# Place Cells as Multi-Scale Position Embeddings: Random Walk Transition Kernels for Path Planning

arXiv: https://arxiv.org/abs/2505.14806 · HTML: https://mlsanity.com/paper/2505.14806/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 82 references that are in the map.
- First posted on arXiv 20 May 2025.

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

- [On Evaluation of Embodied Navigation Agents](https://mlsanity.com/paper/1807.06757/) (2018)
- [Diffusion-based Generation, Optimization, and Planning in 3D Scenes](https://mlsanity.com/paper/2301.06015/) (2023)
- [Neural Diffusion Processes](https://mlsanity.com/paper/2206.03992/) (2022)
- [Emergence of Maps in the Memories of Blind Navigation Agents](https://mlsanity.com/paper/2301.13261/) (2023)
- [Learning Grid Cells as Vector Representation of Self-Position Coupled with Matrix Representation of Self-Motion](https://mlsanity.com/paper/1810.05597/) (2018)
- [Neural population geometry and optimal coding of tasks with shared latent structure](https://mlsanity.com/paper/2402.16770/) (2024)
- [On Path Integration of Grid Cells: Group Representation and Isotropic Scaling](https://mlsanity.com/paper/2006.10259/) (2020)
- [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)
- [Emergence of Grid-like Representations by Training Recurrent Networks with Conformal Normalization](https://mlsanity.com/paper/2310.19192/) (2023)
- [A minimalistic representation model for head direction system](https://mlsanity.com/paper/2411.10596/) (2024)

## Built on by (most cited first, 1 of 1)

- [Learning to Plan from Random Exploration](https://mlsanity.com/paper/2609.38383/) (2026)

## Abstract (arXiv)

The hippocampus supports spatial navigation by encoding cognitive maps through collective place cell activity. We model the place cell population as non-negative spatial embeddings derived from the spectral decomposition of multi-step random walk transition kernels. In this framework, inner product or equivalently Euclidean distance between embeddings encode similarity between locations in terms of their transition probability across multiple scales, forming a cognitive map of adjacency. The combination of non-negativity and inner-product structure naturally induces sparsity, providing a principled explanation for the localized firing fields of place cells without imposing explicit constraints. The temporal parameter that defines the diffusion scale also determines field size, aligning with the hippocampal dorsoventral hierarchy. Our approach constructs global representations efficiently through recursive composition of local transitions, enabling smooth, trap-free navigation and preplay-like trajectory generation. Moreover, theta phase arises intrinsically as the angular relation between embeddings, linking spatial and temporal coding within a single representational geometry.