# A Unified Geometric Framework for Developmental Analysis of Spatial Transcriptomic Data

arXiv: https://arxiv.org/abs/2608.15306 · HTML: https://mlsanity.com/paper/2608.15306/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 68 references that are in the map.
- First posted on arXiv 15 Aug 2026.

## Builds on (most cited first, 11 of 11)

- [Graph Neural Networks: A Review of Methods and Applications](https://mlsanity.com/paper/1812.08434/) (2018)
- [TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics](https://mlsanity.com/paper/2002.04461/) (2020)
- [Proximal Optimal Transport Modeling of Population Dynamics](https://mlsanity.com/paper/2106.06345/) (2021)
- [Poisson Learning: Graph Based Semi-Supervised Learning At Very Low Label Rates](https://mlsanity.com/paper/2006.11184/) (2020)
- [CO-Optimal Transport](https://mlsanity.com/paper/2002.03731/) (2020)
- [Hypergraph Co-Optimal Transport: Metric and Categorical Properties](https://mlsanity.com/paper/2112.03904/) (2021)
- [Efficient Trajectory Inference in Wasserstein Space Using Consecutive Averaging](https://mlsanity.com/paper/2405.19679/) (2024)
- [Multi-Marginal Gromov-Wasserstein Transport and Barycenters](https://mlsanity.com/paper/2205.06725/) (2022)
- [Laplace Learning in Wasserstein Space](https://mlsanity.com/paper/2511.13229/) (2025)
- [Conic Formulations of Transport Metrics for Unbalanced Measure Networks and Hypernetworks](https://mlsanity.com/paper/2508.10888/) (2025)
- [Learning to Orchestrate Agents under Uncertainty](https://mlsanity.com/paper/2605.27073/) (2026)

## Abstract (arXiv)

High-throughput single-cell and spatial transcriptomic technologies provide high-resolution snapshots of heterogeneous cellular states, but their destructive nature prevents repeated measurements of the same cells over time. Consequently, temporal and spatial dynamics must be inferred from independently sampled, unaligned cell populations, making it challenging to reconstruct developmental trajectories. Optimal transport (OT) offers a geometric framework for aligning cell populations and inferring developmental trajectories, but many existing approaches focus on modeling the evolution of distributions of cells in gene expression space rather than the relational structure encoded by gene expression networks. To address this limitation, we introduce a geometric framework for analyzing the spatiotemporal evolution of gene expression networks through embeddings in Gromov--Wasserstein (GW) space. By representing each developmental stage as a graph combining gene expression and spatial proximity, our approach enables comparisons of network structure across time, continuous interpolation between developmental stages via GW geodesics, and quantification of network-level changes using Ollivier-Ricci curvature. We evaluate our framework on a spatiotemporal transcriptomic textitDrosophila dataset and show that GW geodesic interpolations reproduce main trends in curvature dynamics observed in empirical gene expression networks. Agreement with higher-order Co-Optimal Transport (COOT) distances, which jointly represent spatial and temporal information, further validates the framework and suggests that hypernetwork representations successfully record salient biological changes across time. In general, our approach provides a unified geometric approach to study dynamically evolving biological networks.