scholarly journals Duality and Dimensionality Reduction Discrete Line Generation Algorithm for a Triangular Grid

2018 ◽  
Vol 7 (10) ◽  
pp. 391 ◽  
Author(s):  
Lingyu Du ◽  
Qiuhe Ma ◽  
Jin Ben ◽  
Rui Wang ◽  
Jiahao Li
Keyword(s):  
Dimensionality Reduction ◽  
Path Model ◽  
Triangular Grid ◽  
Weak Duality ◽  
Vector Data ◽  
Hexagonal Grid ◽  
Generation Algorithm ◽  
Discrete Line ◽  
Hexagonal Grids ◽  
Line Generation

Vectors are a key type of geospatial data, and their discretization, which involves solving the problem of generating a discrete line, is particularly important. In this study, we propose a method for constructing a discrete line mathematical model for a triangular grid based on a “weak duality” hexagonal grid, to overcome the drawbacks of existing discrete line generation algorithms for a triangular grid. First, a weak duality relationship between triangular and hexagonal grids is explored. Second, an equivalent triangular grid model is established based on the hexagonal grid, using this weak duality relationship. Third, the two-dimensional discrete line model is solved by transforming it into a one-dimensional optimal wandering path model. Finally, we design and implement the dimensionality reduction generation algorithm for a discrete line in a triangular grid. The results of our comparative experiment indicate that the proposed algorithm has a computation speed that is approximately 10 times that of similar existing algorithms; in addition, it has better fitting effectiveness. Our proposed algorithm has broad applications, and it can be used for real-time grid transformation of vector data, discrete global grid system (DGGS), and other similar applications.

Dimensionality reduction of tensor data based on local linear embedding and mode product

2021 ◽  
pp. 1-18
Author(s):  
Ting Gao ◽  
Zhengming Ma ◽  
Wenxu Gao ◽  
Shuyu Liu
Keyword(s):  
Dimensionality Reduction ◽  
Learning Algorithm ◽  
Vector Data ◽  
Global Features ◽  
Global Mode ◽  
Local Linear Embedding ◽  
Local Linear ◽  
Clustering And Classification ◽  
Linear Embedding ◽  
Tensor Data

There are three contributions in this paper. (1) A tensor version of LLE (short for Local Linear Embedding algorithm) is deduced and presented. LLE is the most famous manifold learning algorithm. Since its proposal, various improvements to LLE have kept emerging without interruption. However, all these achievements are only suitable for vector data, not tensor data. The proposed tensor LLE can also be used a bridge for various improvements to LLE to transfer from vector data to tensor data. (2) A framework of tensor dimensionality reduction based on tensor mode product is proposed, in which the mode matrices can be determined according to specific criteria. (3) A novel dimensionality reduction algorithm for tensor data based on LLE and mode product (LLEMP-TDR) is proposed, in which LLE is used as a criterion to determine the mode matrices. Benefiting from local LLE and global mode product, the proposed LLEMP-TDR can preserve both local and global features of high-dimensional tenser data during dimensionality reduction. The experimental results on data clustering and classification tasks demonstrate that our method performs better than 5 other related algorithms published recently in top academic journals.

A Line Generation Algorithm on 3D Face-Centered Cubic Grid

2011 ◽  
Vol 33 (12) ◽  
pp. 2407-2416 ◽  
Author(s):  
Li-Jun HE ◽  
Yong-Kui LIU ◽  
Shi-Chang SUN
Keyword(s):  
Face Centered Cubic ◽  
Generation Algorithm ◽  
3D Face ◽  
Face Centered ◽  
Line Generation

Dimensionality reduction for tensor data based on projection distance minimization and hilbert-schmidt independence criterion maximization1

2021 ◽  
Vol 40 (5) ◽  
pp. 10307-10322
Author(s):  
Weichao Gan ◽  
Zhengming Ma ◽  
Shuyu Liu
Keyword(s):  
Dimensionality Reduction ◽  
High Dimensional ◽  
Tensor Algebra ◽  
Reduction Algorithm ◽  
Subspace Projection ◽  
Vector Data ◽  
Projection Distance ◽  
Distance Minimization ◽  
Tensor Data ◽  
Independence Criterion

Tensor data are becoming more and more common in machine learning. Compared with vector data, the curse of dimensionality of tensor data is more serious. The motivation of this paper is to combine Hilbert-Schmidt Independence Criterion (HSIC) and tensor algebra to create a new dimensionality reduction algorithm for tensor data. There are three contributions in this paper. (1) An HSIC-based algorithm is proposed in which the dimension-reduced tensor is determined by maximizing HSIC between the dimension-reduced and high-dimensional tensors. (2) A tensor algebra-based algorithm is proposed, in which the high-dimensional tensor are projected onto a subspace and the projection coordinate is set to be the dimension-reduced tensor. The subspace is determined by minimizing the distance between the high-dimensional tensor data and their projection in the subspace. (3) By combining the above two algorithms, a new dimensionality reduction algorithm, called PDMHSIC, is proposed, in which the dimensionality reduction must satisfy two criteria at the same time: HSIC maximization and subspace projection distance minimization. The proposed algorithm is a new attempt to combine HSIC with other algorithms to create new algorithms and has achieved better experimental results on 8 commonly-used datasets than the other 7 well-known algorithms.

AN IMPLEMENTATION OF THE PROCEDURAL GENERATION OF UNSTRUCTURED QUADRANGULAR GRID ALGORITHM BASED ON A STRUCTURED TRIANGULAR GRID IN REGULAR HEXAGONS ON THE PLANE

2021 ◽  
pp. 88-95
Author(s):  
Н.М. Чернышов ◽  
О.В. Авсеева
Keyword(s):  
Coordinate System ◽  
Real Time ◽  
Triangular Grid ◽  
Large Area ◽  
Hexagonal Grid ◽  
Procedural Generation ◽  
Grid Algorithm

Работа посвящена реализации алгоритма процедурной генерации нерегулярной четырехугольной сетки, позволяющего рассчитывать сетку для большой области в реальном времени. При генерации используются кубическая система координат, в которой строится регулярная треугольная сетка для каждой ячейки шестиугольной сетки, процедура релаксации четырехугольной сетки. This work is devoted to the implementation of an algorithm for procedural generation of an unstructured quadrangular grid, which allows to calculate the grid for a large area in real time. When building the grid, a cubic coordinate system, in which a structured triangular grid is built for each cell of a hexagonal grid, and a relaxation of the quadrangular grid algorithm are used.

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