The Geometry of Projection Heads: Conditioning, Invariance, and Collapse

Muon motivates designing optimizer geometry around the function of each parameter block and uses the spectral norm for hidden linear layers. For the language-model head, the spectral norm is not a faithful measure of functional change. Softmax removes shared logit shifts, whereas the spectral norm can assign arbitrarily large size to updates that change no output probability. We therefore treat the LM head and softmax as one module and derive an update geometry for their composition. Hilbert's projective distance respects this invariance as it measures the largest change in pairwise log odds. For an update $S$ with token rows $s_i^\top$, we show that the largest Hilbert distance over $\left\lVert h\right\rVert_2\leq H$ is exactly $H D(S)$, where $D(S)=\max_{i<j}\left\lVert s_i - s_j\right\rVert_2$ is the Euclidean row diameter. This diameter replaces the spectral norm in the resulting Muon-style steepest descent problem. An exact solution is possible, but its direct formulation contains one $d$-dimensional vector variable for every token pair. For a vocabulary size of approximately $50$k, this means more than one billion token pairs, making the calculation impractical at every training step. We instead impose a stronger common-ball constraint and derive projected RowNorm as an $O(Vd)$ solution. For the exact RowNorm oracle, we prove that its first-order decrease is at least $1/\sqrt{2}$ of the exact diameter-constrained optimum. With Muon on the backbone, experiments across three seeds at 190M, 380M, and 640M parameters show that RowNorm reduces mean final step diameters and empirical Hilbert RMS perturbations by factors of $45$--$60$ and $12$--$15$, respectively, with only a $0.0057$--$0.0153$ increase in mean final validation loss.

The query and key projections $\WQ,\WK$ in attention are almost always trained by Euclidean optimizers with no constraint on their geometry. We constrain them to the Stiefel manifold and optimize them there with a Riemannian Adam that carries one scalar second moment per frame, caps its step by a trust region, and retracts polarly. Four propositions prove this update is steepest descent in the embedded metric, independent of gradient scale, well conditioned, and exactly $\mathrm{O}(d)$-equivariant, each certified numerically in \texttt{float64}. A fifth supplies the mechanism: weight decay has \emph{identically zero} Riemannian gradient on $\St(d,r)$, since $W = W I_r$ lies in the normal space, so the learned attention geometry survives the collapse cycles that decay drives through the rest of the model. On modular arithmetic grokking, a single run holds $97.0\%$ validation accuracy at epoch 20,000 against the baseline's $61.1\%$---an unstable endpoint we report as evidence for the mechanism rather than as an effect size. On CIFAR-10 patches the same rule gains $\mathbf{+8.98}$,pp over 12 paired starts ($t{=}60.6$, $12/12$), and the gap widens with data rather than eroding. The step rule earns this: a fixed-step Riemannian update is degree one in the gradient, so it moves $24$--$40\times$ less per step than an identically shaped AdamW matrix---its frames barely leave their initialization, and freezing them outright costs only $0.28$,pp. An ablation credits the whole gain to making the step scale free, and nothing measurable to the projector or to equivariance. A negative result sharpens the account: gauge removal cannot motivate the method, because a direction along which the loss is invariant carries no gradient at all.

Methods operating on Vision Transformer (ViT) feature spaces typically rely on Euclidean distance or cosine similarity. This assumes that every direction is equally meaningful, but there is no reason to believe the true task geometry has this property. The task-sensitive geometry of the feature space is given by the pullback metric $g(F) = J(F)^\top J(F)$, where $J$ is the Jacobian of the decoder's output fed to a task-specific distance, with respect to the features. Storing the full $g$ is infeasible at modern scales, and for dense outputs such as depth maps even forming $J$ is impractical. We show that whether a low-rank approximation of this metric can be learned depends on the model-decoder pair, and we characterize this with a matrix-free diagnostic $κ_{cap}(r)$ computable with a low number of Jacobian-vector products. For tractable pairs, we develop the Spectral Pullback Network (SPN), which learns a low-rank version of the metric from randomized power iteration, and we distill it into a $310$K-parameter importance head that predicts token importance directly from the features. When the Jacobian spectrum is too spread out for a low-rank approximation, passing the decoder's input features through a VAE bottleneck can restore tractability. Across DPT, DINOv2, CLIP, and VGGT backbones, $κ_{cap}(r)$ predicts which learned-metric architectures are viable. The importance head reaches Spearman $ρ= 0.998$ on DINOv2 CLS, and our geometric token pruning reduces the additional depth error of ToMe-based token selection by $25\%$ on DPT depth at prune ratio $0.5$, without fine-tuning the ViT. Project page: https://cyberiada.github.io/TaskInducedViTs/