Expressivity of Transformers: A Tropical Geometry Perspective

TL;DR

This paper introduces a tropical geometry framework to analyze the expressivity of transformers, revealing their exact spatial partitioning capabilities and combinatorial complexity growth with network parameters.

Contribution

It models self-attention as a tropical rational map, establishes bounds on linear regions, and demonstrates the stability of these partitions under soft attention.

Findings

Transformers evaluate to Power Voronoi Diagrams in the zero-temperature limit.

Multi-head self-attention increases polyhedral complexity exponentially with the number of heads.

Number of linear regions scales as Θ(N^{d_model}L), showing combinatorial explosion.

Abstract

To quantify the geometric expressivity of transformers, we introduce a tropical geometry framework to characterize their exact spatial partitioning capabilities. By modeling self-attention as a vector-valued tropical rational map, we prove it evaluates exactly to a Power Voronoi Diagram in the zero-temperature limit. Building on this equivalence, we establish a combinatorial rationale for Multi-Head Self-Attention (MHSA): via the Minkowski sum of Newton polytopes, multi-head aggregation expands the polyhedral complexity to $\mathcal{O}(N^H)$, overcoming the $\mathcal{O}(N)$ bottleneck of single heads. Extending this to deep architectures, we derive the first tight asymptotic bounds on the number of linear regions in transformers ($\Theta(N^{d_{\text{model}}L})$), demonstrating a combinatorial explosion driven intrinsically by sequence length $N$, ambient embedding dimension $d_{\text{model}}$, and network depth $L$. Importantly, we guarantee that this idealized polyhedral skeleton is geometrically stable: finite-temperature soft attention preserves these topological partitions via exponentially tight differential approximation bounds.