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

• URL: http://arxiv.org/abs/2606.19491v1
• Date: Wed, 17 Jun 2026 18:28:37 GMT
• Title: Algebraic Dead Directions in LayerNorm Transformers: A Forward-Pass-Only Diagnostic at LLM Scale
• Abstract summary: Inverse-scale direction $-1/|-1|$ of the LayerNorm affine is an exact kernel of the post-final-norm centred activation covariance.<n>It is read from the LN scale parameter alone, with no forward or backward pass and no eigensolve.
• Score: 5.67642958049511
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

Related papers

• Round-Trip Consistency: Bidirectional Diffusion Models Can Predict Their Own Rollout Errors [51.56484100374058]
  Autoregressive models accumulate error over long rollouts, yet at deployment there is no ground truth to measure it against.<n>We train a single conditional latent diffusion model that steps a dynamical system forward or backward in time via a direction flag.<n>We show that this bidirectionality supplies a measurement-free test-time error signal.
  arXiv  Detail & Related papers  (2026-08-01T13:49:46Z)
• Dead-Direction Signatures: A Cheap Spectral Reading of Singular Complexity [5.67642958049511]
  Singular learning theory characterises the complexity of a deep network through the geometry of its loss singularities.<n>We introduce DeadDirection Signatures (DDS), a family of cheap closed-form spectral readings of singular structure.<n>Each reads a network's activation matrix or per-sample-gradient FisherGram at a chosen layer, replacing the posterior chain spectral linear algebra.
  arXiv  Detail & Related papers  (2026-06-19T06:49:09Z)
• Dead Directions: Geometric Singular Learning [0.0]
  Singular learning theory and information geometry have studied the same parameter spaces in mostly separate vocabularies.<n>We bridge them through one primitive, the dead direction: a unit vector along which the Fisher metric degenerates.<n>A selection rule on smooth fibres translates this rate into Watanabe's single-direction contribution to the real log canonical threshold.<n>We then lift this rate to a deep network: a multi-layer K-FAC factorisation writes each Fisher block as a product of activation- and gradient-side rates.
  arXiv  Detail & Related papers  (2026-06-04T09:54:08Z)
• Hidden-State Privacy Has an Empty Middle [51.56484100374058]
  We prove a complementary Fisher-ball lower bound: every full-rank Gaussian release at $O(1)$ Fisher utility admits a direction whose Mahalanobis signal grows linearly in hidden width.<n>A split-memory transformer trained from scratch reaches $G_mathrmMah in [20, 33]$ at 90M and maintains a $6$--$24times$ advantage over same-budget GPT baselines from 30M to 1B at a fixed-token language loss penalty.
  arXiv  Detail & Related papers  (2026-05-21T20:12:09Z)
• Where Pretraining writes and Alignment reads: the asymmetry of Transformer weight space [2.2940141855172036]
  Cross-entropy pretraining and preference alignment update the same transformer weights, but leave geometrically distinct traces.<n>We characterise this asymmetry with a relative-subspace-fraction probe that tracks how weight deltas align with residual-stream activation subspaces.
  arXiv  Detail & Related papers  (2026-05-15T20:00:59Z)
• Dimensional Criticality at Grokking Across MLPs and Transformers [2.652953665748039]
  Abrupt transitions between distinct dynamical regimes are a hallmark of complex systems.<n>We introduce textbfTDU--OFC (Thresholded Diffusion Update--Olami-Feder-Christensen), an offline avalanche probe.<n>Across Transformers trained on modular addition and generalizations trained on XOR, we discover a localized crossing of the diffusion baseline $D=1$.
  arXiv  Detail & Related papers  (2026-04-06T13:43:20Z)
• Robust Layerwise Scaling Rules by Proper Weight Decay Tuning [50.11170157029911]
  In modern scale-invariant architectures, training quickly enters an degrading-governed steady state.<n>We introduce a weight-decay scaling rule for AdamW that preserves sublayer gain across widths.<n>Our results extend $mu$P beyond the near-init regime by explicitly controlling the steady-state scales set by parameters.
  arXiv  Detail & Related papers  (2025-10-17T02:58:35Z)
• Emergence and scaling laws in SGD learning of shallow neural networks [64.48316762675141]
  We study the complexity of online gradient descent (SGD) for learning a two-layer neural network with $P$ neurons on isotropic Gaussian data.<n>We provide a precise analysis of SGD dynamics for the training of a student two-layer network to minimize the mean squared error (MSE) objective.
  arXiv  Detail & Related papers  (2025-04-28T16:58:55Z)
• Improved Algorithm for Adversarial Linear Mixture MDPs with Bandit Feedback and Unknown Transition [71.33787410075577]
  We study reinforcement learning with linear function approximation, unknown transition, and adversarial losses. We propose a new algorithm that attains an $widetildeO(dsqrtHS3K + sqrtHSAK)$ regret with high probability.
  arXiv  Detail & Related papers  (2024-03-07T15:03:50Z)
• Transformers as Support Vector Machines [54.642793677472724]
  We establish a formal equivalence between the optimization geometry of self-attention and a hard-margin SVM problem. We characterize the implicit bias of 1-layer transformers optimized with gradient descent. We believe these findings inspire the interpretation of transformers as a hierarchy of SVMs that separates and selects optimal tokens.
  arXiv  Detail & Related papers  (2023-08-31T17:57:50Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.