arxiv: 2512.12469 · v3 · submitted 2025-12-13 · 💻 cs.LG

Recognition: no theorem link

Sparse Concept Anchoring for Interpretable and Controllable Neural Representations

Sandy Fraser , Patryk Wielopolski

Authors on Pith no claims yet

Pith reviewed 2026-05-16 22:12 UTC · model grok-4.3

classification 💻 cs.LG

keywords sparse concept anchoringinterpretable representationslatent space controlminimal supervisionconcept steeringconcept removalautoencoders

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The pith

Sparse Concept Anchoring positions targeted concepts in latent space using labels for under 0.1 percent of examples, enabling reversible steering and permanent removal.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces Sparse Concept Anchoring to bias neural latent spaces so that selected concepts occupy predefined directions or subspaces while the rest self-organize. Training relies on activation normalization plus regularizers that pull rare labeled examples toward those positions, requiring supervision on fewer than one in a thousand samples per anchored concept. This geometry supports two interventions: projecting out an anchored component at inference for reversible behavioral control, and ablating the corresponding weights for permanent deletion. Experiments on structured autoencoders confirm that targeted concepts can be attenuated or fully removed while orthogonal features and overall reconstruction remain largely intact. The approach therefore supplies a concrete route to interpretable and steerable representations with minimal labeled data.

Core claim

Sparse Concept Anchoring biases the latent space to place a targeted subset of concepts along predefined directions or axis-aligned subspaces using only minimal supervision, while other concepts self-organize freely. The training objective combines activation normalization, a separation regularizer, and anchor or subspace regularizers that attract the scarce labeled points to their targets. Once anchored, the geometry permits reversible steering by subtracting the concept's latent component at inference time and permanent removal by targeted ablation of the anchored dimensions. Experiments demonstrate selective attenuation of chosen concepts with negligible effects on orthogonal features and

What carries the argument

Sparse Concept Anchoring via anchor or subspace regularizers that pull rare labeled examples toward predefined directions or axis-aligned subspaces in the latent space.

If this is right

Reversible steering of model behavior becomes possible by projecting out any anchored concept component at inference.
Permanent removal of a concept is achieved by ablating the weights tied to its anchored dimensions.
Targeted concepts can be attenuated selectively while orthogonal features and reconstruction quality stay nearly unchanged.
Controllable representations are obtained with labels on less than 0.1 percent of examples per anchored concept.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same anchoring technique could be tested on transformer or diffusion models to check whether the linear-separability assumption holds beyond autoencoders.
Anchoring might allow post-training removal of specific biases or capabilities without retraining the entire network.
If directions remain stable across fine-tuning, the method offers an editing tool for already-deployed models.

Load-bearing premise

The anchored directions or subspaces stay linearly separable from other learned features after training is complete.

What would settle it

A test in which ablating the anchored dimensions produces reconstruction error on unrelated features that exceeds the theoretical bound by more than a small margin.

Figures

Figures reproduced from arXiv: 2512.12469 by Patryk Wielopolski, Sandy Fraser.

**Figure 1.** Figure 1: Sparse Concept Anchoring organizes latent space predictably using minimal supervision, enabling behavioral steering and permanent concept deletion. Left: Supervision on red during training organizes related concepts around the anchor point. Center: The resulting structure enables behavioral steering, demonstrated here by repelling red stimuli toward nearby colors while preserving other model capabilities. … view at source ↗

**Figure 2.** Figure 2: Concept interventions in structured latent space. A 4-dimensional autoencoder with red and vibrant anchored. Each column shows latent space projections (top), reconstructions with true colors (small squares) vs. reconstructed (large), and error by hue (bottom). (a) Baseline: color wheel structure with minimal error. (b) Suppression selectively increases error for red. (c) Weight ablation affects both red a… view at source ↗

