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I-Complexity And Discrete Derivative Of Logarithms: A Symmetry-Based Explanation, Vladik Kreinovich, Jaime Nava Aug 2011

I-Complexity And Discrete Derivative Of Logarithms: A Symmetry-Based Explanation, Vladik Kreinovich, Jaime Nava

Departmental Technical Reports (CS)

In many practical applications, it is useful to consider Kolmogorov complexity K(s) of a given string s, i.e., the shortest length of a program that generates this string. Since Kolmogorov complexity is, in general, not computable, it is necessary to use computable approximations K_~(s) to K(s). Usually, to describe such an approximations, we take a compression algorithm and use the length of the compressed string as K_~(s). This approximation, however, is not perfect: e.g., for most compression algorithms, adding a single bit to the string $s$ can drastically change the value K_~(s) -- while …

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Theoretical Explanation Of Bernstein Polynomials' Efficiency: They Are Optimal Combination Of Optimal Endpoint-Related Functions, Jaime Nava, Vladik Kreinovich Jul 2011

Theoretical Explanation Of Bernstein Polynomials' Efficiency: They Are Optimal Combination Of Optimal Endpoint-Related Functions, Jaime Nava, Vladik Kreinovich

Departmental Technical Reports (CS)

In many applications of interval computations, it turned out to be beneficial to represent polynomials on a given interval [x_-, x₊] as linear combinations of Bernstein polynomials (x- x_-)^k * (x₊ - x)^n-k. In this paper, we provide a theoretical explanation for this empirical success: namely, we show that under reasonable optimality criteria, Bernstein polynomials can be uniquely determined from the requirement that they are optimal combinations of optimal polynomials corresponding to the interval's endpoints.

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Orthogonal Bases Are The Best: A Theorem Justifying Bruno Apolloni's Heuristic Neural Network Idea, Jaime Nava, Vladik Kreinovich Jun 2011

Orthogonal Bases Are The Best: A Theorem Justifying Bruno Apolloni's Heuristic Neural Network Idea, Jaime Nava, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the main problems with neural networks is that they are often very slow in learning the desired dependence. To speed up neural networks, Bruno Apolloni proposed to othogonalize neurons during training, i.e., to select neurons whose output functions are orthogonal to each other. In this paper, we use symmetries to provide a theoretical explanation for this heuristic idea.

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Full-Text Articles in Engineering

I-Complexity And Discrete Derivative Of Logarithms: A Symmetry-Based Explanation, Vladik Kreinovich, Jaime Nava

Departmental Technical Reports (CS)

Theoretical Explanation Of Bernstein Polynomials' Efficiency: They Are Optimal Combination Of Optimal Endpoint-Related Functions, Jaime Nava, Vladik Kreinovich

Departmental Technical Reports (CS)

Orthogonal Bases Are The Best: A Theorem Justifying Bruno Apolloni's Heuristic Neural Network Idea, Jaime Nava, Vladik Kreinovich

Departmental Technical Reports (CS)