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Class-Imbalanced Graph Learning without Class Rebalancing
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Support Vector Machine (SVM) is originally proposed as a binary classification model, and it has already achieved great success in different applications. In reality, it is more often to solve a problem which has more than two classes. So, it is natural to extend SVM to a multi-class classifier. There have been many works proposed to construct a multi-class classifier based on binary SVM, such as one versus all strategy, one versus one strategy and Weston's multi-class SVM. One versus all strategy and one versus one strategy split the multi-class problem to multiple binary classification subproblems, and we need to train multiple binary classifiers. Weston's multi-class SVM is formed by ensuring risk constraints and imposing a specific regularization, like Frobenius norm. It is not derived by maximizing the margin between hyperplane and training data which is the motivation in SVM. In this paper, we propose a multi-class SVM model from the perspective of maximizing margin between training points and hyperplane, and analyze the relation between our model and other related methods. In the experiment, it shows that our model can get better or compared results when comparing with other related methods.more » « less
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Abstract The Hilbert class polynomial has as roots the j -invariants of elliptic curves whose endomorphism ring is a given imaginary quadratic order. It can be used to compute elliptic curves over finite fields with a prescribed number of points. Since its coefficients are typically rather large, there has been continued interest in finding alternative modular functions whose corresponding class polynomials are smaller. Best known are Weber’s functions, which reduce the size by a factor of 72 for a positive density subset of imaginary quadratic discriminants. On the other hand, Bröker and Stevenhagen showed that no modular function will ever do better than a factor of 100.83. We introduce a generalization of class polynomials, with reduction factors that are not limited by the Bröker–Stevenhagen bound. We provide examples matching Weber’s reduction factor. For an infinite family of discriminants, their reduction factors surpass those of all previously known modular functions by a factor at least 2.more » « less
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We contribute to the theory of orders of number fields. We define a notion of ray class group associated to an arbitrary order in a number field together with an arbitrary ray class modulus for that order (including Archimedean data), constructed using invertible fractional ideals of the order. We show existence of ray class fields corresponding to the class groups. These ray class groups (resp., ray class fields) specialize to classical ray class groups (resp., fields) of a number field in the case of the maximal order, and they specialize to ring class groups (resp., fields) of orders in the case of trivial modulus. We give exact sequences for simultaneous change of order and change of modulus. As a consequence, we identify the ray class field of an order with a given modulus as a specific subfield of a ray class field of the maximal order with a larger modulus. We also uniquely describe each ray class field of an order in terms of the splitting behavior of primes.more » « less
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