Consideration on the learning efficiency of multiple-layered neural networks with linear units.

In the last two decades, remarkable progress has been done in singular learning machine theories on the basis of algebraic geometry. These theories reveal that we need to find resolution maps of singularities for analyzing asymptotic behavior of state probability functions when the number of data increases. In particular, it is essential to construct normal crossing divisors of average log loss functions.

Learning Coefficient of Vandermonde Matrix-Type Singularities in Model Selection.

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.

Monitoring of cholesterol oxidation in a lipid bilayer membrane using streptolysin O as a sensing and signal transduction element.

Streptolysin O (SLO), which recognizes sterols and forms nanopores in lipid membranes, is proposed as a sensing element for monitoring cholesterol oxidation in a lipid bilayer. The structural requirements of eight sterols for forming nanopores by SLO confirmed that a free 3-OH group in the β-configuration of sterols is required for recognition by SLO in a lipid bilayer. The extent of nanopore formation by SLO in lipid bilayers increased in the order of cholestanol View Article and Find Full Text PDF

Learning coefficient of generalization error in Bayesian estimation and vandermonde matrix-type singularity.

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 ).

Asymptotic analysis of Bayesian generalization error with Newton diagram.

Statistical learning machines that have singularities in the parameter space, such as hidden Markov models, Bayesian networks, and neural networks, are widely used in the field of information engineering. Singularities in the parameter space determine the accuracy of estimation in the Bayesian scenario. The Newton diagram in algebraic geometry is recognized as an effective method by which to investigate a singularity.

Stochastic complexities of reduced rank regression in Bayesian estimation.

Reduced rank regression extracts an essential information from examples of input-output pairs. It is understood as a three-layer neural network with linear hidden units. However, reduced rank approximation is a non-regular statistical model which has a degenerate Fisher information matrix.