Algebraic Dead Directions in LayerNorm Transformers: A Forward-Pass-Only Diagnostic at LLM Scale

Tejas Pradeep Shirodkar; P. J. Narayanan

cs.LGJun 17, 2026

Algebraic Dead Directions in LayerNorm Transformers: A Forward-Pass-Only Diagnostic at LLM Scale

Authors: Tejas Pradeep Shirodkar, P. J. Narayanan

Organizations: IIIT, Hyderabad

Abstract

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.

Explore similar work

Jul 1, 2026cs.LG

Measuring Dead Directions: Decomposing and Classifying Singular Structure off Canonical Alignment

We give a descent-free, alignment-free measurement of singular structure on trained networks. At a single frozen checkpoint the read recovers the order

k

of each dead direction from the directional-Fisher rate, the master invariant from which the per-direction learning coefficient

1/(2k)

follows exactly, in whatever basis the optimizer left. The same read classifies each direction, separating a genuine singularity, whose order the architecture fixes, from a flat gauge symmetry; the directional-Fisher magnitude settles the cases the order cannot. A pluggable detector supplies the directions for transformer, convolutional, and normalisation layers. The read recovers the architecture-predicted order across constructed cells and trained networks, including a fine-tuned vision transformer whose dead structure is the LayerNorm-kernel gauge and a from-scratch one whose compressed MLP forms a node-death at its activation order. Where the singular structure enumerates, the per-direction orders assemble, through the typed intersection of the loci, into the global coefficient

(λ, m)

matching the closed form. The method removes the canonical-alignment and descent preconditions of the underlying rate result, turning order-recovery into a deterministic, architecture-general reading. We then map its reach into the Watanabe triple: the order determines the universal singular fluctuation

ν(k)

, though a trained network's realized

ν

falls below it as the live structure absorbs the dead direction's data fluctuation, and the multiplicity recovers from the dominant structure under a single-locus assumption.

Tejas Pradeep Shirodkar

Fisher Information Order

Ask PDF

May 30, 2026cs.LG

Looped Transformers with Layer Normalization Provably Learn the Power Method

Transformers have achieved remarkable success across a wide range of applications, and a growing body of work suggests that part of their strength comes from their ability to learn and execute algorithmic procedures. However, our understanding of how transformers learn such algorithms remains limited, especially in the presence of layer normalization (LN). In this work, we study principal component prediction as a concrete testbed for understanding the training dynamics of transformers with LN. We prove that a looped linear transformer with LN, trained by gradient descent, converges to a solution that implements the power method, with each self-attention layer performing one power iteration. Notably, the model is trained only for principal component prediction, rather than being explicitly supervised to implement the power method. Our finding thus reveals an "algorithmic implicit bias" of looped transformers with LN: principal-component prediction can in principle be achieved by many mechanisms, yet gradient descent selects one that realizes the power method. We further provide a concrete comparison between transformers with and without LN: even with layerwise guidance from power iterations, a transformer without LN cannot exactly learn the power method, whereas the corresponding transformer with LN can, leading to a provable performance gap in principal component prediction. Our results provide, to our knowledge, the first theoretical analysis of the training dynamics of looped and single-layer transformers with LN, and shed light on the role of LN in transformer models.

Lyumin Wu, Chenyang Zhang, Yuan Cao

Transformer Layers Transformer Architectures Batch Normalization Principal Component Analysis Gradient Descent Power Method

Ask PDF

Aug 10, 2026cs.LG

Why Post-Norm Transformers Collapse: Attention Amplification and Gradient Repair Failure

Deep decoder-only Transformers often replace the original Post-Norm architecture with Pre-Norm variants because Post-Norm training is highly sensitive to warmup and learning rate under conventional initialization schemes. Although prior work has identified rank collapse and gradient vanishing as related symptoms, it remains poorly understood how causal attention creates high-similarity representations and why training dynamics fail to repair them. We give a two-stage analysis of Post-Norm rank collapse using token similarity as a scalar state variable. First, at initialization, causal attention acts approximately as a prefix-averaging operator that increases token similarity across depth, while the SwiGLU branch contributes only a smaller damping effect. Second, once training enters a high-similarity regime, growth of pre-normalization residual norms makes the RMSNorm backward factor contractive; under mild conditions, gradients to earlier layers decay geometrically. As a complementary result, we characterize the properties of a collapsed network: its best predictor is frequency distribution with relatively high loss floor, and gradients in collapsed layers vanish at frequency distribution. Experiments on 48-layer decoder-only Transformers trained on C4 dataset match the predicted initialization-time similarity growth and collapse-time gradient contraction, and show that collapsed runs stay near the predicted frequency loss. Together, these results distinguish the forward similarity amplification and backward repair incapacity in Post-Norm collapse, while also characterizing the behavior of collapsed networks.

Xingjian Wang, Qingyu Han, Xiaodong Luo +1

Batch Normalization Transformer Encoder Transformer Architectures Causal Attention Gradient Flow Low-Rank Factorization+1

Ask PDF