Constructing Frequency Domains on Graphs in Near-Linear Time

Analysis of big data has become an increasingly relevant area of research, with data often represented on discrete networks both constructed and organic. While for structured domains, there exist intuitive definitions of signals and frequencies, the definitions are much less obvious for data sets associated with a given network. Often, the eigenvectors of an induced graph Laplacian are used to construct an orthogonal set of low-frequency vectors. For larger graphs, however, the computational cost of creating such structures becomes untenable, and the quality of the approximation is adequate only for signals near the span of the set. We propose a construction of a full basis of frequencies with computational complexity that is near-linear in time and linear in storage. Using this frequency domain, we can compress data sets on unstructured graphs more robustly and accurately than spectral-based constructions.