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Unsupervised source separation involves unraveling an unknown set of sourcesignals recorded through a mixing operator, with limited prior knowledge aboutthe sources, and only access to a dataset of signal mixtures. This problem isinherently ill-posed and is further challenged by the variety of time-scalesexhibited by sources in time series data. Existing methods typically rely on apreselected window size that limits their capacity to handle multi-scalesources. To address this issue, instead of operating in the time domain, wepropose an unsupervised multi-scale clustering and source separation frameworkby leveraging wavelet scattering covariances that provide a low-dimensionalrepresentation of stochastic processes, capable of distinguishing betweendifferent non-Gaussian stochastic processes. Nested within this representationspace, we develop a factorial Gaussian-mixture variational autoencoder that istrained to (1) probabilistically cluster sources at different time-scales and(2) independently sample scattering covariance representations associated witheach cluster. Using samples from each cluster as prior information, weformulate source separation as an optimization problem in the waveletscattering covariance representation space, resulting in separated sources inthe time domain. When applied to seismic data recorded during the NASA InSightmission on Mars, our multi-scale nested approach proves to be a powerful toolfor discriminating between sources varying greatly in time-scale, e.g.,minute-long transient one-sided pulses (known as ``glitches'') and structuredambient noises resulting from atmospheric activities that typically last fortens of minutes. These results provide an opportunity to conduct furtherinvestigations into the isolated sources related to atmospheric-surfaceinteractions, thermal relaxations, and other complex phenomena.

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