Effects of Irregular Topology in Spherical Self-Organizing Maps

We explore the effect of different topologies on properties of self-organizing maps (SOM). We suggest several diagnostics for measuring topology-induced errors in SOM and use these in a comparison of four different topologies. The results show that SOM is less sensitive to localized irregularities in the network structure than the literature may otherwise suggest. Further, the results support the use of spherical topologies as a solution to the boundary problem in traditional SOM.

Learning shape correspondence with anisotropic convolutional neural networks

Establishing correspondence between shapes is a fundamental problem in geometry processing, arising in a wide variety of applications. The problem is especially difficult in the setting of non-isometric deformations, as well as in the presence of topological noise and missing parts, mainly due to the limited capability to model such deformations axiomatically. Several recent works showed that invariance to complex shape transformations can be learned from examples. In this paper, we introduce an intrinsic convolutional neural network architecture based on anisotropic diffusion kernels, which we term Anisotropic Convolutional Neural Network (ACNN). In our construction, we generalize convolutions to non-Euclidean domains by constructing a set of oriented anisotropic diffusion kernels, creating in this way a local intrinsic polar representation of the data (`patch'), which is then correlated with a filter. Several cascades of such filters, linear, and non-linear operators are stacked to form a deep neural network whose parameters are learned by minimizing a task-specific cost. We use ACNNs to effectively learn intrinsic dense correspondences between deformable shapes in very challenging settings, achieving state-of-the-art results on some of the most difficult recent correspondence benchmarks

Path finding on a spherical SOM using the distance transform and floodplain analysis

Data visualization has become an important tool for analyzing very complex data. In particular, spatial visualization enables users to view data in a intuitive manner. It has typically been used to externalize clusters and their relationships which exist in highly complex multidimensional data. We envisage that not only cluster formation and relationships but also other types of information, such as temporal changes of datum, can be extracted through the spatialization. In this paper, we investigate an application of trajectory/path analysis carried out using a Self-Organizing Map as a spatialization method. We propose an application of distance transformations to the Geodesic Self-Organizing Map. This new approach allows a user to visually inspect the trajectory of multidimensional knowledge pieces on a two-dimensional space. The trajectories discovered through this approach are essentially the shortest paths between two points on the Self-Organizing Map. However, those paths might go outside of the input dataspace due to the connectivity of neurons imposed by the grid structure. We also present a method to find the shortest path, which falls within the input dataspace using simple floodplain analysis

Path finding on a spherical self-organizing map using distance transformations

Spatialization methods create visualizations that allow users to analyze high-dimensional data in an intuitive manner and facilitates the extraction of meaningful information. Just as geographic maps are simpli ed representations of geographic spaces, these visualizations are esssentially maps of abstract data spaces that are created through dimensionality reduction. While we are familiar with geographic maps for path planning/ nding applications, research into using maps of high-dimensional spaces for such purposes has been largely ignored. However, literature has shown that it is possible to use these maps to track temporal and state changes within a high-dimensional space. A popular dimensionality reduction method that produces a mapping for these purposes is the Self-Organizing Map. By using its topology preserving capabilities with a colour-based visualization method known as the U-Matrix, state transitions can be visualized as trajectories on the resulting mapping. Through these trajectories, one can gather information on the transition path between two points in the original high-dimensional state space. This raises the interesting question of whether or not the Self-Organizing Map can be used to discover the transition path between two points in an n-dimensional space. In this thesis, we use a spherically structured Self-Organizing Map called the Geodesic Self-Organizing Map for dimensionality reduction and the creation of a topological mapping that approximates the n-dimensional space. We rst present an intuitive method for a user to navigate the surface of the Geodesic SOM. A new application of the distance transformation algorithm is then proposed to compute the path between two points on the surface of the SOM, which corresponds to two points in the data space. Discussions will then follow on how this application could be improved using some form of surface shape analysis. The new approach presented in this thesis would then be evaluated by analyzing the results of using the Geodesic SOM for manifold embedding and by carrying out data analyses using carbon dioxide emissions data