# Learning Grid Cells as Vector Representation of Self-Position Coupled with Matrix Representation of Self-Motion

arXiv: https://arxiv.org/abs/1810.05597 · HTML: https://mlsanity.com/paper/1810.05597/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Oct 2018.

## Built on by (most cited first, 18 of 18)

- [A Review of Location Encoding for GeoAI: Methods and Applications](https://mlsanity.com/paper/2111.04006/) (2021)
- [Sphere2Vec: A General-Purpose Location Representation Learning over a Spherical Surface for Large-Scale Geospatial Predictions](https://mlsanity.com/paper/2306.17624/) (2023)
- [TorchSpatial: A Location Encoding Framework and Benchmark for Spatial Representation Learning](https://mlsanity.com/paper/2406.15658/) (2024)
- [RANGE: Retrieval Augmented Neural Fields for Multi-Resolution Geo-Embeddings](https://mlsanity.com/paper/2502.19781/) (2025)
- [Self-Supervised Learning of Representations for Space Generates Multi-Modular Grid Cells](https://mlsanity.com/paper/2311.02316/) (2023)
- [Learning non-Markovian Decision-Making from State-only Sequences](https://mlsanity.com/paper/2306.15156/) (2023)
- [On Conformal Isometry of Grid Cells: Learning Distance-Preserving Position Embedding](https://mlsanity.com/paper/2405.16865/) (2024)
- [Conformal Isometry of Lie Group Representation in Recurrent Network of Grid Cells](https://mlsanity.com/paper/2210.02684/) (2022)
- [Learning Neural Representation of Camera Pose with Matrix Representation of Pose Shift via View Synthesis](https://mlsanity.com/paper/2104.01508/) (2021)
- [Sphere2Vec: Multi-Scale Representation Learning over a Spherical Surface for Geospatial Predictions](https://mlsanity.com/paper/2201.10489/) (2022)
- [Emergence of Grid-like Representations by Training Recurrent Networks with Conformal Normalization](https://mlsanity.com/paper/2310.19192/) (2023)
- [nD-RoPE: A Generalized RoPE for n-Dimensional Position Embedding](https://mlsanity.com/paper/2606.12146/) (2026)
- [Tessellating The Earth](https://mlsanity.com/paper/2606.27514/) (2026)
- [Emergence of Double-slit Interference by Representing Visual Space in Artificial Neural Networks](https://mlsanity.com/paper/2205.10081/) (2022)
- [Place Cells as Multi-Scale Position Embeddings: Random Walk Transition Kernels for Path Planning](https://mlsanity.com/paper/2505.14806/) (2025)
- [ProM3E: Probabilistic Masked MultiModal Embedding Model for Ecology](https://mlsanity.com/paper/2511.02946/) (2025)
- [SAMCNet for Spatial-configuration-based Classification: A Summary of Results](https://mlsanity.com/paper/2112.12219/) (2021)
- [GridPE: A Grid Cell-Inspired Unified Position Embedding for Arbitrary-Dimensional Spaces](https://mlsanity.com/paper/2406.07049/) (2024)

## Abstract (arXiv)

This paper proposes a representational model for grid cells. In this model, the 2D self-position of the agent is represented by a high-dimensional vector, and the 2D self-motion or displacement of the agent is represented by a matrix that transforms the vector. Each component of the vector is a unit or a cell. The model consists of the following three sub-models. (1) Vector-matrix multiplication. The movement from the current position to the next position is modeled by matrix-vector multiplication, i.e., the vector of the next position is obtained by multiplying the matrix of the motion to the vector of the current position. (2) Magnified local isometry. The angle between two nearby vectors equals the Euclidean distance between the two corresponding positions multiplied by a magnifying factor. (3) Global adjacency kernel. The inner product between two vectors measures the adjacency between the two corresponding positions, which is defined by a kernel function of the Euclidean distance between the two positions. Our representational model has explicit algebra and geometry. It can learn hexagon patterns of grid cells, and it is capable of error correction, path integral and path planning.