arXiv subjects

Ryuichi Sakamoto

Publications and source records attributed to Ryuichi Sakamoto.

3 recordsLinked to original sources

arXiv2020

The $h^{*}$-polynomial of the cut polytope of $K_{2,m}$ in the lattice spanned by its vertices

The cut polytope of a graph is an important object in several fields, such as functional analysis, combinatorial optimization, and probability. For example, Sturmfels and Sullivant showed that the toric ideals of cut polytopes are useful in algebraic statistics. In the theory of lattice polytopes, the $h^{*}$-polynomial is one of the most important invariants. The necessary and sufficient condition in terms of graphs that the $h^{*}$-polynomial of a cut polytope is palindromic is known. However, except for trees, there are no classes of graphs for which the $h^{*}$-polynomial of their cut polytope is explicitly specified. In the present paper, we determine the $h^{*}$-polynomial of the cut polytope of complete bipartite graph $K_{2,m}$ using the theory of Gröbner bases of toric ideals.

math.CO↗

arXiv2019

Lexicographic and reverse lexicographic quadratic Gröbner bases of cut ideals

Hibi conjectured that if a toric ideal has a quadratic Gröbner basis, then the toric ideal has either a lexicographic or a reverse lexicographic quadratic Gröbner basis. In this paper, we present a cut ideal of a graph that serves as a counterexample to this conjecture. We also discuss the existence of a quadratic Gröbner basis of a cut ideal of a cycle. Nagel and Petrović claimed that a cut ideal of a cycle has a lexicographic quadratic Gröbner basis using the results of Chifman and Petrović. However, we point out that the results of Chifman and Petrović used by Nagel and Petrović are incorrect for cycles of length greater than or equal to 6. Hence the existence of a quadratic Gröbner basis for the cut ideal of a cycle (a ring graph) is an open question. We also provide a lexicographic quadratic Gröbner basis of a cut ideal of a cycle of length less than or equal to 7.

math.AC↗

arXiv2019

Generation High resolution 3D model from natural language by Generative Adversarial Network

We present a method of generating high resolution 3D shapes from natural language descriptions. To achieve this goal, we propose two steps that generating low resolution shapes which roughly reflect texts and generating high resolution shapes which reflect the detail of texts. In a previous paper, the authors have shown a method of generating low resolution shapes. We improve it to generate 3D shapes more faithful to natural language and test the effectiveness of the method. To generate high resolution 3D shapes, we use the framework of Conditional Wasserstein GAN. We propose two roles of Critic separately, which calculate the Wasserstein distance between two probability distribution, so that we achieve generating high quality shapes or acceleration of learning speed of model. To evaluate our approach, we performed quantitive evaluation with several numerical metrics for Critic models. Our method is first to realize the generation of high quality model by propagating text embedding information to high resolution task when generating 3D model.

cs.GR↗