Read the book: «Triangulation in neural network»

© Vitaly Fartushnov, 2025

ISBN 978-5-0067-5897-1

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Triangulation in neural network

Triangulation in the context of neural networks is the use of a geometric method of dividing a set of points into triangles to solve various problems of data processing, structure analysis, or building connections between network elements.

Main applications of triangulation in neural networks

– Generating connections between neurons: Delaunay triangulation is used to determine the topology of connections between neurons based on the spatial arrangement of the input data. For example, the weights of feedback connections between neurons can be formed based on the distances between them calculated from a triangulation grid ^[1].

– Data preprocessing: Before training the neural network, data points can be triangulated to identify local structures and neighborhoods, which allows geometric relationships between objects to be taken into account.

– Segmentation and surfacing: In 3D modeling and point cloud processing tasks, neural networks can use triangulation to construct surfaces and meshes, which is important for object reconstruction and shape analysis [2] [3].

Example: Delaunay triangulation in neural networks

In one approach, after constructing a Delaunay triangulation from the input set of points, each neuron’s neighbors in the triangulation are determined. Based on the distances to the neighbors, a scaling constant is calculated, and the weights of the connections between neurons are set taking into account these distances and the constant. This allows the local geometry of the data to be taken into account when forming the network structure ^[1].

Modern Methods: Learned Triangulation

In modern studies, such as PointTriNet, triangulation is integrated directly into the neural network architecture as a differentiable layer. Here, two networks are used: one classifies whether a candidate triangle should be included in the final triangulation, and the other proposes new candidates. This approach allows for the automatic construction of optimal triangulations for point clouds in 3D space, which is useful for computer vision and 3D reconstruction tasks [2] [3].

Benefits of Using Triangulation

– Taking into account local data geometry

– Optimization of the structure of connections in a neural network

– Improving the quality of segmentation and modeling of complex objects

Conclusion

Triangulation in neural networks is a tool that allows for efficient modeling and analysis of spatial data structures, as well as for constructing more informative and adaptive topologies of connections between neurons, which is especially relevant for tasks related to processing spatial and geometric data [2] [1] [3].

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Delaunay triangulation in neural networks

Delaunay triangulation is a method of partitioning a set of points in a plane (or space) into triangles (or higher-dimensional simplices) such that no point in the original set falls inside the circumscribed circle (or sphere) of any triangle (simplex) [4] [5] [6]. This approach ensures a partition with the largest possible minimum angles, which avoids «thin» and degenerate triangles [7] [5].

Application in neural networks

In neural networks, Delaunay triangulation is used to construct a graph data structure, where points (e.g., point cloud elements or feature vectors) become graph vertices, and edges are determined by the triangulation results. This allows:

– It is optimal to form local connections between data elements, which is critical for tasks related to the processing of geometric information, 3D modeling and surface reconstruction [8] [9].

– Avoid degenerate connections: Due to the Delaunay properties, connections between points do not form «thin» triangles, which improves the stability and quality of information transfer in graph neural networks [7] [5] [9].

– Improve the graph structure: For example, in modern graph neural networks (GNNs), Delaunay triangulation is used to «restructure» the original graph to combat problems such as over-squashing and over-smoothing ^[9].

Example: DeepDT

DeepDT proposes to use Delaunay triangulation to construct a graph from a point cloud, after which the neural network learns to classify tetrahedrons (in 3D) as belonging or not belonging to the surface of an object. The point features are aggregated according to the structure specified by the Delaunay triangulation, which allows for complex spatial relationships to be taken into account and improves the quality of surface reconstruction ^[8].

Algorithmic details

– Efficient algorithms such as divide and conquer, incremental edge flip approach, and Boyer—Watson algorithm are used to construct the Delaunay triangulation [4] [6].

– In graph neural networks, edges between vertices are determined based on the results of triangulation, not the original topology, which can significantly change the nature of information dissemination ^[9].

Briefly about the advantages

– Maximizing the minimum angle: avoiding degenerate triangles [7] [5].

– Optimal local structure: efficient aggregation of information between neighbors ^[9].

– Flexibility: Can be used in 2D, 3D and higher dimensions [4] [8].

Conclusion:

Delaunay triangulation in neural networks is a tool for constructing efficient and robust graph structures, especially relevant for problems related to geometry, 3D reconstruction and point cloud processing. It allows forming connections between data elements taking into account their spatial arrangement, which improves the quality of training and the final results of models [8] [9].

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How Delaunay Triangulation Helps Avoid «Thin’ Triangles in Neural Networks

Delaunay triangulation helps to avoid «skinny» (narrow, elongated) triangles due to its main geometric property: it maximizes the minimum angle among all angles of all constructed triangles [10] [11] [12]. This means that in any other possible triangulation for the same set of points, the smallest angle will not be larger than in the Delaunay triangulation. This approach prevents the appearance of triangles with very small angles, i.e. «skinny» triangles, which can negatively affect the accuracy and stability of calculations in neural networks, especially when working with geometric or graph structures.

