Revisiting Anisotropy in Language Transformers: The Geometry of Learning Dynamics

Raphael Bernas; Fanny Jourdan; Antonin Poché; Céline Hudelot

cs.CLApr 9, 2026

Revisiting Anisotropy in Language Transformers: The Geometry of Learning Dynamics

Authors: Raphael Bernas, Fanny Jourdan, Antonin Poché, Céline Hudelot

Organizations: MICS, CentraleSupelec, Université Paris-Saclay · IRT Saint Exupéry, Toulouse, France · Mila - Quebec AI Institute, Montreal, Canada · IRIT Toulouse, Toulouse, France

Abstract

Since their introduction, Transformer architectures have dominated Natural Language Processing (NLP). However, recent research has highlighted an inherent anisotropy phenomenon in these models, presenting a significant challenge to their geometric interpretation. Previous theoretical studies on this phenomenon are rarely grounded in the underlying representation geometry. In this paper, we extend them by deriving geometric arguments for how frequency-biased sampling attenuates curvature visibility and why training preferentially amplify tangent directions. Empirically, we then use concept-based mechanistic interpretability during training, rather than only post hoc, to fit activation-derived low-rank tangent proxies and test them against ordinary backpropagated true gradients. Across encoder-style and decoder-style language models, we find that these activation-derived directions capture both unusually large gradient energy and a substantially larger share of gradient anisotropy than matched-rank normal controls, providing strong empirical support for a tangent-aligned account of anisotropy.

Figures & tables

Figure 1: Temporal dynamics of embedding variance across model architectures. Left: Training trajectories for the top-5 most frequent versus bottom-5 least frequent tokens in Pythia -410m, illustrating the tighter confinement of frequent tokens. Middle: Mean Euclidean distance to centroid vs. log-frequency for Pythia (160m, 410m, 1b, 1.4b), SmoLLM2 (360m, 1.7b), and EuroBERT (210m, 610m). Causal decoders exhibit a strong negative correlation (Pearson $\rho\approx-0.5$ ), confirming frequency-induced variance collapse. Rightmost: Detailed Min/Max distance statistics for EuroBERT -610m. Although the mean trend is attenuated, the distributional view still shows a weaker frequency-dependent concentration pattern.

EARLY
Metric	EuroBERT-210m				EuroBERT-610m				Gaperon-1.5B				moderncamembert-base				OLMo-1B
	$T^{1}_{\mathrm{in}}{}^{*}$	$T^{1}_{\mathrm{out}}$	$T^{m}$	$T^{\ell}$	$T^{1}_{\mathrm{in}}{}^{*}$	$T^{1}_{\mathrm{out}}$	$T^{m}$	$T^{\ell}$	$T^{1}_{\mathrm{in}}{}^{*}$	$T^{1}_{\mathrm{out}}$	$T^{m}$	$T^{\ell}$	$T^{1}_{\mathrm{in}}{}^{*}$	$T^{1}_{\mathrm{out}}$	$T^{m}$	$T^{\ell}$	$T^{1}_{\mathrm{in}}{}^{*}$	$T^{1}_{\mathrm{out}}$	$T^{m}$	$T^{\ell}$
$E_{r}$	7.42	53.1	20.6	18.6	10.7	68.6	21.3	20.6	239	1.4e3	46.8	43.3	225	64	66.2	30.1	364	1.2e3	58.7	66.1
$P_{E^{null}}$	4.8e-2	4.8e-2	4.8e-2	4.8e-2	4.8e-2	4.8e-2	9.1e-2	8.7e-2	4.8e-2	4.8e-2	4.8e-2	4.8e-2	4.8e-2	4.8e-2	4.8e-2	4.8e-2	4.8e-2	4.8e-2	4.8e-2	4.8e-2
$\Delta I^{\star}_{T}\%$	1.92	3.32	39.8	38.4	1.06	2.98	24.1	22.7	62.1	89.6	3.97	4.15	5.68	8.6	37.1	9.51	85.1	104	16.1	11.3
$\Delta I^{\star}_{N}\%$	-0.157	-0.026	-0.398	-0.373	-0.217	-0.015	-0.207	-0.165	-0.189	-0.052	-0.043	-0.05	-0.006	-0.062	-0.233	-0.09	-0.186	-0.066	-0.098	-0.068

Table 1: Cross-model tangent-alignment summary for the shared early-context tangent fit, transposed and split into two stacked five-model panels. Early and late statistics appear as row blocks.

