The Geometric Wall: Manifold Structure Predicts Layerwise Sparse Autoencoder Scaling Laws

Abstract: Sparse autoencoders (SAEs) operationalise the linear representation hypothesis: they reconstruct model activations as sparse linear combinations of interpretable dictionary atoms, on the implicit assumption that activation space is well approximated by a globally linear structure. Their reconstruction error varies sharply across layers in ways that existing scaling laws, fitted at single layers, do not explain. We argue that this variation is the empirical trace of a geometric mismatch: where the activation manifold is curved and its intrinsic dimension varies across layers, no sparse linear dictionary can match it uniformly, and the SAE's width-sparsity scaling becomes a layer-dependent function of manifold structure rather than a single universal law. We conduct the first cross-layer SAE scaling study, fitting and regressing on 844 residual-stream Gemma Scope SAE checkpoints across 68 layers of Gemma 2 2B and 9B. Stage 1 fits a per-layer scaling-law surface; Stage 2 regresses the fitted parameters and the derived per-layer width exponents on four layerwise geometric summaries. We find that manifold geometry predicts the per-layer width exponent in both models, and that the same regression coefficients learnt on one model predict the other model's per-layer exponents under cross-model transfer, indicating a transferable geometric law. At the showcase layers where richer width grids permit identification of the asymptotic floor, we find that the fitted floor tracks the layerwise geometric ordering: higher curvature and intrinsic dimension correspond to higher floor, consistent with the irreducible second-order residual that any sparse linear approximation of a curved manifold must leave behind. SAEs thus encounter not a finite-resource ceiling but a geometry-dependent wall, set by the manifold they are trying to reconstruct.

• The paper shows that multi-scale curvature is the strongest predictor of layerwise SAE width-scaling exponents, achieving leave-one-out R² values of 0.976 in Gemma 2 2B and 0.869 in 9B across 844 checkpoints.
• The paper finds that geometry-dependent reconstruction floors persist even with unlimited dictionary width, with intrinsic dimension and curvature creating separate limits that standard Euclidean reconstruction cannot remove at fixed sparsity.
• The paper demonstrates that curvature-based scaling predictions transfer across Gemma 2 model sizes with R² values of 0.920 from 2B to 9B and 0.970 from 9B to 2B, while noting that broader architectures and SAE types remain untested.

Overview

This paper asks whether the width–sparsity scaling law for sparse autoencoders (SAEs), established by Gao et al. at a single layer of GPT-4, holds uniformly across the depth of a LLM, and if not, what predicts its layerwise variation. The authors' answer is that activation manifold geometry — intrinsic dimension, multi-scale curvature, tangent variation, and within-layer heterogeneity — systematically conditions the per-layer scaling law. They term this dependence a "geometric wall": a geometry-dependent asymptotic reconstruction floor that no amount of dictionary width can eliminate at fixed sparsity, as distinct from a finite-resource ceiling.

The empirical basis is the first cross-layer SAE scaling study: 844 publicly released Gemma Scope JumpReLU checkpoints spanning all 26 layers of Gemma 2 2B and all 42 layers of 9B on the residual stream. The analysis proceeds in two stages: Stage 1 fits a per-layer no-floor scaling surface at every layer; Stage 2 regresses the fitted parameters and derived per-layer width exponents on four layerwise geometric summaries. A complementary calibration refits the full six-parameter with-floor surface at six showcase layers where richer width grids make the asymptotic floor identifiable.

Geometric framework

The paper grounds its account in pullback information geometry. The natural metric on next-token distributions is the Fisher–Rao metric; each layer's predictive map $F_\ell$ pulls this metric back to hidden space via $$\mathbf{G}_\ell(h) = J_\ell(h)^\top \boldsymbol{\Sigma}_{F_\ell(h)} J_\ell(h)$$. The framework identifies two sources of mismatch between SAE practice and this geometry: a basis–manifold mismatch, since a flat $k$-sparse linear dictionary cannot conform to a curved manifold under any metric, and an objective–metric mismatch, since standard $\ell_2$ reconstruction optimises ambient Euclidean distance rather than pullback distance. The empirical geometric summaries used throughout are Euclidean $k$-NN proxies (TWO-NN for intrinsic dimension, multi-scale PCA residuals for curvature, Grassmannian chordal distances for tangent variation), whose relationship to the pullback quantities is argued to be metric-invariant or gauge-related rather than computed directly — a concession the authors state plainly, deferring direct Jacobian computation as computationally prohibitive.

The mechanistic prediction follows from differential geometry: when principal curvatures are non-zero, the tangent plane departs from the manifold by a second-order term proportional to curvature times squared local distance. In the small-$k$ regime where the scaling law applies, a $k$-sparse combination spans a $k$-dimensional flat affine subspace, so even infinitely dense atoms leave an irreducible residual. Higher curvature raises this residual directly; higher intrinsic dimension shrinks the share of local tangent structure any finite atom budget covers. Both effects raise the asymptotic floor and depress the width-scaling rate.

