Machine learning and data processing techniques relying on covariance information are widespread as they identify meaningful patterns in unsupervised and unlabeled settings. As a prominent example, Principal Component Analysis (PCA) projects data points onto the eigenvectors of their covariance matrix, capturing the directions of maximum variance. This mapping, however, falls short in two directions: it fails to capture information in low-variance directions, relevant when, e.g., the data contains high-variance noise; and it provides unstable results in low-sample regimes, especially when covariance eigenvalues are close. CoVariance Neural Networks (VNNs), i.e., graph neural networks using the covariance matrix as a graph, show improved stability to estimation errors and learn more expressive functions in the covariance spectrum than PCA, but require training and operate in a labeled setup. To get the benefits of both worlds, we propose Covariance Scattering Transforms (CSTs), deep untrained networks that sequentially apply filters localized in the covariance spectrum to the input data and produce expressive hierarchical representations via nonlinearities. We define the filters as covariance wavelets that capture specific and detailed covariance spectral patterns. We improve CSTs' computational and memory efficiency via a pruning mechanism, and we prove that their error due to finite-sample covariance estimations is less sensitive to close covariance eigenvalues compared to PCA, improving their stability. Our experiments on age prediction from cortical thickness measurements on 4 datasets collecting patients with neurodegenerative diseases show that CSTs produce stable representations in low-data settings, as VNNs but without any training, and lead to comparable or better predictions w.r.t. more complex learning models.

In this paper, we propose a non-parametric task-agnostic representation learning method for cell complexes (CCs). Specifically, we propose a scattering transform for CCs that extends geometric scattering to CCs. In addition, we introduce scattering transforms for product cell complexes (PCCs), which are a particular type of CCs that can be factored as the Cartesian product of two smaller simplicial complexes (SCs). We utilize the intrinsic product structure of PCCs to develop separable filters across the two constituent factors of PCCs. The proposed scattering transform for PCCs is computationally efficient, as it can be expressed in terms of scattering on the individual SCs. The effectiveness of the proposed model is demonstrated through experiments on various tasks, such as trajectory classification, missing edge feature estimation, and edge feature prediction, on both synthetic and real-world PCCs.