# Signal Processing on Product Spaces

arXiv: https://arxiv.org/abs/2303.10495 · HTML: https://mlsanity.com/paper/2303.10495/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Mar 2023.

## Built on by (most cited first, 3 of 3)

- [Imputation of Time-varying Edge Flows in Graphs by Multilinear Kernel Regression and Manifold Learning](https://mlsanity.com/paper/2409.05135/) (2024)
- [Kernel Regression of Multi-Way Data via Tensor Trains with Hadamard Overparametrization: The Dynamic Graph Flow Case](https://mlsanity.com/paper/2509.22197/) (2025)
- [Kernel Regression with Tensor Trains and Hadamard Overparameterization](https://mlsanity.com/paper/2607.17390/) (2026)

## Abstract (arXiv)

We establish a framework for signal processing on product spaces of simplicial and cellular complexes. For simplicity, we focus on the product of two complexes representing time and space, although our results generalize naturally to products of simplicial complexes of arbitrary dimension. Our framework leverages the structure of the eigenmodes of the Hodge Laplacian of the product space to jointly filter along time and space. To this end, we provide a decomposition theorem of the Hodge Laplacian of the product space, which highlights how the product structure induces a decomposition of each eigenmode into a spatial and temporal component. Finally, we apply our method to real world data, specifically for interpolating trajectories of buoys in the ocean from a limited set of observed trajectories.