A Symmetry-Matching Approach to Blind-Spot Reduction in Sparse Autoencoders

Csaba Balogh

cs.LGOct 17, 2026

A Symmetry-Matching Approach to Blind-Spot Reduction in Sparse Autoencoders

Authors: Csaba Balogh

Organizations: National Directorate General for Hospitals, Budapest, Hungary · Data Science in Health Programme, Semmelweis University, Budapest, Hungary

Abstract

Language models can treat semantically distinct inputs as interchangeable at the representation level, creating blind spots that standard sparse autoencoder (SAE) objectives do not target. In safety-critical settings such as clinical dosage extraction, legal clause interpretation, or financial amount verification, such failures can propagate silently into downstream decisions. We study this problem as one of feature-basis orientation. Building on a symmetry-matching view from algebraic error-detection theory, we add a differentiable $V_{\mathrm{Gini}}$ regularisation term to SAE training that penalises uneven perturbation response within a specified family. Across GPT-2, Gemma 2, and Qwen 2.5, a frozen multi-seed evaluation shows that $V$ -regularisation raises both relative and absolute lower-tail responses while reducing response-profile imbalance, with no material general-text reconstruction degradation. JumpReLU and minimum description length (MDL) objectives remain close to the Standard SAE profile under the same protocol. Family-geometry controls show that the effect is not explained by template duplication or a single dominant perturbation family, and held-out probes plus null-calibrated separation tests evaluate how far the representation-level change carries to downstream discriminability. A zero-shot audit on out-of-distribution clinical radiology minimal pairs finds the relative weak-tail lift in all evaluated model-by-family cells, not perfectly rank-aligned with the trained statistic, and a held-out family audit finds it on family kinds never used in training. Together, the results show that perturbation-aware SAE training can reorient feature bases toward safety-relevant input changes while leaving downstream validation as a separate empirical step.

Figures & tables

Object	Measures	Role
$\operatorname{Stab}(G,F)\cap E=\{e\}$	Absence of global task-relevant invariance	Algebraic reference condition
$D_{i}$	Per-pair relative code displacement	Base sampled response
$S_{0}$	Exact sampled zero-response set	Sampled exact-response boundary
$S_{\gamma}$	Low-response margin set	Operational response threshold
$C_{\gamma}$	Coverage above a fixed margin	Mass above a chosen threshold
$V_{\mathrm{Gini}}$	Response-profile imbalance	SAE training objective

Table 1: Hierarchy of perturbation-response quantities used in this paper.

Model	$\Delta L_{20}(D)$	$\Delta L_{20}(\\|\Delta z\\|)$	$\Delta V_{\mathrm{Gini}}$	Positive families	$\Delta$ OWT NMSE
GPT-2	$+0.0562\pm 0.0010$	$+5.3584\pm 0.1202$	$-0.2919\pm 0.0008$	$1.0000\pm 0.0000$	$+0.0000\pm 0.0000$
Gemma 2	$+0.1301\pm 0.0042$	$+10.3454\pm 1.1553$	$-0.2322\pm 0.0001$	$1.0000\pm 0.0000$	$-0.0004\pm 0.0007$
Qwen 2.5	$+0.1316\pm 0.0006$	$+2.4650\pm 0.1129$	$-0.2432\pm 0.0001$	$1.0000\pm 0.0000$	$+0.0006\pm 0.0011$

Table 2: Frozen multi-seed Standard versus

V

-regularised SAE comparison. Values are mean

\pm

standard deviation across three seeds. Positive

\Delta L_{20}(D)

and

\Delta L_{20}(\|\Delta z\|)

indicate lower-tail response lifting; negative

\Delta V_{\mathrm{Gini}}

indicates reduced response-profile imbalance. The OWT reconstruction change is reported as NMSE difference.

Figure 1: Empirical cumulative distribution functions (CDFs) of per-pair perturbation response $D$ over all sixteen perturbation families. For each model, the red line marks the 20th percentile of the Standard SAE response distribution, defining a model-specific low-response region. Lower CDF values to the left of the cutoff indicate fewer perturbation pairs remaining in the weak-response region under $V$ -regularisation. The step structure of the V-reg curves reflects the resolution of each model’s code space. GPT-2 small concentrates pairs on fewer discrete response levels, producing visible steps, while the higher-dimensional Gemma 2 spreads responses almost continuously.