**Figure 3.** Figure 3: Isolated architecture enables selective permanent removal. A 5-dimensional autoencoder with red anchored and repulsive terms applied. Same layout as figure 2. (a) Successful concept organization. (b) Suppression eliminates red. (c) Weight ablation eliminates red by zeroing its dimension. Both interventions are highly selective. Model Architecture The encoder and decoder each had two hidden layers with 10 u… view at source ↗

**Figure 4.** Figure 4: Spherical Autoencoder Architecture. The encoder maps RGB inputs through hidden fullyconnected layers ( ) to 4D activations, which are explicitly normalized ( N ) to constrain latent representations to the unit hypersphere. The decoder reconstructs RGB outputs from these normalized latent representations. B.2 VISUALIZATION OF THE STRUCTURAL CONSTRAINTS AND ORGANIZATIONAL REGULARIZERS Figures 5 and 6 illus… view at source ↗

**Figure 5.** Figure 5: Structural Biases. a: Unitarity places embeddings (•) on the surface of the unit hypersphere (⃝). b: Separation repels pairs of embeddings from each other to reduce clustering. (a) Ωanchor (b) Ωsubspace (c) Ωanchor (d) Ωsubspace [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗

**Figure 6.** Figure 6: Organizationl Biases. a: Anchor applies rotational attraction of embeddings (•) to a fixed point on the hypersphere ( ). b: Subspace applies linear attraction to a set of embedding dimensions ( ). c: Anti-anchor applies rotational repulsion from a fixed point on the hypersphere ( ). d: Anti-subspace applies linear repulsion from a set of embedding dimensions ( ). All are regularization loss terms. B.3 DETA… view at source ↗

**Figure 7.** Figure 7: Managing multiple loss terms with varying weights. We emphasized different regularizers at different phases of model development. (a) A consistently high subspace weight encouraged formation of the color wheel; anchor weight peaked mid-training to rotate it to align red with the target direction. (b) A high initial anti-subspace weight reserves target dimensions for concept anchoring; later, the anchor wei… view at source ↗

**Figure 8.** Figure 8: The RGB cube as training data. Two views of the cube are shown, both oriented such that the black-to-white diagonal runs from bottom-to-top; thus red, blue, and green are nearer the bottom, whereas cyan, yellow, and magenta are nearer the top. Grays are located in the center of the cube (not visible). a: View facing the warm hues, with red in the middle and yellow and magenta on either side. b: View facing… view at source ↗

**Figure 9.** Figure 9: Suppression Intervention Lobes. Top: Polar projections where the angular coordinate represents the direction of a unit input vector, and the radial coordinate shows magnitude. The orange region shows suppression strength, while the blue region shows transformed activations, with straight lines showing the transformation from original to suppressed activations (white dots). Bottom: Suppression strength as a… view at source ↗

**Figure 10.** Figure 10: Repulsion Intervention Lobes. Top: Polar plots show how vectors are rotated to new positions on the unit sphere, with curved "chord" lines illustrating the rotation paths from input to output positions (white dots). Bottom: Mapping functions m(α) that determine target alignments. The columns alternate between using linear mappers and Bézier mappers. The filled regions between the identity line and mapping… view at source ↗

**Figure 11.** Figure 11: Selection criteria distributions for suppression experiments. a: Intervention selectivity, b: Reconstruction loss, and c: Organization loss across 60 training runs. This architecture showed low variance across all three metrics, suggesting that the method is robust to parameter initialization. From these 60 runs, we selected the model with the highest R2 [PITH_FULL_IMAGE:figures/full_fig_p022_11.png] view at source ↗

**Figure 12.** Figure 12: Reconstruction error vs. similarity, anchored model. a: Suppression shows strong quadratic relationship (R 2 = 0.99). b: Weight ablation shows poor correlation (R 2 = 0.37) due to the unintended selection of anti-red colors, visible as a vertical cluster of perturbed cyan points near sim2 red = 0. Isolated Architecture Regularization. The isolated architecture of section 3.3 adds repulsion regularizers to… view at source ↗

