Biological agents navigate familiar environments not by recomputing a route for each new goal, but by reusing a map learned once and reading off the path as goals change. Existing artificial cognitive‑map models mimic this reuse, yet their guidance is not explicitly grounded in additive heterogeneous edge costs, and they often rely on state‑indexed high‑rank spectral constructions that cause storage to explode as the environment grows.
To address these issues we introduce BCM (Bellman‑Grounded Cognitive Map). Its main contributions are:
- Bellman self‑supervised objective: minimize $$\mathcal{L}= \sum_{s}\bigl\|V(s)-\min_{a}[c(s,a)+V(s')]\bigr\|^{2}$$ which writes local edge costs $c(s,a)$ directly into the map, allowing instant queries of the shortest weighted path for any goal.
- Compact coordinate encoding: represent each state with a low‑dimensional vector $\mathbf{z}_s\in\mathbb{R}^d$, replacing the high‑rank spectral matrix and achieving sublinear memory growth.
On weighted grids with up to $N=1600$ nodes, BCM attains 100% success and a mean Gap of only $5\%$ relative to exact Dijkstra search, whereas a connectivity‑based spectral baseline suffers about $45\%$ Gap. Importantly, when the graph size increases from $N=400$ to $N=3600$, memory usage grows sublinearly while performance remains competitive, demonstrating scalability to large, complex environments.
In summary, BCM embeds additive route costs into a compact, reusable cognitive‑map representation, bridging biological flexibility and optimal path planning.
Review