Model	$\Delta L_{20}(D)$	Positive families	$\Delta L_{20}(\\|\Delta z\\|)$	$\Delta L_{20}(\mathrm{decoded})$	$\Delta L_{20}(g)$
GPT-2	$+0.055845$	$16/16$	$+5.4349$	$+0.000368$	$-0.159311$
Gemma 2	$+0.133151$	$16/16$	$+11.0187$	$+0.067886$	$-0.151532$
Qwen 2.5	$+0.131023$	$16/16$	$+2.4443$	$+0.040904$	$-0.107768$

Table 3: Lower-tail response and absolute code displacement on the sixteen-family perturbation set. Values are V-reg minus Standard for the corresponding checkpoint pair. Positive

\Delta L_{20}(D)

and

\Delta L_{20}(\|\Delta z\|)

indicate that the weak lower tail is lifted in relative and absolute SAE-code response. The gain-normalised lower tail records the input-complexity-normalised variant.

Model	Metric	Fixed-family interval	Family-resampled interval
Gemma 2	$\Delta L_{20}(D)$	$+0.1332\,[+0.1256,+0.1361]$	$[+0.0891,+0.1811]$
GPT-2	$\Delta L_{20}(D)$	$+0.0558\,[+0.0527,+0.0567]$	$[+0.0417,+0.0699]$
Qwen 2.5	$\Delta L_{20}(D)$	$+0.1310\,[+0.1248,+0.1328]$	$[+0.0868,+0.1792]$
Gemma 2	$\Delta L_{20}(\\|\Delta z\\|)$	$+11.0187\,[+9.8028,+11.7425]$	$[+6.2071,+16.3177]$
GPT-2	$\Delta L_{20}(\\|\Delta z\\|)$	$+5.4349\,[+5.1541,+5.5791]$	$[+4.0521,+6.9244]$
Qwen 2.5	$\Delta L_{20}(\\|\Delta z\\|)$	$+2.4443\,[+2.1110,+2.7675]$	$[+0.9824,+4.1303]$

Table 4: Template-cluster and family-resampled uncertainty for the lower-tail effects. Fixed intervals resample template clusters within each of the sixteen families. Family-resampled intervals additionally resample families and are wider, but remain positive for the relative lower-tail effect.

Model	$\Delta$ AUROC (abs. code)	Fixed-family 95% CI	Family-resampled 95% CI
Gemma 2	$+0.037$	$[+0.013,+0.063]$	$[-0.014,+0.087]$
Qwen 2.5	$+0.048$	$[+0.024,+0.077]$	$[+0.007,+0.091]$
GPT-2	$-0.010$	$[-0.049,+0.024]$	$[-0.078,+0.064]$

Table 5: Null-calibrated separation. Family-macro AUROC difference (V-reg minus Standard) for absolute SAE-code distance against meaning-preserving nuisance controls. Fixed-family intervals resample template clusters within each family; family-resampled intervals also resample families. Intervals are 95% confidence intervals (CIs). Gemma 2 and Qwen 2.5 are positive under the fixed-family interval; GPT-2 is uncertain in point direction.

Figure 2: Frozen multi-seed baseline extension across Standard, JumpReLU, MDL, and

V

-regularised SAEs. The non-perturbation-aware objectives stay close to Standard on lower-tail response and response-profile imbalance, whereas

V

-regularisation separates on both axes across GPT-2, Gemma 2, and Qwen 2.5.

