Interference Beyond Geometry in Concept Extraction

Valérie Costa; Bahareh Tolooshams

cs.LGSep 28, 2026

Interference Beyond Geometry in Concept Extraction

Authors: Valérie Costa, Bahareh Tolooshams

Organizations: University of Alberta, Amii · Harvard University · University of Alberta, Amii Canada CIFAR AI Chair

Abstract

Interference is commonly treated as geometric overlap between learned features. We introduce effective interference, which combines feature geometry and code statistics to capture realized interactions, distinguishing constructive from destructive interference and frequent weak interactions from rare strong ones. Under local fixed-support assumptions, we characterize how architectural constraints shape interference through four mechanisms: feature orthogonalization, bias compensation, gain adaptation, and encoder-decoder separation. Experiments with sparse autoencoders show that constrained architectures selectively reduce overlap among co-active features, while bias, gain, and encoder freedom allow constructive cross-contributions to remain. Together, these results show that interference in learned representations depends not only on feature geometry, but also on how features are used and on the architecture that produces their codes.

Figures & tables

Figure 1: Effective interference is a joint property of concept geometry and code statistics. It factors as $I_{ij}=\rho_{ij}\pi_{ij}M_{ij}$ , where $\rho_{ij}={\bm{d}}_{i}^{\top}{\bm{d}}_{j}$ captures the geometric overlap between dictionary elements ${\bm{d}}_{i},{\bm{d}}_{j}$ , while $\pi_{ij}M_{ij}=\mathbb{E}[z_{i}z_{j}]$ captures how their corresponding codes $z_{i},z_{j}$ are used across the data, through co-activation frequency $\pi_{ij}$ and conditional magnitude $M_{ij}$ .

Figure 2: Geometric and effective interference have distinct training dynamics. A) Global geometric overlap rises, whereas B) effective interference falls from its peak. Consistent with Proposition 4.3 , this reflects selective orthogonalization of co-active pairs, reducing realized interactions without globally orthogonalizing the dictionary. C) Residual coupling declines but remains nonzero, so fixed-support code optimality ( Proposition 4.1 ) is approached only approximately (see Figure 7 for joint distribution of decoder orientation $\rho_{ij}$ and co-activation-weighted magnitude $\pi_{ij}M_{ij}$ ).

Figure 3: Architectural freedom shifts co-active feature pairs toward constructive decoder interactions. Joint distributions of orientation $\rho_{ij}$ and co-activation-weighted magnitude $\pi_{ij}M_{ij}$ . In the tied, normalized, bias-free model, pairs with large $\pi_{ij}M_{ij}$ concentrate near $\rho_{ij}=0$ , consistent with local orthogonalizing pressure. With bias, encoder-gain freedom, or an untied encoder, some of these pairs extend toward positive $\rho_{ij}$ . This pattern recurs across other SAEs ( Figure 8 ).

Figure 6: SAE reconstruction contains substantial signed effective interference. Constructive $\mathcal{I}_{+}$ , destructive $\mathcal{I}_{-}$ , and net $\mathcal{I}_{\mathrm{net}}$ contributions across architectural variants and SAE families. Sizable positive and negative terms can cancel, so small net interference does not imply weak interactions. Relative to the tied case, architectural freedom generally shifts the balance toward higher interference.

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

	$\mathrm{MMCS}$	median	$\mathrm{MMCS}_{\|\cdot\|}$	$>\!0.9$	$>\!0.7$
JumpReLU	$0.711$	$0.785$	$0.712$	$24.8\%$	$61.1\%$
TopK	$0.548$	$0.533$	$0.552$	$8.9\%$	$31.2\%$
BatchTopK	$0.436$	$0.335$	$0.444$	$6.0\%$	$18.5\%$
Matryoshka	$0.425$	$0.332$	$0.433$	$3.1\%$	$15.8\%$

Appendix

Table 1: Cross-seed dictionary agreement for the runs in Figure 2 , averaged over seed pairs

(0,1)

(0,2)

, and

(1,2)

	TopK			BatchTopK			Matryoshka			JumpReLU
Constraint	FVU	L0	Dead	FVU	L0	Dead	FVU	L0	Dead	FVU	L0	Dead
(i) Tied	.118	40.0	0.00	.124	40.9	0.00	.126	40.7	0.00	.100	45.6	0.02
(ii) Tied + Bias	.086	40.0	0.00	.085	40.1	0.00	.091	40.1	0.00	.086	45.5	0.48
(iii) Tied + Gain	.091	40.0	0.00	.092	40.0	0.12	.096	40.2	0.01	.087	45.4	0.62
(iv) Tied + Gain + Bias	.085	40.0	0.01	.084	40.0	0.02	.090	40.1	0.01	.085	45.4	0.90
(v) Untied	.083	40.0	0.54	.084	40.8	5.41	.088	40.6	6.05	.083	45.0	17.62

Appendix

Table 2: Holdout evaluation after 200M training tokens. “Dead” is the percentage of features that never activate on the holdout set.

Figure 7: Transition of interference factors as training progress. This figure complements Figure 2 .

Figure 8: Consistent trend on architectural freedom shifting co-active feature pairs toward constructive decoder interactions . This figure complements Figure 3 .

Figure 9: The untying mechanism to separate constructive decoder interactions from encoder cross-talk is consistent across SAEs, including TopK, BatchTopK, Matryoshka, and JumpReLU. This figure complements Figure 5 .

Figure 10: Interference across sparsity and dictionary width. TopK SAEs trained on Pythia-160M, varying constraints, support size $k$ , and width $p$ . Left: varying $k$ Right: varying $p$ .

Figure 11: Effective interference on MNIST. The same qualitative pattern as in language models appears: interference is lowest in the most constrained tied setting and increases as architectural degrees of freedom are added. This figure complements Figure 6 .

Figure 12: Decoder orientation and co-activation on MNIST. Relaxing the constraints broadens the distribution of decoder correlations and increases co-activation among non-orthogonal feature pairs. This figure complements Figure 3 .

Panel of Figure 13 — Figure 13: Effective interference for independently trained SAEBench SAEs. Gemma-2-2B layer 12 (left) and Pythia-160M layer 8 (right), across the four architectures.

Panel of Figure 14 — Figure 14: Decoder orientation and co-activation for independently trained SAEBench SAEs. Gemma-2-2B layer 12 (top) and Pythia-160M layer 8 (bottom).

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Sparse Autoencoders (SAEs) have emerged as a powerful paradigm for disentangling feature superposition in transformer-based architectures, enabling precise control via activation steering. However, the theoretical foundations of compositional steering -- the simultaneous activation of distinct semantic latents -- remain under-explored. The prevailing Linear Representation Hypothesis often abstracts away non-linear interference effects that arise in overcomplete dictionaries. We present a geometric framework for analyzing the instability of feature unions. Modeling the activation space as a high-dimensional sparse cone manifold, we derive an asymptotic compositional-collapse threshold under a spherical dictionary model, characterized by the Gaussian mean width (statistical dimension) of the signal cone. We further show that, in the high-bias regime, ReLU rectification converts microscopic correlation-induced variance fluctuations into a systematic drift that accumulates under composition, yielding interference growth consistent with a ratchet effect. We validate the predicted scaling trends on structured semantic features extracted from CLEVR, where hierarchical correlations accelerate the transition relative to random baselines. Together, our results highlight geometric constraints on the scalability of union-based steering and motivate composition mechanisms that explicitly manage interference beyond naive linear superposition.

Abstract

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