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

A new study measures the linearity of transformer feed-forward blocks, finding that linear recoverability (R²_lin) varies widely across blocks and is a learned property, not an architectural one. The analysis of GPT-2, Pythia-160m, and llama-160m shows that some blocks are nearly linear while others are strongly nonlinear, with implications for model compression and understanding transformer computation.

arXiv:2606.19379v1 Announce Type: new Abstract: 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.