Looped Transformers with Layer Normalization Provably Learn the Power Method

In the paper we show that there is an analogy between the operations occurring in a layer of a transformer (projections and layer normalizations, disregarding the feedforward neural network) and a step in the power method. Coherently with this analogy, we show that passing through a layer the tokens tend to be tilted towards the principal eigenvector of a matrix which is the product of the output and value weight matrices of that layer. In the special case of a transformer with shared weights (i.e., in which all layers have identical weights) then the alignment with this principal eigenvector is particularly evident empirically, and can also be shown analytically. The analogy also suggests a method to steer the output of the transformer towards an arbitrary desired direction in token space.

Pretrained transformers sit near singular minima of the loss, where the Fisher information metric degenerates along dead directions: directions in parameter space along which the directional Fisher vanishes. Locating such a direction normally needs a forward pass and an eigendecomposition of activations, or a sampling-based complexity estimate; none returns a direction computable from the network's parameters alone. We give one, for LayerNorm transformers. The inverse-scale direction $γ^{-1}/\|γ^{-1}\|$ of the LayerNorm affine is an exact algebraic kernel of the post-final-norm centred activation covariance, for any input distribution, and induces a corresponding dead direction in parameter space. It is read from the LN scale parameter alone, with no forward or backward pass and no eigensolve: the cheapest dead-direction read, specific to LayerNorm. We test it on $14$ pretrained transformers ($9$ LayerNorm, $5$ RMSNorm; $160$M-$35$B; language and vision objectives). At random initialisation the predicted direction matches the measured bottom singular direction (one forward pass, direct SVD) to four decimal places on $9/9$ LayerNorm models, and is correctly absent on $5/5$ RMSNorm models, which lack the mean-subtraction projector that creates it. On the trained checkpoint the covariance eigenvalue along this direction deepens by ${\sim}10^3\times$ and further dead directions open; the random-init-to-trained gap is a one-forward-pass, per-checkpoint readout of singular structure along the predicted coordinate. Two consequences follow in closed form: the residual stream's smallest singular value is preserved block-to-block on $13/14$ transformers measured on their own input distribution, the one exception (Gemma$4$-$31$B) a genuine dead direction the same read pinpoints; and the kernel direction's presence classifies a transformer's normalisation from the parameters alone.

In pre-LayerNorm looped transformers, LayerNorm inside the recurrent block acts as an implicit gain controller: by coupling the block's local Lipschitz constant inversely to the activation scale, it renders the recurrence Jacobian non-normal -- asymptotically contractive at every verified fixed point even where its operator norm exceeds 1 -- so the true stability budget is the spectral margin, not an operator-norm bound. That margin depletes as the carry $ρ\to 1$, and a minority of initializations never converge to a fixed point at all, so the diagonal carry constraint $ρ(\bar{A}) < 1$ is necessary but not sufficient for convergence of the full recurrence. Training experiments across six tasks, including a controlled ablation, reveal that the linear carry is not the depth-memory mechanism: gradient descent routes memory through the block's more expressive nonlinear recurrence and leaves the stability-constrained carry at rest -- the carry's role is stabilization, not memory. We characterize the boundary of this claim: on tasks with axis-aligned per-channel structure, gradient descent does recruit the carry. All results are derived analytically and verified in a from-scratch, CPU-scale implementation; verification at larger scale is needed.