Coinductive guide to inductive transformer heads

TL;DR

This paper presents a novel algebraic framework using combinatorial Hopf algebras to unify and analyze transformer models, revealing their linear time-invariant nature and intrinsic loss function gradients within layers.

Contribution

It introduces a coinductive perspective on transformer heads, modeling them with Hopf algebra concepts, and explains how attention mechanisms compute generalized convolutions without explicit backward passes.

Findings

Transformers can be viewed as linear time-invariant systems.

Attention computes a generalized convolution transform.

Layers inherently contain loss function gradients, eliminating the need for backpropagation.

Abstract

We argue that all building blocks of transformer models can be expressed with a single concept: combinatorial Hopf algebra. Transformer learning emerges as a result of the subtle interplay between the algebraic and coalgebraic operations of the combinatorial Hopf algebra. Viewed through this lens, the transformer model becomes a linear time-invariant system where the attention mechanism computes a generalized convolution transform and the residual stream serves as a unit impulse. Attention-only transformers then learn by enforcing an invariant between these two paths. We call this invariant Hopf coherence. Due to this, with a degree of poetic license, one could call combinatorial Hopf algebras "tensors with a built-in loss function gradient". This loss function gradient occurs within the single layers and no backward pass is needed. This is in contrast to automatic differentiation which happens across the whole graph and needs a explicit backward pass. This property is the result of the fact that combinatorial Hopf algebras have the surprising property of calculating eigenvalues by repeated squaring.