Topology-Preserving Projection Pursuit via Neural Networks
ABSTRACT: Data in high dimensional subspaces introduce obstacles in analysis and visualization which are not present in lower dimensions. To combat these issues, we use dimension reduction, a statistical tool for mapping data from high to low dimensions while retaining as much information as possible. This method reduces data to a manageable size for storage and computation, cleans data of redundancy, and allows for ease of visualization. Topology-preserving dimension reduction in particular allows us to maintain topological features and the shape of the dataset. We are motivated by such an existing linear method that uses orthogonal projections to preserve k-th order holes and automatic differentiation to find the optimal projection. In our research, we deviate from linear maps and generate nonlinear projections via neural networks. We measure the accuracy of our algorithm by computing the distance between persistence diagrams of the original and projected data. Our aim is to improve the performance of topology-preserving dimension reduction methods by discovering relationships in data that were otherwise impossible with linear methods.Report coming soon...