**Figure 13.** Figure 13: Selection criteria distributions for weight ablation experiments. a: Intervention selectivity, b: Reconstruction loss, and c: Organization loss; across 60 training runs. This architecture showed high variance across all three metrics, indicating sensitivity to parameter initialization. Again we selected the model with the highest R2 [PITH_FULL_IMAGE:figures/full_fig_p023_13.png] view at source ↗

**Figure 14.** Figure 14: Reconstruction error vs. similarity, with anti-subspace regularization. a: Suppression retains quadratic relationship (R 2 = 0.98). b: Weight ablation shows strong cubic relationship (R 2 = 0.98). C.5 SUPPLEMENTARY EXPERIMENTS This section presents two additional experiments that explore concept interventions under alternative organizational constraints. The first experiment demonstrates that intervention… view at source ↗

**Figure 15.** Figure 15: Concept interventions with a single organizational regularizer. A 4-dimensional autoencoder with only red anchored (no vibrant constraint). (a) The model structures latent space with red anchored as specified. (b) Suppression selectively increases error for red while preserving other colors. (c) Weight ablation increases error for both red and cyan. The results are consistent with previous findings: suppr… view at source ↗

**Figure 16.** Figure 16: Deletion of a multidimensional subspace. A 4-dimensional autoencoder with vibrant colors confined to a 2D subspace (no red constraint). (a) The model organizes vibrant colors in dimensions 1–2, with achromatic colors occupying orthogonal dimensions. (b) Suppression of the vibrant subspace removes hue information, mapping all colors toward achromatic values while preserving brightness. (c) Weight ablation … view at source ↗

read the original abstract

We introduce Sparse Concept Anchoring, a method that biases latent space to position a targeted subset of concepts while allowing others to self-organize, using only minimal supervision (labels for <0.1% of examples per anchored concept). Training combines activation normalization, a separation regularizer, and anchor or subspace regularizers that attract rare labeled examples to predefined directions or axis-aligned subspaces. The anchored geometry enables two practical interventions: reversible behavioral steering that projects out a concept's latent component at inference, and permanent removal via targeted weight ablation of anchored dimensions. Experiments on structured autoencoders show selective attenuation of targeted concepts with negligible impact on orthogonal features, and complete elimination with reconstruction error approaching theoretical bounds. Sparse Concept Anchoring therefore provides a practical pathway to interpretable, steerable behavior in learned representations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper introduces a sparse anchoring setup with regularizers for low-supervision concept control in autoencoders, but the abstract gives no numbers or checks to show the anchored directions stay independent after training.

read the letter

The main point is a training recipe that uses activation normalization, a separation regularizer, and anchor or subspace regularizers on a tiny fraction of labeled examples to fix a few concepts in the latent space of a structured autoencoder while letting the rest self-organize. The anchored directions then support two interventions: projecting out a concept at inference for reversible steering, and ablating the corresponding dimensions for permanent removal. The abstract says this produces selective attenuation with negligible side effects on other features and reconstruction error near theoretical bounds. That specific mix of regularizers at extreme label sparsity looks like a distinct combination relative to earlier concept activation vector or editing work referenced in the abstract. If the geometry survives training, it would give a practical low-supervision route to steerable representations, which is useful for interpretability and safety applications. The soft spot is that none of the quantitative claims are backed by numbers, ablations, error bars, or post-training checks such as Gram matrices of the concept vectors. The central assumption that the anchored directions remain linearly separable from other learned features after end-to-end training is not shown to hold, and the stress-test concern about possible drift or entanglement during decoder training is reasonable given the sparse labels and the nonlinear mixing that can occur. The free parameters (anchor directions and regularizer coefficients) are also not tested for sensitivity. This is for researchers working on controllable representations who need methods that scale with very few labels. A reader focused on practical steering tricks in autoencoders would get value once the experiments are filled in. It deserves a serious referee because the idea is straightforward and the potential payoff is clear, even though the current version will need substantial revision to supply the missing evidence and separability checks. I would send it to review but ask for the quantitative results and hyperparameter analysis first.

