No Equivariant Architecture Covers All Equivariant Attention

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 ($Θ(N^{d_{\text{model}}L})$), 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.

We develop a rigorous statistical theory of multi-head attention (MHA) as an ensemble of Nadaraya-Watson (NW) kernel regression estimators. Building on the algebraic identity between single-head softmax attention and the NW estimator, we prove that MHA is a structured ensemble of H NW estimators, each operating in a distinct learned projection subspace of the key space. We derive an explicit Bias-Variance-Covariance decomposition of the MHA mean squared error, showing that variance reduction depends not merely on the number of heads H but fundamentally on the decorrelation of head outputs. Decorrelation is governed by the principal angles between learned projection subspaces: orthogonal projections yield maximum variance reduction; aligned projections yield none. We introduce the Head Diversity Index (HDI), a computable spectral measure of inter-head decorrelation, and prove that MHA mean squared error is monotonically decreasing in HDI. This provides the first rigorous theoretical explanation for the empirically observed specialization of attention heads. Under a fixed total-dimension budget D = H * d_k, we solve the optimal head-dimension allocation problem, deriving the MSE-minimizing pair (H*, d_k*) from data distribution and regression smoothness. The solution yields a new architectural scaling law: the optimal per-head dimension grows logarithmically with training set size, while the optimal number of heads grows nearly linearly with the total budget D. Our framework unifies three strands of prior work: the NW theory of single-head attention, the general weighting theory for ensemble learning, and the decorrelation-variance-reduction isomorphism between biological and computational ensembles. Multi-head attention is the Transformer's instantiation of a universal principle: identical agents plus diversity-enforcing mechanisms yields emergent optimality.

We prove that a multi-head scaled dot product attention can be viewed as a parameter identification strategy. The ratio of unidentified parameters to the total number of parameters scales like the reciprocal of the number of heads ($1/2 \to 1/(2H)$), meaning models with more heads are structurally more identified. A subtle side effect of the mathematics observation that attention can never be fully identified. Similarly we also show that some bias terms can have no effect on softmax-based attention layers in both the single- and multiple-head settings, though this is mostly a curiosity that should have a marginal effect on model size and model training/prediction efficiency. We also touch on modern improvements to transformers including RoPE and GQA from this perspective, illustrating how those as well can improve the ratio of meaningful'' parameters to all parameters. Simple numerical examples demonstrate that training can indeed involve updates that overlap model-invariant subspaces that arise from a lack of identification. As part of our experiments we use a rebalancing'' approach that can ``fix'' updates that overlap unindentified subspaces but do not try to present evidence this should actually be adopted. Instead we simply view our numerical results as exploring and confirming the theoretical results. As a whole we discuss a purely mathematical/statistical explanation, identification, for why specific architectural choices in transformers may have improved performance.