# How Linear Is a Transformer Feed-Forward Block? Per-Block Linear Recoverability Is Learned, Not Architectural

arXiv: https://arxiv.org/abs/2606.19379 · HTML: https://mlsanity.com/paper/2606.19379/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 17 references that are in the map.
- First posted on arXiv 12 Jun 2026.

## Builds on (most cited first, 11 of 11)

- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)
- [Pointer Sentinel Mixture Models](https://mlsanity.com/paper/1609.07843/) (2016)
- [GLU Variants Improve Transformer](https://mlsanity.com/paper/2002.05202/) (2020)
- [Pythia: A Suite for Analyzing Large Language Models Across Training and Scaling](https://mlsanity.com/paper/2304.01373/) (2023)
- [Understanding intermediate layers using linear classifier probes](https://mlsanity.com/paper/1610.01644/) (2016)
- [Transformer Feed-Forward Layers Are Key-Value Memories](https://mlsanity.com/paper/2012.14913/) (2020)
- [LLM.int8(): 8-bit Matrix Multiplication for Transformers at Scale](https://mlsanity.com/paper/2208.07339/) (2022)
- [Revisiting Model Stitching to Compare Neural Representations](https://mlsanity.com/paper/2106.07682/) (2021)
- [Deep Polynomial Neural Networks](https://mlsanity.com/paper/2006.13026/) (2020)
- [Higher-Order Factorization Machines](https://mlsanity.com/paper/1607.07195/) (2016)

## Abstract (arXiv)

Transformer feed-forward networks (FFNs) are often treated as nonlinear stores of computation, yet how nonlinear a trained FFN block actually is has rarely been measured. We treat each FFN as a position-wise input-to-output map and split it into the exact least-squares linear approximation plus a residual. The held-out variance the closed-form linear map explains defines a block's linear recoverability (R^2_lin), an optimiser-free measure of its linearity. Across all twelve blocks of GPT-2, Pythia-160m, and llama-160m, R^2_lin is highly heterogeneous and non-monotone with depth, ranging from near-linear (>0.99) to strongly nonlinear (<0.3) between adjacent blocks, and is not set by the activation function: same-width GELU models GPT-2 and Pythia-160m have sharply different profiles, so recoverability is a learned property of individual trained blocks, not an architectural one. A low-rank bilinear probe of the residual recovers only a few points of R^2, with gain uncorrelated with residual nonlinearity: the unrecovered computation is not a single position-wise product but higher-order or distributed structure. The measurement also serves as a targeted compression signal: recoverable blocks admit large single-layer replacements (GPT-2's early FFN at 8x fewer parameters for +0.77 perplexity), while low-recoverability blocks flag where this is unsafe. It further exposes a methodological pitfall: trained linear baselines can badly under-converge on ill-conditioned transformer activations, so we report the exact closed-form least-squares ceiling throughout.