# Estimating Latent Population Flows from Aggregated Data via Inversing Multi-Marginal Optimal Transport

arXiv: https://arxiv.org/abs/2212.14527 · HTML: https://mlsanity.com/paper/2212.14527/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 29 references that are in the map.
- First posted on arXiv 30 Dec 2022.

## Builds on (most cited first, 10 of 10)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Learning to Match via Inverse Optimal Transport](https://mlsanity.com/paper/1802.03644/) (2018)
- [Multi-marginal optimal transport and probabilistic graphical models](https://mlsanity.com/paper/2006.14113/) (2020)
- [Learning Stochastic Behaviour from Aggregate Data](https://mlsanity.com/paper/2002.03513/) (2020)
- [SISTA: learning optimal transport costs under sparsity constraints](https://mlsanity.com/paper/2009.08564/) (2020)
- [Inference with Aggregate Data: An Optimal Transport Approach](https://mlsanity.com/paper/2003.13933/) (2020)
- [Scalable computation of dynamic flow problems via multi-marginal graph-structured optimal transport](https://mlsanity.com/paper/2106.14485/) (2021)
- [Estimating ensemble flows on a hidden Markov chain](https://mlsanity.com/paper/1905.09119/) (2019)
- [Learning Hidden Markov Models from Aggregate Observations](https://mlsanity.com/paper/2011.11236/) (2020)
- [Consistently Estimating Markov Chains with Noisy Aggregate Data](https://mlsanity.com/paper/1604.04182/) (2016)

## Built on by (most cited first, 6 of 6)

- [A Variational Autoencoder for Neural Temporal Point Processes with Dynamic Latent Graphs](https://mlsanity.com/paper/2312.16083/) (2023)
- [Negative-Binomial Randomized Gamma Markov Processes for Heterogeneous Overdispersed Count Time Series](https://mlsanity.com/paper/2402.18995/) (2024)
- [Poisson-Gamma Modeling of Inter-Relational Dependencies in Dynamic Knowledge Graphs](https://mlsanity.com/paper/2607.02872/) (2026)
- [A Poisson-Gamma Dynamic Factor Model with Time-Varying Transition Dynamics](https://mlsanity.com/paper/2402.16297/) (2024)
- [Scaling up Dynamic Edge Partition Models via Stochastic Gradient MCMC](https://mlsanity.com/paper/2403.00044/) (2024)
- [Poisson-Gamma Dynamical Systems with Time-varying Transition Dynamics](https://mlsanity.com/paper/2609.00896/) (2026)

## Abstract (arXiv)

We study the problem of estimating latent population flows from aggregated count data. This problem arises when individual trajectories are not available due to privacy issues or measurement fidelity. Instead, the aggregated observations are measured over discrete-time points, for estimating the population flows among states. Most related studies tackle the problems by learning the transition parameters of a time-homogeneous Markov process. Nonetheless, most real-world population flows can be influenced by various uncertainties such as traffic jam and weather conditions. Thus, in many cases, a time-homogeneous Markov model is a poor approximation of the much more complex population flows. To circumvent this difficulty, we resort to a multi-marginal optimal transport (MOT) formulation that can naturally represent aggregated observations with constrained marginals, and encode time-dependent transition matrices by the cost functions. In particular, we propose to estimate the transition flows from aggregated data by learning the cost functions of the MOT framework, which enables us to capture time-varying dynamic patterns. The experiments demonstrate the improved accuracy of the proposed algorithms than the related methods in estimating several real-world transition flows.