The Leray Dimension of a Convex Code

TL;DR

This paper introduces the Leray dimension for convex codes, linking algebraic topology and commutative algebra to improve bounds on the minimal embedding dimension of neural codes.

Contribution

It defines the Leray dimension using Betti numbers from minimal free resolutions, providing a new algebraic method to bound neural code embedding dimensions.

Findings

Leray dimension offers stronger bounds than previous methods.

Betti numbers from minimal free resolutions encode topological information.

The algebraic computation of Leray dimension is demonstrated with examples.

Abstract

Convex codes were recently introduced as models for neural codes in the brain. Any convex code $\C$ has an associated minimal embedding dimension $d(\C)$, which is the minimal Euclidean space dimension such that the code can be realized by a collection of convex open sets. In this work we import tools from combinatorial commutative algebra in order to obtain better bounds on $d(\C)$ from an associated simplicial complex $\Delta(\C)$. In particular, we make a connection to minimal free resolutions of Stanley-Reisner ideals, and observe that they contain topological information that provides stronger bounds on $d(\C)$. This motivates us to define the Leray dimension $d_L(\C),$ and show that it can be obtained from the Betti numbers of such a minimal free resolution. We compare $d_L(\C)$ to two previously studied dimension bounds, obtained from Helly's theorem and the simplicial homology of $\Delta(\C)$. Finally, we show explicitly how $d_L(\C)$ can be computed algebraically, and illustrate this with examples.