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].
