# GRIL: A 2-parameter Persistence Based Vectorization for Machine Learning

arXiv: https://arxiv.org/abs/2304.04970 · HTML: https://mlsanity.com/paper/2304.04970/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 59 references that are in the map.
- First posted on arXiv 11 Apr 2023.

## Builds on (most cited first, 25 of 25)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [XGBoost: A Scalable Tree Boosting System](https://mlsanity.com/paper/1603.02754/) (2016)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Weisfeiler and Leman Go Neural: Higher-order Graph Neural Networks](https://mlsanity.com/paper/1810.02244/) (2018)
- [TUDataset: A collection of benchmark datasets for learning with graphs](https://mlsanity.com/paper/2007.08663/) (2020)
- [Improving Graph Neural Network Expressivity via Subgraph Isomorphism Counting](https://mlsanity.com/paper/2006.09252/) (2020)
- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures](https://mlsanity.com/paper/1904.09378/) (2019)
- [Deep Learning with Topological Signatures](https://mlsanity.com/paper/1707.04041/) (2017)
- [Topological Graph Neural Networks](https://mlsanity.com/paper/2102.07835/) (2021)
- [A Topology Layer for Machine Learning](https://mlsanity.com/paper/1905.12200/) (2019)
- [Graph Filtration Learning](https://mlsanity.com/paper/1905.10996/) (2019)
- [A Topological Regularizer for Classifiers via Persistent Homology](https://mlsanity.com/paper/1806.10714/) (2018)
- [Community Detection on Networks with Ricci Flow](https://mlsanity.com/paper/1907.03993/) (2019)
- [A Refined Laser Method and Faster Matrix Multiplication](https://mlsanity.com/paper/2010.05846/) (2020)
- [Understanding the Representation Power of Graph Neural Networks in Learning Graph Topology](https://mlsanity.com/paper/1907.05008/) (2019)
- [Interactive Visualization of 2-D Persistence Modules](https://mlsanity.com/paper/1512.00180/) (2015)
- [A Kernel for Multi-Parameter Persistent Homology](https://mlsanity.com/paper/1809.10231/) (2018)
- [ToDD: Topological Compound Fingerprinting in Computer-Aided Drug Discovery](https://mlsanity.com/paper/2211.03808/) (2022)
- [PersGNN: Applying Topological Data Analysis and Geometric Deep Learning to Structure-Based Protein Function Prediction](https://mlsanity.com/paper/2010.16027/) (2020)
- [Generalized Persistence Diagrams for Persistence Modules over Posets](https://mlsanity.com/paper/1810.11517/) (2018)
- [Fast Computation of Zigzag Persistence](https://mlsanity.com/paper/2204.11080/) (2022)
- [Computing Generalized Rank Invariant for 2-Parameter Persistence Modules via Zigzag Persistence and its Applications](https://mlsanity.com/paper/2111.15058/) (2021)

## Built on by (most cited first, 13 of 13)

- [Stable Vectorization of Multiparameter Persistent Homology using Signed Barcodes as Measures](https://mlsanity.com/paper/2306.03801/) (2023)
- [Graphcode: Learning from multiparameter persistent homology using graph neural networks](https://mlsanity.com/paper/2405.14302/) (2024)
- [A Framework for Fast and Stable Representations of Multiparameter Persistent Homology Decompositions](https://mlsanity.com/paper/2306.11170/) (2023)
- [D-GRIL: End-to-End Topological Learning with 2-parameter Persistence](https://mlsanity.com/paper/2406.07100/) (2024)
- [EMP: Effective Multidimensional Persistence for Graph Representation Learning](https://mlsanity.com/paper/2401.13713/) (2024)
- [Point-Level Topological Representation Learning on Point Clouds](https://mlsanity.com/paper/2406.02300/) (2024)
- [Quasi Zigzag Persistence: A Topological Framework for Analyzing Time-Varying Data](https://mlsanity.com/paper/2502.16049/) (2025)
- [TopInG: Topologically Interpretable Graph Learning via Persistent Rationale Filtration](https://mlsanity.com/paper/2510.05102/) (2025)
- [A Persistent Homology Design Space for 3D Point Cloud Deep Learning](https://mlsanity.com/paper/2604.04299/) (2026)
- [From Persistence to Survival: Hypothesis Testing, Effect Sizes and Vectorisation for Topological Features](https://mlsanity.com/paper/2606.11911/) (2026)
- [xPerT: Extended Persistence Transformer](https://mlsanity.com/paper/2410.14193/) (2024)
- [Heat Field Signatures: From Point Clouds to Smooth Geometry](https://mlsanity.com/paper/2609.07975/) (2026)
- [COMPLEX: A Closed-Form Certified Embedding of Multiparameter Persistence Modules](https://mlsanity.com/paper/2609.22012/) (2026)

## Abstract (arXiv)

1-parameter persistent homology, a cornerstone in Topological Data Analysis (TDA), studies the evolution of topological features such as connected components and cycles hidden in data. It has been applied to enhance the representation power of deep learning models, such as Graph Neural Networks (GNNs). To enrich the representations of topological features, here we propose to study 2-parameter persistence modules induced by bi-filtration functions. In order to incorporate these representations into machine learning models, we introduce a novel vector representation called Generalized Rank Invariant Landscape (GRIL) for 2-parameter persistence modules. We show that this vector representation is 1-Lipschitz stable and differentiable with respect to underlying filtration functions and can be easily integrated into machine learning models to augment encoding topological features. We present an algorithm to compute the vector representation efficiently. We also test our methods on synthetic and benchmark graph datasets, and compare the results with previous vector representations of 1-parameter and 2-parameter persistence modules. Further, we augment GNNs with GRIL features and observe an increase in performance indicating that GRIL can capture additional features enriching GNNs. We make the complete code for the proposed method available at https://github.com/soham0209/mpml-graph.