Semi-Supervised Classification via Hypergraph Convolutional Extreme Learning Machine

Extreme Learning Machine (ELM) is characterized by simplicity, generalization ability, and computational efficiency. However, previous ELMs fail to consider the inherent high-order relationship among data points, resulting in being powerless on structured data and poor robustness on noise data. This paper presents a novel semi-supervised ELM, termed Hypergraph Convolutional ELM (HGCELM), based on using hypergraph convolution to extend ELM into the non-Euclidean domain. The method inherits all the advantages from ELM, and consists of a random hypergraph convolutional layer followed by a hypergraph convolutional regression layer, enabling it to model complex intraclass variations. We show that the traditional ELM is a special case of the HGCELM model in the regular Euclidean domain. Extensive experimental results show that HGCELM remarkably outperforms eight competitive methods on 26 classification benchmarks.

A new twofold Cornacchia-type algorithm and its applications

<p style='text-indent:20px;'>We focus on exploring more potential of Longa and Sica's algorithm (ASIACRYPT 2012), which is an elaborate iterated Cornacchia algorithm that can compute short bases for 4-GLV decompositions. The algorithm consists of two sub-algorithms, the first one in the ring of integers <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula> and the second one in the Gaussian integer ring <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{Z}[i] $\end{document}</tex-math></inline-formula>. We observe that <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{Z}[i] $\end{document}</tex-math></inline-formula> in the second sub-algorithm can be replaced by another Euclidean domain <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{Z}[\omega] $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M5">\begin{document}$ (\omega = \frac{-1+\sqrt{-3}}{2}) $\end{document}</tex-math></inline-formula>. As a consequence, we design a new twofold Cornacchia-type algorithm with a theoretic upper bound of output <inline-formula><tex-math id="M6">\begin{document}$ C\cdot n^{1/4} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M7">\begin{document}$ C = \frac{3+\sqrt{3}}{2}\sqrt{1+|r|+|s|} $\end{document}</tex-math></inline-formula> with small values <inline-formula><tex-math id="M8">\begin{document}$ r, s $\end{document}</tex-math></inline-formula> given by the curves.</p><p style='text-indent:20px;'>The new twofold algorithm can be used to compute <inline-formula><tex-math id="M9">\begin{document}$ 4 $\end{document}</tex-math></inline-formula>-GLV decompositions on two classes of curves. First it gives a new and unified method to compute all <inline-formula><tex-math id="M10">\begin{document}$ 4 $\end{document}</tex-math></inline-formula>-GLV decompositions on <inline-formula><tex-math id="M11">\begin{document}$ j $\end{document}</tex-math></inline-formula>-invariant <inline-formula><tex-math id="M12">\begin{document}$ 0 $\end{document}</tex-math></inline-formula> elliptic curves over <inline-formula><tex-math id="M13">\begin{document}$ \mathbb{F}_{p^2} $\end{document}</tex-math></inline-formula>. Second it can be used to compute the <inline-formula><tex-math id="M14">\begin{document}$ 4 $\end{document}</tex-math></inline-formula>-GLV decomposition on the Jacobian of the hyperelliptic curve defined as <inline-formula><tex-math id="M15">\begin{document}$ \mathcal{C}/\mathbb{F}_{p}:y^{2} = x^{6}+ax^{3}+b $\end{document}</tex-math></inline-formula>, which has an endomorphism <inline-formula><tex-math id="M16">\begin{document}$ \phi $\end{document}</tex-math></inline-formula> with the characteristic equation <inline-formula><tex-math id="M17">\begin{document}$ \phi^2+\phi+1 = 0 $\end{document}</tex-math></inline-formula> (hence <inline-formula><tex-math id="M18">\begin{document}$ \mathbb{Z}[\phi] = \mathbb{Z}[\omega] $\end{document}</tex-math></inline-formula>). As far as we know, none of the previous algorithms can be used to compute the <inline-formula><tex-math id="M19">\begin{document}$ 4 $\end{document}</tex-math></inline-formula>-GLV decomposition on the latter class of curves.</p>

A note on r-precious ring

An element of a ring [Formula: see text] is said to be [Formula: see text]-precious if it can be written as the sum of a von Neumann regular element, an idempotent element and a nilpotent element. If all the elements of a ring [Formula: see text] are [Formula: see text]-precious, then [Formula: see text] is called an [Formula: see text]-precious ring. We study some basic properties of [Formula: see text]-precious rings. We also characterize von Neumann regular elements in [Formula: see text] when [Formula: see text] is a Euclidean domain and by this argument, we produce elements that are [Formula: see text]-precious but either not [Formula: see text]-clean or not precious.

$\omega$-Euclidean domain and Laurent series

It is proved that a commutative domain $R$ is $\omega$-Euclidean if and only if the ring of formal Laurent series over $R$ is $\omega$-Euclidean domain. It is also proved that every singular matrice over ring of formal Laurent series $R_{X}$ are products of idempotent matrices if $R$ is $\omega$-Euclidean domain.