# Two-stage Denoising Diffusion Model for Source Localization in Graph Inverse Problems

arXiv: https://arxiv.org/abs/2304.08841 · HTML: https://mlsanity.com/paper/2304.08841/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 15 of its 32 references that are in the map.
- First posted on arXiv 18 Apr 2023.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 15 of 15)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generation](https://mlsanity.com/paper/2203.02923/) (2022)
- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)
- [Autoregressive Denoising Diffusion Models for Multivariate Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2101.12072/) (2021)
- [Score-based Generative Modeling of Graphs via the System of Stochastic Differential Equations](https://mlsanity.com/paper/2202.02514/) (2022)
- [Position-aware Graph Neural Networks](https://mlsanity.com/paper/1906.04817/) (2019)
- [Accurate Learning of Graph Representations with Graph Multiset Pooling](https://mlsanity.com/paper/2102.11533/) (2021)
- [DyDiff-VAE: A Dynamic Variational Framework for Information Diffusion Prediction](https://mlsanity.com/paper/2106.03251/) (2021)
- [Source Localization of Graph Diffusion via Variational Autoencoders for Graph Inverse Problems](https://mlsanity.com/paper/2206.12327/) (2022)
- [Catch'Em All: Locating Multiple Diffusion Sources in Networks with Partial Observations](https://mlsanity.com/paper/1611.06963/) (2016)
- [An Invertible Graph Diffusion Neural Network for Source Localization](https://mlsanity.com/paper/2206.09214/) (2022)
- [Inf-VAE: A Variational Autoencoder Framework to Integrate Homophily and Influence in Diffusion Prediction](https://mlsanity.com/paper/2001.00132/) (2020)

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

- [Around the World in 80 Timesteps: A Generative Approach to Global Visual Geolocation](https://mlsanity.com/paper/2412.06781/) (2024)
- [Learning Regularization for Graph Inverse Problems](https://mlsanity.com/paper/2408.10436/) (2024)
- [Structure-prior Informed Diffusion Model for Graph Source Localization with Limited Data](https://mlsanity.com/paper/2502.17928/) (2025)

## Abstract (arXiv)

Source localization is the inverse problem of graph information dissemination and has broad practical applications. However, the inherent intricacy and uncertainty in information dissemination pose significant challenges, and the ill-posed nature of the source localization problem further exacerbates these challenges. Recently, deep generative models, particularly diffusion models inspired by classical non-equilibrium thermodynamics, have made significant progress. While diffusion models have proven to be powerful in solving inverse problems and producing high-quality reconstructions, applying them directly to the source localization is infeasible for two reasons. Firstly, it is impossible to calculate the posterior disseminated results on a large-scale network for iterative denoising sampling, which would incur enormous computational costs. Secondly, in the existing methods for this field, the training data itself are ill-posed (many-to-one); thus simply transferring the diffusion model would only lead to local optima. To address these challenges, we propose a two-stage optimization framework, the source localization denoising diffusion model (SL-Diff). In the coarse stage, we devise the source proximity degrees as the supervised signals to generate coarse-grained source predictions. This aims to efficiently initialize the next stage, significantly reducing its convergence time and calibrating the convergence process. Furthermore, the introduction of cascade temporal information in this training method transforms the many-to-one mapping relationship into a one-to-one relationship, perfectly addressing the ill-posed problem. In the fine stage, we design a diffusion model for the graph inverse problem that can quantify the uncertainty in the dissemination. The proposed SL-Diff yields excellent prediction results within a reasonable sampling time at extensive experiments.