There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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Convex Analysisの二,三の進展について

关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

Economics and Information Theory

At initialization, artificial neural networks (ANNs) are equivalent to Gaussian processes in the infinite-width limit, thus connecting them to kernel methods. We prove that the evolution of an ANN during training can also be described by a kernel: during gradient descent on the parameters of an ANN, the network function (which maps input vectors to output vectors) follows the so-called kernel gradient associated with a new object, which we call the Neural Tangent Kernel (NTK). This kernel is central to describe the generalization features of ANNs. While the NTK is random at initialization and varies during training, in the infinite-width limit it converges to an explicit limiting kernel and stays constant during training. This makes it possible to study the training of ANNs in function space instead of parameter space. Convergence of the training can then be related to the positive-definiteness of the limiting NTK. We then focus on the setting of least-squares regression and show that in the infinite-width limit, the network function follows a linear differential equation during training. The convergence is fastest along the largest kernel principal components of the input data with respect to the NTK, hence suggesting a theoretical motivation for early stopping. Finally we study the NTK numerically, observe its behavior for wide networks, and compare it to the infinite-width limit.

/pdf/neural-tangent-kernel-convergence-and-generalization-in-3ks0qgekbr.pdf

Neural Tangent Kernel: Convergence and Generalization in Neural Networks

In this paper we introduce the concepts of Stolarsky and Gini
divergence measures and establish a number of basic properties. Some comparison
results in the same class or between different classes are also given.

Stolarsky and Gini Divergence Measures in Information Theory

An Ostrowski type integral inequality for convex functions and applications for quadrature rules and integral means are given. A refinement and a counterpart result for Hermite-Hadamard inequalities are obtained and some inequalities for pdf's and (HH)-divergence measure are also mentioned.

An Ostrowski Type Inequality for Convex Functions

Some companions of Gruss inequality in 2-inner prod- uct spaces and applications for determinantal integral inequalities are given.

2-inner product spaces and applications for determinantal integral inequalities

In this paper some new inequalities of Schwarz and Buzano type for n-tuples of vectors in inner product spaces are given. Applications for norm and numerical radius inequalities for n-tuples of bounded linear operators and for functions of normal operators defined by power series with nonnegative coefficients are also provided.

Sever S Dragomir

Papers

Stolarsky and Gini Divergence Measures in Information Theory

An Ostrowski Type Inequality for Convex Functions

2-inner product spaces and applications for determinantal integral inequalities

Inequalities Of Schwarz Type For n-Tuples Of Vectors In Inner Product Spaces With Applications

Reverses of Callebaut discrete inequality via some results due to Zhuang