arXiv subjects

Shengli Tan

Publications and source records attributed to Shengli Tan.

3 recordsLinked to original sources

arXiv2024

The Poincar\'e Problem for a foliated surface

Let $\mathcal F$ be a foliation on a smooth projective surface $S$ over the complex number $\mathbb{C}$. We introduce three birational non-negative invariants $c_1^2(\mathcal F)$, $c_2(\mathcal F)$ and $\chi(\mathcal F)$, called the Chern numbers. If the foliation $\mathcal F$ is not of general type, the first Chern number $c_1^2(\mathcal F)=0$, and $c_2(\mathcal F)=\chi(\mathcal F)=0$ except when $\mathcal F$ is induced by a non-isotrivial fibration of genus $g=1$. If $\mathcal F$ is of general type, we obtain a slope inequality when $\mathcal F$ is algebraically integral. As a corollary, $\mathcal F$ is always transcendental if the slope is less than $2$. On the other hand, we also prove three sharp Noether type inequalities if $\mathcal F$ is of general type. As applications, we obtain a criterion for foliations to be transcendental using Noether type inequalities, and we also give a partial positive answer to the question on the lower bound on the volume of a foliation of general type.

math.AG↗

arXiv2024

Non-existence of Shimura curves of Mumford type generically in the non-hyperelliptic locus

We show that there does not exist any Shimura curve with strictly maximal Higgs field generically in the Torelli locus of non-hyperelliptic curves of genus $g\geq 4$. In particular, Shimura curves of Mumford type are not generically in the Torelli locus of non-hyperelliptic curves of genus $g\geq 4$.

math.AG↗

arXiv2021

Training Generative Adversarial Networks with Adaptive Composite Gradient

The wide applications of Generative adversarial networks benefit from the successful training methods, guaranteeing that an object function converges to the local minima. Nevertheless, designing an efficient and competitive training method is still a challenging task due to the cyclic behaviors of some gradient-based ways and the expensive computational cost of these methods based on the Hessian matrix. This paper proposed the adaptive Composite Gradients (ACG) method, linearly convergent in bilinear games under suitable settings. Theory and toy-function experiments suggest that our approach can alleviate the cyclic behaviors and converge faster than recently proposed algorithms. Significantly, the ACG method is not only used to find stable fixed points in bilinear games as well as in general games. The ACG method is a novel semi-gradient-free algorithm since it does not need to calculate the gradient of each step, reducing the computational cost of gradient and Hessian by utilizing the predictive information in future iterations. We conducted two mixture of Gaussians experiments by integrating ACG to existing algorithms with Linear GANs. Results show ACG is competitive with the previous algorithms. Realistic experiments on four prevalent data sets (MNIST, Fashion-MNIST, CIFAR-10, and CelebA) with DCGANs show that our ACG method outperforms several baselines, which illustrates the superiority and efficacy of our method.

cs.LG↗