Martian time-series unraveled: A multi-scale nested approach with factorial variational autoencoders

Unsupervised source separation involves unraveling an unknown set of source signals recorded through a mixing operator, with limited prior knowledge about the sources, and only access to a dataset of signal mixtures. This problem is inherently ill-posed and is further challenged by the variety of timescales exhibited by sources. Existing methods typically rely on a preselected window size that determines their operating timescale, limiting their capacity to handle multi-scale sources. To address this issue, we propose an unsupervised multi-scale clustering and source separation framework by leveraging wavelet scattering spectra that provide a low-dimensional representation of stochastic processes, capable of distinguishing between different non-Gaussian stochastic processes. Nested within this representation space, we develop a factorial Gaussian-mixture variational autoencoder that is trained to (1) probabilistically cluster sources at different timescales and (2) independently sample scattering spectra representations associated with each cluster. As the final stage, using samples from each cluster as prior information, we formulate source separation as an optimization problem in the wavelet scattering spectra representation space, aiming to separate sources in the time domain. When applied to the entire seismic dataset recorded during the NASA InSight mission on Mars, containing sources varying greatly in timescale, our multi-scale nested approach proves to be a powerful tool for disentangling such different sources, e.g., minute-long transient one-sided pulses (known as ``glitches'') and structured ambient noises resulting from atmospheric activities that typically last for tens of minutes. These results provide an opportunity to conduct further investigations into the isolated sources related to atmospheric-surface interactions, thermal relaxations, and other complex phenomena.