Signal Processing on the Permutahedron: Tight Spectral Frames for Ranked Data Analysis

Ranked data sets, where m judges/voters specify a preference ranking of n objects/candidates, are increasingly prevalent in contexts such as political elections, computer vision, recommender systems, and bioinformatics. The vote counts for each can be viewed an n! vector lying on the permutahedron, which is Cayley graph symmetric group with vertices labeled by permutations edge when two differ adjacent transposition. Leveraging combinatorial representation theory recent progress signal processing graphs, we investigate novel, scalable transform method to interpret exploit structure ranked data. We represent permutahedron using overcomplete dictionary atoms, captures both smoothness information about (typically focus spectral decomposition methods processing) structural symmetry from theory). These atoms have more naturally interpretable than any known basis signals they form Parseval frame, ensuring beneficial numerical properties energy preservation. develop specialized algorithms open software that take advantage improve scalability proposed method, making it applicable high-dimensional found applications.