# Graph Signal Processing over a Probability Space of Shift Operators

arXiv: https://arxiv.org/abs/2108.09192 · HTML: https://mlsanity.com/paper/2108.09192/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Aug 2021.

## Built on by (most cited first, 5 of 5)

- [Graph Neural Networks with a Distribution of Parametrized Graphs](https://mlsanity.com/paper/2310.16401/) (2023)
- [Graph Distribution-valued Signals: A Wasserstein Space Perspective](https://mlsanity.com/paper/2509.25802/) (2025)
- [Spectral Convergence of Complexon Shift Operators](https://mlsanity.com/paper/2309.07169/) (2023)
- [Rotation-Parameterized Graph Fractional Fourier Transform: Definition, Properties, and Optimal Filtering](https://mlsanity.com/paper/2511.16111/) (2025)
- [Filter Learning for Subgraphs: Algebras and Performance Risk Bounds](https://mlsanity.com/paper/2607.21263/) (2026)

## Abstract (arXiv)

Graph signal processing (GSP) uses a shift operator to define a Fourier basis for the set of graph signals. The shift operator is often chosen to capture the graph topology. However, in many applications, the graph topology may be unknown a priori, its structure uncertain, or generated randomly from a predefined set for each observation. Each graph topology gives rise to a different shift operator. In this paper, we develop a GSP framework over a probability space of shift operators. We develop the corresponding notions of Fourier transform, MFC filters, and band-pass filters, which subsumes classical GSP theory as the special case where the probability space consists of a single shift operator. We show that an MFC filter under this framework is the expectation of random convolution filters in classical GSP, while the notion of bandlimitedness requires additional wiggle room from being simply a fixed point of a band-pass filter. We develop a mechanism that facilitates mapping from one space of shift operators to another, which allows our framework to be applied to a rich set of scenarios. We demonstrate how the theory can be applied by using both synthetic and real datasets.