E

-r is the matched-normal energy ratio;

E

-null

p

, the Monte Carlo normal-space null;

\Delta I^{*}\%

100\times\Delta\mathrm{Iso}^{*}/\mathrm{Iso}^{*}_{\mathrm{base}}

, reported separately for tangent (T) and matched-normal (N) removal; and Iso

T>N

p

, the one-sided sign test for

\Delta\mathrm{Iso}^{*}_{T}>\Delta\mathrm{Iso}^{*}_{N,\det}

. Starred layers mark the primary transformer-layer test; effect-size rows are bold and p-values plain. For Pythia,

E^{0}_{\mathrm{in}}

E^{0}_{\mathrm{out}}

are the embedding input and post-embedding-dropout pre-transformer probes, and

T^{1}_{\mathrm{in}},T^{1}_{\mathrm{out}},T^{m},T^{\ell}

denote first-layer input, first-layer output, middle-layer input, and last-layer input.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 2: Curvature bias and its attenuation. (a) Under uniform radial sampling, directions with higher curvature $\kappa$ reach larger intrinsic distances $r$ at the same ambient radius $t$ , causing over-representation of curved regions. (b) When the sampling density $g(t)$ concentrates at small $t$ , all directions yield similar intrinsic displacements and the curvature bias is attenuated.

	EuroBERT-210M								Pythia-410M								SmolLM2-360M
Metric	10k-60k	70k-120k	130k-180k	190k-240k	250k-300k	310k-360k	370k-420k	430k-480k	10k-22k	26k-38k	42k-54k	58k-70k	78k-90k	94k-106k	110k-122k	126k-138k	160k-640k	800k-1.28M	1.44M-1.92M	2.08M-2.56M
$\mathrm{ID}_{\mathrm{Graph}}$	22 21 22	22 10 24	22 7 24	24 9 23	23 10 20	25 10 20	24 9 21	24 12 22	19 37 6	24 27 5	22 28 4	21 25 4	19 22 4	21 23 4	20 25 5	19 24 5	18 19 11	18 25 7	17 27 5	15 23 5
$\mathrm{ID}_{2\mathrm{nn}}$	54.56 52.08 60.58	63.18 33.19 57.36	70.28 23.93 52.52	65.69 28.18 49.62	68.02 32.23 47.95	63.79 31.47 46.67	62.41 32.26 42.93	67.51 34.10 45.47	52.24 76.46 16.07	49.32 74.92 15.48	49.04 68.65 15.13	44.80 65.89 11.93	53.78 64.83 11.36	46.90 62.67 12.21	49.64 64.69 11.81	44.94 68.03 11.56	46.82 73.38 30.43	45.02 72.84 21.72	45.40 76.89 19.09	43.69 63.82 15.30
$PCA_{70\%}$	76 25 56	90 1 69	98 1 30	96 1 43	95 1 32	96 17 25	94 1 30	93 1 34	88 101 1	91 101 1	93 101 1	94 101 1	95 101 1	95 101 1	96 101 1	96 101 1	49 101 1	49 101 1	48 101 1	53 101 1
$r_{k}^{\mathrm{eff}}$	166.7 164.0 210.7	194.5 18.9 187.5	202.2 3.0 165.4	198.0 3.1 134.4	198.6 4.3 124.4	196.4 4.1 118.8	195.0 8.8 99.8	196.9 7.2 119.3	141.5 261.4 4.4	136.5 243.9 2.2	133.6 234.4 1.8	134.0 223.6 1.8	131.7 221.3 1.8	129.7 209.7 1.9	129.3 207.1 2.3	128.0 204.4 2.2	119.2 235.4 30.5	105.7 243.3 2.3	104.9 238.3 1.6	87.6 213.8 1.9
$\mathrm{Sim}_{\mathrm{PCA}}$	0.514 0.348 0.230	0.851 0.596 0.387	0.810 0.663 0.516	0.852 0.578 0.314	0.897 0.568 0.350	0.847 0.419 0.237	0.571 0.244 0.335	0.825 0.567 0.302	0.956 0.887 0.988	0.985 0.903 0.923	0.988 0.952 0.952	0.991 0.951 0.983	0.995 0.932 0.973	0.997 0.960 0.996	0.998 0.983 0.991	0.998 0.984 0.998	0.935 0.612 0.540	0.971 0.740 0.780	0.964 0.874 0.688	0.922 0.884 0.627

Appendix

Table 2: Layerwise geometry across training checkpoints (Semi-NMF concept decomposition). Each cell reports the first , intermediate , and last transformer layers (top

\to

bottom; indices omitted because model depths differ). Salmon shading is min-max normalized per model and metric (stronger = larger value). Last-layer entries for

c_{W}

in the provided logs are shown as “-” since their is no weight to correlate with. In

PCA_{70\%}

, a value above the decomposition threshold is noted 101 (which implies a need for more than 100 features to explain 70% variance, thus a better estimation in those cases is the effective rank

r_{k}^{\mathrm{eff}}

Figure 3: Embedding-update tangent and normal enrichments across token frequency. For each model, we compute tangent and normal enrichment ratios on the embedding update $dW=W_{t+n}-W_{t}$ , normalized so that the random baseline is $1$ . The left panel in each model reports tangent enrichment; the right panel reports normal enrichment. Curves are plotted against $\log_{10}(\mathrm{freq})$ and colored by token rarity group, which makes the frequency dependence visible while preserving within-group variability. Frequent tokens tend to exhibit the strongest tangent enrichment, whereas normal enrichment remains near, or slightly below, the random baseline. The effect is visibly weaker for the EuroBERT models, consistent with the milder frequency-concentration pattern in Figure 1 .