In neural networks, this property is important because «thin» triangles lead to uneven distribution of neighbors and can worsen the aggregation of features between nodes, as well as increase numerical errors. Delaunay triangulation, by avoiding such triangles, provides a more uniform and stable structure of connections between nodes, which has a positive effect on the quality of information transfer and model training [10] [11] [12].

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Implementation of an Oscillatory Chaotic Neural Network Using NVIDIA CUDA Technology to Solve Clustering Problems

Oscillatory chaotic neural networks (OCNNs) are a class of artificial neural networks in which the dynamics of individual neurons are described by oscillatory and chaotic processes. Such networks are particularly effective for clustering tasks, since they are able to detect complex data structures due to synchronization and desynchronization of neurons.

Features of implementation on NVIDIA CUDA

– Parallelization of computations: CUDA allows for efficient parallelization of computations involving the dynamics of a large number of oscillators, which is critical for modeling large-scale chaotic neural networks [13] [14] [15].

– Organization of flows: Various flow schemes (X-flow, Y-flow) are used to calculate the output values of neurons and the synchronization matrix. The optimal number of flows is considered to be no more than half of the maximum number of video card flows, which ensures minimal calculation time [13] [14].

– Buffering and memory: Various options for storing data in memory have been proposed for storing the synchronization matrix, taking into account the network size, which allows for efficient use of video card resources [13] [14].

– Analysis of results: Synchronization between neurons is analyzed using undirected graphs and disjoint set systems, which allows identifying clusters in the data [13] [14].

Stages of solving the clustering problem

– Network initialization: Setting oscillator parameters and initial conditions.

– Parallel simulation of dynamics: At each time step, the states of the oscillators and their interactions are calculated using CUDA cores.

– Calculation of the synchronization matrix: Pairs of neurons that are in a synchronized state are determined.

– Cluster selection: Based on synchronization analysis, clusters are formed – groups of neurons that oscillate in a coordinated manner.

– Post-processing: Clusters are compared with the original data to interpret the results.

Practical significance

Implementation of OHNS on GPU allows:

– Significantly speed up the processing of large volumes of data through massive parallelism [13] [14] [15].

– To effectively solve clustering problems, especially in cases of complex, nonlinear and heterogeneous data, where classical methods may be ineffective ^[16].

– Apply various algorithmic and structural solutions to optimize network performance and GPU memory usage [13] [14].

Conclusion:

The implementation of oscillatory chaotic neural network using NVIDIA CUDA technology provides high performance and efficiency in solving clustering problems, especially for complex and large datasets. This approach combines the advantages of chaotic dynamics to reveal complex structures and parallel computing to speed up processing [13] [14] [15] [16].

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Analysis of Oscillatory Chaotic Neural Network Method Using CUDA for Solving Projection Problems

The essence of the method

The method is based on the use of an oscillatory chaotic neural network (OCNN) implemented using NVIDIA CUDA technology to accelerate computations. OCNN is a single-layer, recurrent, fully connected network, where the dynamics of neurons is described by a logistic mapping, and the clustering result is revealed by synchronizing the output signals of neurons in time. To improve performance and the possibility of application to big data, the network is implemented on a GPU using CUDA [17] [18] [19] [20].

Algorithm of work taking into account projection tasks

– Data preparation and projection transformations

– The input data is a set of points that can be pre-projected into the desired space (for example, when solving a dimensionality reduction problem or finding clusters in projections).

– Construction of Delaunay triangulation

– Delaunay triangulation is used to determine the topology of connections between neurons. It allows identifying local neighborhoods between points in the projection, which is especially important for correctly accounting for geometric relationships in the projection space. The scaling constant for the connection weights is also calculated based on the triangulation ^[17].

– Formation of the weight matrix

– The weight coefficients between neurons are calculated using a formula taking into account the Euclidean distances between points (neurons) and a scaling constant obtained from triangulation. This ensures the sensitivity of the network to the spatial relationships between points in the projection.

– Parallel modeling of network dynamics

– Calculations of the dynamics of neurons (their output values at each step) and the analysis of synchronization between them are implemented on the GPU using CUDA. Different flow organization schemes (X- and Y-flows) are used to optimize the work depending on the network size and the capabilities of the video card. This allows for efficient processing of large projection problems [17] [18] [19] [20].

– Synchronization analysis and cluster selection

– After the iterations are completed, the synchronization matrix between neurons is analyzed. Using graph theory methods (search for connectivity components) or a system of disjoint sets, clusters of synchronized neurons are selected, which correspond to groups of points in the projection space.

Advantages of the method for projection tasks

– Accounting for local geometry: Using Delaunay triangulation ensures correct detection of local neighborhoods in projections, which is critical for complex multidimensional data.

– High performance: Implementation on CUDA allows to significantly speed up the processing of large data arrays, which is important for projection tasks with a large number of points.