Model	Objective	$\Delta L_{20}(D)$	$\Delta L_{20}(\\|\Delta z\\|)$	$\Delta V_{\mathrm{Gini}}$	Positive families
GPT-2	JumpReLU	$-0.0001\pm 0.0002$	$+0.0498\pm 0.0026$	$-0.0009\pm 0.0002$	$0.354\pm 0.308$
	MDL	$-0.0003\pm 0.0002$	$+0.0315\pm 0.0040$	$+0.0001\pm 0.0002$	$0.063\pm 0.108$
	V-reg	$+0.0562\pm 0.0010$	$+5.3584\pm 0.1202$	$-0.2919\pm 0.0008$	$1.000\pm 0.000$
Gemma 2	JumpReLU	$-0.0005\pm 0.0001$	$+0.1767\pm 0.0063$	$-0.0001\pm 0.0001$	$0.042\pm 0.036$
	MDL	$-0.0006\pm 0.0001$	$+0.1084\pm 0.0011$	$-0.0001\pm 0.0000$	$0.042\pm 0.072$
	V-reg	$+0.1301\pm 0.0042$	$+10.3454\pm 1.1553$	$-0.2322\pm 0.0001$	$1.000\pm 0.000$

Table 6: Frozen multi-seed baseline extension. Deltas are each objective minus the Standard SAE, mean

\pm

standard deviation across three seeds, under the same protocol and caches as Table 2 . JumpReLU and MDL stay near Standard, while

V

-regularisation separates on every model.

Family	GPT-2	Gemma 2 2B	Qwen 2.5 3B
Negation	0.000	0.000	0.000
Typo	0.017	0.000	0.000
Synonym	0.019	0.000	0.002
Number swap	0.036	0.000	0.005
Semantic sub.	0.048	0.000	0.001
Word order	0.046	0.002	0.000

Table 7: Joint

V

-regularised SAE (

\lambda_{2}=0.1

, all six families simultaneously, 5,000 steps) on three models. Raw reconstruction MSE in parentheses. Values are descriptive single-run point estimates; entries printed as 0.000 are rounded to three decimals (Appendix Q ), and multi-seed variability for the primary lower-tail endpoints is reported in Table 2 .

Model	Layer	Std SAE	V-reg	Reduction	$V{<}0.05$	$V{<}0.005$
GPT-2 small	12	0.314	0.002	$-99.3\%$	16/16	15/16
Qwen 2.5 3B	18	0.239	0.000	$-100\%$	16/16	16/16

Table 8: Secondary low-imbalance summary for sixteen-family joint

V

-regularised SAE runs. “Std SAE” and “V-reg” are the mean

V_{\mathrm{Gini}}

over all sixteen families; “

V{<}0.05

” and “

V{<}0.005

” are descriptive low-imbalance counts. This summary covers the two models for which the sixteen-family joint run was conducted; Gemma 2 is evaluated under the frozen multi-seed protocol in Table 2 . Source files results/joint_eval_{gpt2,qwen-2.5-3b}.json .

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

Model	Family	$m$	$\Delta L_{20}(D)$	$\Delta\overline{D}(W_{S})$ [95% CI]	$D_{i}\uparrow$	$\Delta\overline{\\|\Delta z\\|}(W_{S})$ [95% CI]
GPT-2	negation	60	$+0.004215$	$+0.004736$ [ $+0.003720,+0.005731$ ]	0.917	$+0.6337$ [ $+0.5151,+0.7537$ ]
GPT-2	laterality	60	$+0.000945$	$+0.000995$ [ $+0.000886,+0.001104$ ]	0.983	$+0.1202$ [ $+0.1053,+0.1348$ ]
GPT-2	severity	60	$+0.003231$	$+0.004157$ [ $+0.003558,+0.004791$ ]	1.000	$+0.5350$ [ $+0.4577,+0.6183$ ]
GPT-2	anat. dir.	27	$+0.000807$	$+0.000949$ [ $+0.000771,+0.001149$ ]	1.000	$+0.1161$ [ $+0.0954,+0.1388$ ]
Qwen 2.5	negation	16	$+0.009887$	$+0.009887$ [ $+0.008415,+0.011205$ ]	1.000	$-0.1080$ [ $-0.1963,-0.0271$ ]
Qwen 2.5	laterality	16	$+0.000888$	$+0.000960$ [ $+0.000769,+0.001181$ ]	1.000	$-0.0066$ [ $-0.0132,-0.0008$ ]

Appendix

Table 9: Standard-defined OpenI weak-set evaluation.

m=|W_{S}|

;

\Delta L_{20}(D)

uses independently selected own tails, while

\Delta\overline{D}(W_{S})

evaluates the fixed Standard weak set. Intervals are conditional paired 95% bootstrap intervals on the fixed set. “

D_{i}\uparrow

” is the fraction of its pairs with larger relative response under

V

-regularisation. The final column is the absolute code-displacement change on the same relative-

D

weak set.