Two-stage methodology

Stage 1 fits the floor-free log-linearisation $$\log L = a_0 + \beta_n \log n + \beta_k \log k + \gamma \log n \log k$$ per layer, identifiable from the two backbone widths shared across all layers (16K/65K for 2B; 16K/128K for 9B), with PCHIP interpolation across realised sparsities. The derived per-layer exponent is $$\alpha_\ell(k) = -(\beta_{n,\ell} + \gamma_\ell \log k)$$, evaluated at $$\mathbf{G}_\ell(h) = J_\ell(h)^\top \boldsymbol{\Sigma}_{F_\ell(h)} J_\ell(h)$$0. Because only two widths are available at most layers, the six-parameter with-floor form is not identifiable except at showcase layers ($$\mathbf{G}_\ell(h) = J_\ell(h)^\top \boldsymbol{\Sigma}_{F_\ell(h)} J_\ell(h)$$1 for 2B; $$\mathbf{G}_\ell(h) = J_\ell(h)^\top \boldsymbol{\Sigma}_{F_\ell(h)} J_\ell(h)$$2 for 9B), mirroring Gao et al.'s own progression from no-floor to with-floor fits. The no-floor exponent therefore underestimates the asymptotic with-floor exponent by a quantified floor-attenuation identity; both channels push the apparent exponent in the same direction under higher curvature.

Stage 2 regresses per-layer targets on the four geometric summaries via OLS against a nested hypothesis hierarchy: H0 (geometry-invariant constant, implicitly assumed by single-layer scaling-law extrapolation), H1 (single feature), H2 (lowest-collinearity pair), and H_full. Validation uses closed-form leave-$$\mathbf{G}_\ell(h) = J_\ell(h)^\top \boldsymbol{\Sigma}_{F_\ell(h)} J_\ell(h)$$3-layer-out cross-validation via the hat matrix, exhaustive over holdout subsets, plus layer-permutation nulls and F-test ladders. Geometry estimation and error measurement use disjoint C4 document partitions, with WikiText-103 cross-corpus checks.

Results

Geometry predicts the per-layer width exponent. Multi-scale curvature is the dominant single predictor of $$\mathbf{G}_\ell(h) = J_\ell(h)^\top \boldsymbol{\Sigma}_{F_\ell(h)} J_\ell(h)$$4 in both models. At 9B, H1$$\mathbf{G}_\ell(h) = J_\ell(h)^\top \boldsymbol{\Sigma}_{F_\ell(h)} J_\ell(h)$$5 achieves LOO $$\mathbf{G}_\ell(h) = J_\ell(h)^\top \boldsymbol{\Sigma}_{F_\ell(h)} J_\ell(h)$$6 versus $$\mathbf{G}_\ell(h) = J_\ell(h)^\top \boldsymbol{\Sigma}_{F_\ell(h)} J_\ell(h)$$7 for the next-best feature, with H0 rejected at $$\mathbf{G}_\ell(h) = J_\ell(h)^\top \boldsymbol{\Sigma}_{F_\ell(h)} J_\ell(h)$$8; at 2B it reaches LOO $$\mathbf{G}_\ell(h) = J_\ell(h)^\top \boldsymbol{\Sigma}_{F_\ell(h)} J_\ell(h)$$9. All four features individually reject H0 overwhelmingly, but add little beyond curvature because they covary as aspects of the same manifold complexity. The finding is stable across sparsity targets $k$0 and robust to replacing the parametric fit with a non-parametric backbone-secant exponent (agreement within 0.01 at every layer).

Cross-model transfer. Regression coefficients learned on one model predict the other model's per-layer exponents nearly at the test model's in-sample upper bound: transfer $k$1 (H1$k$2, 2B→9B) versus an in-sample bound of 0.929, and $k$3 (9B→2B) versus 0.979. This indicates the geometry-to-scaling mapping is preserved across the 2B→9B size jump within the Gemma 2 family, though whether it extends across architectures remains untested.

Floor–geometry coupling. At the six showcase layers, the fitted asymptotic floors $k$4 rank consistently with the layers' intrinsic-dimension and curvature orderings in both models, preserved across additional sparsity slices. The authors decompose the floor into two channels: an intrinsic-dimension channel recovering Gao et al.'s "spectrum of structure" hypothesis (pure Gaussian noise sits at this channel's limit), and a genuinely additional curvature channel that would predict a non-zero floor even on a fully structured but curved low-dimensional manifold, where the noise-based reading predicts none.

Depth profile alignment. The per-layer $k$5 profile descends through early layers, dips mid-depth, and peaks near relative depth 5/6, landing within ~0.03 of Gao et al.'s reported $k$6 for GPT-4 at the same relative depth — suggestive cross-architecture similarity, but resting on a single anchor and partly attributable to floor attenuation and TopK-versus-JumpReLU differences. Geometry "hurts twice": $k$7 is depressed precisely at middle and final layers where curvature has reached its plateau, the same depths where the floor is elevated.

Limitations

Several constraints bound the claims. The released Gemma Scope family uses a width-dependent training budget (4B/8B/16B tokens), sublinear in width relative to Gao et al.'s convergence-scaled allocation; absolute exponents are conditional on this schedule, though the geometric predictability claim is not. The with-floor calibration is confined to six layers because only two backbone widths are shared elsewhere, so the floor regression is a ranking rather than a formal cross-layer fit. The study covers residual-stream SAEs only, and the geometric summaries are $k$8-NN proxies rather than direct pullback computations. Whether the law holds for TopK or end-to-end-trained SAEs, other architectures, or attention/MLP sublayer sites is left open.

Conclusion

The paper shows that SAE width-sparsity scaling ceases to be universal across depth and becomes a predictable function of activation manifold geometry. Across 844 checkpoints, multi-scale curvature alone explains most cross-layer variation in the width exponent, the same coefficients transfer between model sizes near the theoretical upper bound, and the identifiable asymptotic floors track curvature and intrinsic dimension as predicted by a second-order curvature-residual argument. The result reframes SAE reconstruction limits as geometry-conditioned rather than resource-limited, and implies that interpretability operations treating dictionary atoms as fixed global Euclidean directions inherit a metric choice whose alignment with predictive structure varies systematically by layer.