Unrolling of Simplicial ElasticNet for Edge Flow Signal Reconstruction

The edge flow reconstruction task consists of retreiving edge flow signals from corrupted or incomplete measurements. This is typically solved by a regularized optimization problem on higher-order networks such as simplicial complexes and the corresponding regularizers are chosen based on prior knowledge. Tailoring this prior to the setting of interest can be challenging or it may not even be possible. Thus, we consider to learn this prior knowledge via a model-based deep learning approach. We propose a new regularized optimization problem for the simplicial edge flow reconstruction task, the simplicial ElasticNet, which combines the advantages of the
$\ell _{1}$
and
$\ell _{2}$
norms. We solve the simplicial ElasticNet problem via the multi-block alternating direction method of multipliers (ADMM) algorithm and provide conditions on its convergence. By unrolling the ADMM iterative steps, we develop a model-based neural network with a low requirement on the number of training data. This unrolling network replaces the fixed parameters in the iterative algorithm by learnable weights, thus exploiting the neural network's learning capability while preserving the iterative algorithm's interpretability. We enhance this unrolling network via simplicial convolutional filters to aggregate information from the edge flow neighbors, ultimately, improving the network learning expressivity. Extensive experiments on real-world and synthetic datasets validate the proposed approaches and show considerable improvements over both baselines and traditional non-model-based neural networks.