Solving Zero-Dimensional Systems Through the Rational Univariate Representation

1999 ◽  
Vol 9 (5) ◽  
pp. 433-461 ◽  
Author(s):  
Fabrice Rouillier
Keyword(s):  
Univariate Representation ◽  
Rational Univariate Representation

The Kolmogorov Spline Network for Authentication Data Embedding in Images

2013 ◽  
pp. 96-114
Author(s):  
Pierre-Emmanuel Leni ◽  
Yohan D. Fougerolle ◽  
Frédéric Truchetet
Keyword(s):  
Continuous Functions ◽  
Data Embedding ◽  
Progressive Transmission ◽  
Secret Key ◽  
Transmission Scheme ◽  
Black And White ◽  
Novel Approach ◽  
Univariate Functions ◽  
Univariate Representation ◽  
Superposition Theorem

In 1900, Hilbert declared that high order polynomial equations could not be solved by sums and compositions of continuous functions of less than three variables. This statement was proven wrong by the superposition theorem, demonstrated by Arnol’d and Kolmogorov in 1957, which allows for writing all multivariate functions as sums and compositions of univariate functions. Amongst recent computable forms of the theorem, Igelnik and Parikh’s approach, known as the Kolmogorov Spline Network (KSN), offers several alternatives for the univariate functions as well as their construction. A novel approach is presented for the embedding of authentication data (black and white logo, translucent or opaque image) in images. This approach offers similar functionalities than watermarking approaches, but relies on a totally different theory: the mark is not embedded in the 2D image space, but it is rather applied to an equivalent univariate representation of the transformed image. Using the progressive transmission scheme previously proposed (Leni, 2011), the pixels are re-arranged without any neighborhood consideration. Taking advantage of this naturally encrypted representation, it is proposed to embed the watermark in these univariate functions. The watermarked image can be accessed at any intermediate resolution, and fully recovered (by removing the embedded mark) without loss using a secret key. Moreover, the key can be different for every resolution, and both the watermark and the image can be globally restored in case of data losses during the transmission. These contributions lie in proposing a robust embedding of authentication data (represented by a watermark) into an image using the 1D space of univariate functions based on the Kolmogorov superposition theorem. Lastly, using a key, the watermark can be removed to restore the original image.

scholarly journals On the Univariate Representation of BEKK Models with Common Factors

2016 ◽  
Vol 8 (2) ◽  
Author(s):  
Alain Hecq ◽  
Sébastien Laurent ◽  
Franz C. Palm
Keyword(s):  
Asset Returns ◽  
Multivariate Garch ◽  
Garch Models ◽  
Multivariate Models ◽  
Reduced Rank ◽  
Contagion Effects ◽  
The Matrix ◽  
Ratio Approach ◽  
Multivariate Garch Models ◽  
Univariate Representation

AbstractSimple low order multivariate GARCH models imply marginal processes with a lot of persistence in the form of high order lags. This is not what we find in many situations however, where parsimonious univariate GARCH(1,1) models for instance describe quite well the conditional volatility of some asset returns. In order to explain this paradox, we show that in the presence of common GARCH factors, parsimonious univariate representations can result from large multivariate models generating the conditional variances and conditional covariances/correlations. The diagonal model without any contagion effects in conditional volatilities gives rise to similar conclusions though. Consequently, after having extracted a block of assets representing some form of parsimony, remains the task of determining if we have a set of independent assets or instead a highly dependent system generated with a few factors. To investigate this issue, we first evaluate a reduced rank regressions approach for squared returns that we extend to cross-returns. Second we investigate a likelihood ratio approach, where under the null the matrix parameters have a reduced rank structure. It emerged that the latter approach has quite good properties enabling us to discriminate between a system with seemingly unrelated assets (e.g. a diagonal model) and a model with few common sources of volatility.

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