Subset	$n$	Standard margin	V-reg margin	$\Delta$ margin [95% CI]	Error S $\to$ V
Standard weak 20%	16	0.302	0.358	$+0.055$ [ $-0.087,+0.257$ ]	$1/16\to 0/16$
Remaining 80%	62	0.590	0.575	$-0.014$ [ $-0.123,+0.118$ ]	$7/62\to 2/62$

Appendix

Table 10: Cross-fitted Qwen negation probe on the fixed Standard-defined weak set and remaining pairs. Margin intervals are template-clustered 95% intervals for the paired

V

-regularised-minus-Standard change.

Model	Family	$L_{20}$ Standard	$L_{20}$ V-reg	$\Delta L_{20}$ [95% CI]
GPT-2	negation	0.0157	0.0200	$+0.0042$ [ $+0.0030,+0.0053$ ]
GPT-2	laterality	0.0024	0.0034	$+0.0009$ [ $+0.0008,+0.0011$ ]
GPT-2	severity	0.0044	0.0077	$+0.0032$ [ $+0.0027,+0.0038$ ]
GPT-2	anat. dir.	0.0017	0.0025	$+0.0008$ [ $+0.0007,+0.0010$ ]
Qwen 2.5	negation	0.0298	0.0397	$+0.0099$ [ $+0.0081,+0.0123$ ]
Qwen 2.5	laterality	0.0034	0.0043	$+0.0009$ [ $+0.0007,+0.0012$ ]

Appendix

Table 11: Real-clinical lower-tail response on OpenI minimal pairs, frozen general checkpoints, zero-shot.

L_{20}(D)

is the mean relative code displacement over the weakest quintile of pairs. Intervals are report-clustered 95% intervals of

\Delta L_{20}

Model	Family	Std	Jump	MDL	V-reg	V-reg n
GPT-2	Negation	.219	.222	.223	.000	.267
	Typo	.307	.306	.309	.014	.295
	Synonym	.307	.310	.310	.018	.326
	Number swap	.264	.265	.266	.034	.276
	Semantic sub.	.435	.433	.435	.045	.400
	Word order	.281	.282	.282	.044	.302

Appendix

Table 12:

V_{\mathrm{Gini}}

(raw) across five SAE methods and three models (5,000 steps,

d_{\mathrm{sae}}=4{,}096

, 50 pairs per family). Bold indicates the lowest

V

per family per model. The standard, JumpReLU, and MDL SAEs produce similar single-run profiles; the

V

-regularised SAE reduces response-profile imbalance in this descriptive comparison.

Model	Family	Standard	V-reg	Change
GPT-2	Negation	0.073	0.115	+58%
	Typo	0.067	0.081	+21%
	Synonym	0.043	0.050	+17%
	Number swap	0.021	0.023	+7%
	Semantic sub.	0.045	0.045	0%
	Word order	0.043	0.057	+34%

Appendix

Table 13: Mean relative sensitivity per family (standard vs

V

-regularised SAE). The

V

-regularisation increases mean sensitivity on most families across all three models, confirming that it redistributes perturbation signals more uniformly rather than suppressing them.

Family	GPT-2	Gemma 2	Qwen 2.5
Negation	0.000	0.000	0.000
Typo	0.000	0.000	0.000
Synonym	0.000	0.000	0.000
Number swap	0.000	0.000	0.000
Semantic sub.	0.000	0.000	0.000
Word order	0.000	0.000	0.000

Appendix

Table 14:

V_{\mathrm{Gini}}

(gain) for the V-reg n variant. Values are rounded to three decimals and indicate very low gain imbalance under input-complexity normalisation.

