scholarly journals Euclidean Embeddings and Riemannian Bergman Metrics

2015 ◽  
Vol 26 (1) ◽  
pp. 499-528
Author(s):  
Eric Potash
Keyword(s):  
Euclidean Embeddings ◽  
Bergman Metrics

scholarly journals Novel metric for hyperbolic phylogenetic tree embeddings

2021 ◽  
Vol 6 (1) ◽  
Author(s):  
Hirotaka Matsumoto ◽  
Takahiro Mimori ◽  
Tsukasa Fukunaga
Keyword(s):  
Phylogenetic Tree ◽  
Hyperbolic Space ◽  
Nearest Neighbor ◽  
Cancer Genomics ◽  
Geometric Approach ◽  
Artificial Intelligence Research ◽  
Phylogenetic Methods ◽  
Novel Approach ◽  
Multiple Trees ◽  
Euclidean Embeddings

Abstract Advances in experimental technologies, such as DNA sequencing, have opened up new avenues for the applications of phylogenetic methods to various fields beyond their traditional application in evolutionary investigations, extending to the fields of development, differentiation, cancer genomics, and immunogenomics. Thus, the importance of phylogenetic methods is increasingly being recognized, and the development of a novel phylogenetic approach can contribute to several areas of research. Recently, the use of hyperbolic geometry has attracted attention in artificial intelligence research. Hyperbolic space can better represent a hierarchical structure compared to Euclidean space, and can therefore be useful for describing and analyzing a phylogenetic tree. In this study, we developed a novel metric that considers the characteristics of a phylogenetic tree for representation in hyperbolic space. We compared the performance of the proposed hyperbolic embeddings, general hyperbolic embeddings, and Euclidean embeddings, and confirmed that our method could be used to more precisely reconstruct evolutionary distance. We also demonstrate that our approach is useful for predicting the nearest-neighbor node in a partial phylogenetic tree with missing nodes. Furthermore, we proposed a novel approach based on our metric to integrate multiple trees for analyzing tree nodes or imputing missing distances. This study highlights the utility of adopting a geometric approach for further advancing the applications of phylogenetic methods.

scholarly journals Low distortion euclidean embeddings of trees

1998 ◽  
Vol 106 (1) ◽  
pp. 339-348 ◽  
Author(s):  
Nathan Linial ◽  
Avner Magen ◽  
Michael E. Saks
Keyword(s):  
Low Distortion ◽  
Euclidean Embeddings

scholarly journals Optimal convergence speed of Bergman metrics on symplectic manifolds

2020 ◽  
Vol 18 (4) ◽  
pp. 1091-1126
Author(s):  
Wen Lu ◽  
Xiaonan Ma ◽  
George Marinescu
Keyword(s):  
Symplectic Manifolds ◽  
Convergence Speed ◽  
Optimal Convergence ◽  
Bergman Metrics

scholarly journals On the existence of Kobayashi and Bergman metrics for Model domains

2020 ◽  
Author(s):  
Nikolay Shcherbina
Keyword(s):  
Entire Function ◽  
Continuous Function ◽  
Pseudoconvex Domain ◽  
Plurisubharmonic Function ◽  
Holomorphic Curves ◽  
List Type ◽  
Bergman Metric ◽  
The Core ◽  
Bounded Smooth ◽  
Bergman Metrics

Abstract We prove that for a pseudoconvex domain of the form $${\mathfrak {A}} = \{(z, w) \in {\mathbb {C}}^2 : v > F(z, u)\}$$ A = { ( z , w ) ∈ C 2 : v > F ( z , u ) } , where $$w = u + iv$$ w = u + i v and F is a continuous function on $${\mathbb {C}}_z \times {\mathbb {R}}_u$$ C z × R u , the following conditions are equivalent: The domain $$\mathfrak {A}$$ A is Kobayashi hyperbolic. The domain $$\mathfrak {A}$$ A is Brody hyperbolic. The domain $$\mathfrak {A}$$ A possesses a Bergman metric. The domain $$\mathfrak {A}$$ A possesses a bounded smooth strictly plurisubharmonic function, i.e. the core $$\mathfrak {c}(\mathfrak {A})$$ c ( A ) of $$\mathfrak {A}$$ A is empty. The graph $$\Gamma (F)$$ Γ ( F ) of F can not be represented as a foliation by holomorphic curves of a very special form, namely, as a foliation by translations of the graph $$\Gamma ({\mathcal H})$$ Γ ( H ) of just one entire function $${\mathcal {H}} : {\mathbb {C}}_z \rightarrow {\mathbb {C}}_w$$ H : C z → C w .

Random Euclidean embeddings in spaces of bounded volume ratio

2004 ◽  
Vol 339 (1) ◽  
pp. 33-38 ◽  
Author(s):  
Alexander Litvak ◽  
Alain Pajor ◽  
Mark Rudelson ◽  
Nicole Tomczak-Jaegermann ◽  
Roman Vershynin
Keyword(s):  
Volume Ratio ◽  
Bounded Volume ◽  
Euclidean Embeddings

Euclidean embeddings in spaces of finite volume ratio via random matrices

2005 ◽  
Vol 2005 (589) ◽  
pp. 1-19 ◽  
Author(s):  
A. E. Litvak ◽  
A. Pajor ◽  
M. Rudelson ◽  
N. Tomczak-Jaegermann ◽  
R. Vershynin
Keyword(s):  
Random Matrices ◽  
Finite Volume ◽  
Volume Ratio ◽  
Euclidean Embeddings
Sign in / Sign up

Export Citation Format

Share Document