Joint Mapper and Topological Signal Processing for Point Cloud Classification

Mapper is a widely used method for extracting compact topological descriptors from high-dimensional data by representing them as simplicial complexes. It applies a filter (lens) function to partition the data domain and build a simplicial complex that captures the underlying geometric structure together with lens-induced signals. Recently, Topological Signal Processing (TSP) has emerged as a powerful framework for analyzing and processing signals defined on simplicial complexes. In this paper, we jointly leverage Mapper and TSP for point cloud classification. Mapper provides a data-driven topological domain that encodes the global structure of the cloud, while TSP enables spectral analysis and sparse signal representations on the resulting complex. Using the Fiedler vector of the graph Laplacian as the lens, we construct the Mapper simplicial complex and define signals on its nodes and edges. We then compute sparse edge-signal representations by balancing reconstruction accuracy and sparsity in the simplicial Fourier domain. Numerical experiments on synthetic datasets show high classification performance across geometric shapes with a significant simplification of original data.