Figure 4: Embedding-update tangent and normal enrichments across token frequency. For each model, we compute tangent and normal enrichment ratios on the embedding update $dW=W_{t+n}-W_{t}$ , normalized so that the random baseline is $1$ . The left panel in each model reports tangent enrichment; the right panel reports normal enrichment. Curves are plotted against $\log_{10}(\mathrm{freq})$ and colored by token rarity group, which makes the frequency dependence visible while preserving within-group variability. Frequent tokens tend to exhibit the strongest tangent enrichment, whereas normal enrichment remains near, or slightly below, the random baseline. The effect is visibly weaker for the EuroBERT models, consistent with the milder frequency-concentration pattern in Figure 1 .

Model	Token	Freq.	Early		Stable
			Tangent	Normal	Tangent	Normal
EuroBERT-210m	of	2.40e-02	33.27	0.99	3.72	1.00
	her	1.63e-03	7.40	1.00	3.53	1.00
	lines	1.16e-04	2.79	1.00	2.24	1.00
	jam	6.45e-06	3.26	1.00	2.76	1.00
	chat	2.15e-06	3.63	1.00	2.02	1.00

Appendix

Table 3: Token-level embedding-update enrichments. For each model we report representative tracked tokens spanning the empirical frequency range, together with their frequencies and the corresponding tangent and normal enrichment ratios computed on the embedding update

dW

. Each ratio is normalized by its random-subspace baseline, so that

1

corresponds to chance level. Token selection is guided by the early regime, and the same tokens are then reused in the stable columns. (*) SmolLM2-360m only has 16 available checkpoints starting in the late training phase.

Figure 5: Forest Cover Type Anisotropic Components.

Figure 6: Evolution of gradient-data concept alignment for the embedding layer during training. We report three complementary alignment metrics as a function of the gradient monitoring step: (left) mean correlation over all extracted concepts, (middle) mean correlation restricted to the top-10 highest-aligned concepts, and (right) mean correlation for high-quality alignments (above the spectral gap threshold). Data concepts are compared against gradient-derived concepts extracted from the embedding layer using three decomposition methods: PCA (red), Semi-NMF (blue), and ICA (green). Across all metrics, PCA and Semi-NMF exhibit consistently stronger and more stable alignment than ICA, with Semi-NMF showing particularly strong alignments.

Figure 7: Correlation density maps for the three main components of the model: the first transformer linear head, the second transformer linear head, and the embedding matrix (first layer), shown as a function of training time (x-axis). To construct these maps, we first pass data-derived concepts through the model to obtain an estimate of the concepts learned by the network. Independently, we treat the weight matrices of the corresponding linear layers as data matrices and extract latent concepts using PCA, ICA, and Semi-NMF. The resulting concept weights from each decomposition are used to rank the extracted components (rank 0 corresponds to the highest-weight concept, rank 50 to the lowest). For each rank, we compute the absolute correlation between learned concepts and data concepts and visualize these values as an RGB image, with PCA encoded in red, ICA in green, and Semi-NMF in blue. Brighter colors indicate stronger correlations. Across all layers, higher-ranked concepts consistently exhibit stronger correlations throughout training.

Representation Geometry in Language Models

Representation geometry in language models studies how embeddings and hidden states organize concepts, computation, and behavior.

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Trajectory Geometry of Transformer Representations Across Layers

Vishal Pandey, Gopal Singh, Yacine Mahdid

London, UK · Athens, GR · Montreal, CA

Transformer Interpretability Neural Representation Geometry Transformer Representation Geometry in Language Models

Open paper AskPDF arXiv

Understanding how transformer representations evolve across layers, not merely what they encode, remains an open problem in mechanistic interpretability. We recast the transformer forward pass as a discrete population trajectory through a high-dimensional representation manifold, drawing on geometric tools from computational neuroscience. Rather than probing for pre-specified features, we characterize trajectory geometry using five metrics computed directly in the ambient space: trajectory length, curvature, a semantic convergence index, layerwise cosine similarity, and representational stability. Across three model families (GPT-2, TinyLlama, Qwen2.5) and five controlled prompt families, we report four findings. First, semantically related prompts converge significantly in middle-to-late layers (peak CI 0.41--0.58, p<0.001, Mann-Whitney U), consistent with attractor-like dynamics. Second, reasoning tasks produce trajectories of greater curvature than lexical variations (0.71--0.83 rad vs. 0.27--0.31 rad), suggesting curvature encodes computational complexity. Third, ambiguous tokens exhibit trajectory bifurcation with up to 5.6x representational separation by the final layer, absent in unambiguous controls. Fourth, layerwise cosine similarity reveals a universal three-phase structure: encoding, elaboration, and output preparation, consistent across all three architectures. All four effects vanish under shuffled-layer and random-embedding controls. We release a fully open-source, model-agnostic pipeline and argue that trajectory geometry constitutes a principled, probe-free lens for mechanistic interpretability.

Abstract

Representation Geometry in Language Models

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