Family	GPT-2	Gemma 2	Qwen 2.5
Anatomical dir.	0.61/4.1	0.50/17.4	0.48/18.6
Body part swap	0.60/3.8	0.60/13.8	0.58/14.6
Causal reversal	0.70/5.2	0.50/18.9	0.65/16.6
Condition neg.	0.73/2.1	0.45/17.6	0.61/12.1
Date/time change	0.71/3.1	0.52/18.2	0.66/14.8
Drug name swap	0.56/4.1	0.32/23.0	0.46/18.5

Appendix

Table 15: Input perturbation complexity per family. Each entry reports CV/centered effective rank for hidden-state perturbation magnitudes.

Model	Method	MSE	Frac active
GPT-2	Standard	0.0034	0.352
	JumpReLU	0.0035	0.333
	MDL	0.0065	0.337
	V-reg	0.0069	0.344
	V-reg n	0.0033	0.329
Gemma 2	Standard	0.0130	0.072

Appendix

Table 16: Reconstruction MSE and sparsity (fraction of active features) across five SAE methods and three models. All share

d_{\mathrm{sae}}=4{,}096

and 5,000 steps.

Model	LOTO positive templates	LOFO $\Delta L_{20}(D)$ range	Centered effective-rank range
Gemma 2	$424/424$	$+0.1149$ – $+0.1401$	$13.4$ – $36.7$
GPT-2	$424/424$	$+0.0522$ – $+0.0584$	$2.1$ – $5.2$
Qwen 2.5	$424/424$	$+0.1140$ – $+0.1372$	$10.8$ – $35.8$

Appendix

Table 17: Template-cluster resampling summary. LOTO denotes leave-one-template-out, and LOFO denotes leave-one-family-out.

	GPT-2 small		Qwen 2.5 3B
Family	Std	V-reg	Std	V-reg
Negation	0.219	0.000	0.136	0.000
Typo	0.307	0.000	0.160	0.000
Synonym	0.307	0.000	0.234	0.000
Number swap	0.264	0.000	0.303	0.000
Semantic substitution	0.435	0.035	0.368	0.000

Appendix

Table 18: Per-family

V_{\mathrm{Gini}}

(raw) for the sixteen-family joint

V

-regularised SAE on GPT-2 and Qwen 2.5, standard SAE versus V-reg. The six families above the rule are the original families; the ten below are the auto-generated medical-domain families, none of which informed the metric or training protocol. Both groups reach a low-imbalance regime in this descriptive comparison. Source files results/joint_eval_{gpt2,qwen-2.5-3b}.json .

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Representation Redistribution in Sparse Autoencoders

Ziye Mei, Haotian Ma

Department of Computer Sciences, University of Wisconsin–Madison

Sparse Autoencoders Prompt Sensitivity Language Model Robustness Representation Geometry in Language Models Language Model Safety Evaluation

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Sparse autoencoders (SAEs) decompose language model activations into sparse features, but prompt transformations designed to preserve the underlying request can redistribute activity across them. A feature change observed on one transformed prompt need not recur, and even a recurring change need not affect a downstream computation. We introduce the redistribution sensitivity score (RSS), which identifies large feature responses that recur in a consistent direction, and evaluate the resulting supports on variants excluded from selection. Across 27 pretrained SAEs spanning eight language models, the tested prompt transformations produce a reproducible redistribution profile on these excluded variants. We then separate recurring representation change from signed readout contribution and decision failure. For a fixed linear readout, an exact decomposition shows how response magnitude, readout coefficients, directional alignment, and the clean margin determine downstream consequences. In two frozen safety readouts, combining RSS support erosion with the full clean margin yields recurrent failure AUROCs of 0.935 and 0.958 using median supports of 20 and 19 features, compared with 0.965 and 0.964 using full erosion. RSS therefore provides a way to identify recurring sparse representation changes and quantify their contribution to a downstream readout without equating representation change with failure. Code is available at https://github.com/OpenSAE/Redistribution-Sensitivity-Score

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