Graph recovery from graph wave equation

We propose a method by which to recover an underlying graph from a set of multivariate wave signals that is discretely sampled from a solution of the graph wave equation. Herein, the graph wave equation is defined with the graph Laplacian, and its solution is explicitly given as a mode expansion of the Laplacian eigenvalues and eigenfunctions. For graph recovery, our idea is to extract modes corresponding to the square root of the eigenvalues from the discrete wave signals using the DMD method, and then to reconstruct the graph (Laplacian) from the eigenfunctions obtained as amplitudes of the modes. Moreover, in order to estimate modes more precisely, we modify the DMD method under an assumption that only stationary modes exist, because graph wave functions always satisfy this assumption. In conclusion, we demonstrate the proposed method on the wave signals over a path graph. Since our graph recovery procedure can be applied to non-wave signals, we also check its performance on human joint sensor time-series data.

Geometric Formulation for Discrete Points and its Applications

We introduce a novel formulation for geometry on discrete points. It is based on a universal differential calculus, which gives a geometric description of a discrete set by the algebra of functions. We expand this mathematical framework so that it is consistent with differential geometry, and works on spectral graph theory and random walks. Consequently, our formulation comprehensively demonstrates many discrete frameworks in probability theory, physics, applied harmonic analysis, and machine learning. Our approach would suggest the existence of an intrinsic theory and a unified picture of those discrete frameworks.