimension one.
Kenji Nagata - One of the best experts on this subject based on the ideXlab platform.
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learning coefficient of generalization error in bayesian estimation and Vandermonde Matrix type singularity
2012Co-Authors: Miki Aoyagi, Kenji NagataAbstract:The term algebraic statistics arises from the study of probabilistic models and techniques for statistical inference using methods from algebra and geometry (Sturmfels, 2009). The purpose of our study is to consider the generalization error and stochastic complexity in learning theory by using the log-canonical threshold in algebraic geometry. Such thresholds correspond to the main term of the generalization error in Bayesian estimation, which is called a learning coefficient (Watanabe, 2001a, 2001b). The learning coefficient serves to measure the learning efficiencies in hierarchical learning models. In this letter, we consider learning coefficients for Vandermonde Matrix-type singularities, by using a new approach: focusing on the generators of the ideal, which defines singularities. We give tight new bound values of learning coefficients for the Vandermonde Matrix-type singularities and the explicit values with certain conditions. By applying our results, we can show the learning coefficients of three-layered neural networks and normal mixture models.
Edoardo Signorini - One of the best experts on this subject based on the ideXlab platform.
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on the condition number of the Vandermonde Matrix of the nth cyclotomic polynomial
2020Co-Authors: Antonio J Di Scala, Carlo Sanna, Edoardo SignoriniAbstract:Recently, Blanco-Chacon proved the equivalence between the Ring Learning With Errors and Polynomial Learning With Errors problems for some families of cyclotomic number fields by giving some upper bounds for the condition number $\operatorname{Cond}(V_n)$ of the Vandermonde Matrix $V_n$ associated to the $n$th cyclotomic polynomial. We prove some results on the singular values of $V_n$ and, in particular, we determine $\operatorname{Cond}(V_n)$ for $n = 2^k p^\ell$, where $k, \ell \geq 0$ are integers and $p$ is an odd prime number.
Yongjian Nian - One of the best experts on this subject based on the ideXlab platform.
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an improved azimuth reconstruction method for multichannel sar using Vandermonde Matrix
2017Co-Authors: Pu Cheng, Jianwei Wan, Qin Xin, Zhan Wang, Yongjian NianAbstract:To overcome the contradiction between wide swath and high resolution in synthetic aperture radar systems, a multichannel azimuth reconstruction method is investigated to unambiguously recover the Doppler spectrum. The proposed method is derived from the least squares principle by exploiting a Vandermonde component of the system Matrix. The Vandermonde Matrix is Doppler independent and data independent. Reconstruction filter weightings can be easily achieved, and performance, including signal-to-noise ratio (SNR) and azimuth ambiguity-to-signal ratio, can be explicitly expressed. By well-conditioning the Vandermonde Matrix and coherent processing of all channels, the proposed method improves the reconstruction performance. In simulated reconstruction, compared with the conventional Matrix inversion method, the SNR increases by approximately 30 dB.
Antonio J Di Scala - One of the best experts on this subject based on the ideXlab platform.
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on the condition number of the Vandermonde Matrix of the nth cyclotomic polynomial
2020Co-Authors: Antonio J Di Scala, Carlo Sanna, Edoardo SignoriniAbstract:Recently, Blanco-Chacon proved the equivalence between the Ring Learning With Errors and Polynomial Learning With Errors problems for some families of cyclotomic number fields by giving some upper bounds for the condition number $\operatorname{Cond}(V_n)$ of the Vandermonde Matrix $V_n$ associated to the $n$th cyclotomic polynomial. We prove some results on the singular values of $V_n$ and, in particular, we determine $\operatorname{Cond}(V_n)$ for $n = 2^k p^\ell$, where $k, \ell \geq 0$ are integers and $p$ is an odd prime number.