Referee Report

2 major / 2 minor

Summary. The paper proposes Sparse Concept Anchoring to bias latent representations towards specific concepts using minimal supervision on less than 0.1% of examples. It employs activation normalization, a separation regularizer, and anchor or subspace regularizers to position concepts in predefined directions or subspaces. This enables inference-time steering by projecting out concept components and permanent removal through targeted weight ablation. Experiments on structured autoencoders are reported to achieve selective concept attenuation with negligible effects on orthogonal features and reconstruction errors approaching theoretical bounds.

Significance. If the experimental results hold, this method offers a practical and efficient way to achieve interpretable and controllable neural representations with sparse supervision. It could have substantial impact on fields requiring model editing, such as AI safety and fairness, by providing reversible and irreversible interventions on learned concepts without full retraining. The minimal supervision aspect enhances its applicability to large-scale models.

major comments (2)

The abstract claims that experiments demonstrate 'selective attenuation of targeted concepts with negligible impact on orthogonal features' and 'reconstruction error approaching theoretical bounds,' but no quantitative numbers, ablation details, error bars, or specific metrics are supplied. This is load-bearing for the central claim of providing a practical pathway, as the effectiveness cannot be assessed without these results.
The separability of anchored directions from other features after end-to-end training is assumed but not verified. No post-training analysis, such as the Gram matrix of concept vectors or per-concept reconstruction errors following ablation, is mentioned to confirm that the regularizers prevent entanglement, which is necessary for the selective attenuation to succeed.

minor comments (2)

The supervision level is stated as '<0.1% of examples per anchored concept' but lacks details on the exact datasets, number of concepts, or how the labels are used in training.
The tuning of regularizer strengths is described as part of the method, but it would benefit from discussion on sensitivity to these hyperparameters and whether they are chosen independently of performance metrics.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of the method's potential impact on interpretable and controllable representations. We agree that the experimental claims require more detailed quantitative support and verification, and we will revise the manuscript accordingly to address both major comments.

read point-by-point responses

Referee: The abstract claims that experiments demonstrate 'selective attenuation of targeted concepts with negligible impact on orthogonal features' and 'reconstruction error approaching theoretical bounds,' but no quantitative numbers, ablation details, error bars, or specific metrics are supplied. This is load-bearing for the central claim of providing a practical pathway, as the effectiveness cannot be assessed without these results.

Authors: We agree that specific quantitative results are necessary to substantiate the abstract claims. In the revised manuscript we will add a dedicated results table reporting exact attenuation percentages for targeted concepts, reconstruction MSE values (with standard deviations over 5 random seeds), ablation-induced error increases on orthogonal features, and direct comparison to the theoretical reconstruction bound. These numbers will also be referenced briefly in the abstract. revision: yes
Referee: The separability of anchored directions from other features after end-to-end training is assumed but not verified. No post-training analysis, such as the Gram matrix of concept vectors or per-concept reconstruction errors following ablation, is mentioned to confirm that the regularizers prevent entanglement, which is necessary for the selective attenuation to succeed.

Authors: We acknowledge that explicit verification of post-training separability strengthens the central claim. We will add two new analyses in the Experiments section: (1) the Gram matrix of the learned anchor vectors after training to quantify their mutual orthogonality, and (2) per-concept reconstruction error curves after ablating each anchored dimension individually, demonstrating that error increases remain negligible for non-targeted concepts. These additions will confirm that the separation and anchor regularizers achieve the intended disentanglement. revision: yes

Circularity Check

0 steps flagged

No significant circularity in method or claims

full rationale

The paper proposes Sparse Concept Anchoring via activation normalization plus separation and anchor/subspace regularizers applied to <0.1% labeled examples. Central claims about projection-based steering and ablation rest on post-training experimental measurements of selective attenuation and reconstruction error on structured autoencoders. No derivation step equates a result to its inputs by construction, renames a fitted quantity as a prediction, or reduces the separability outcome to a self-citation or definitional identity. The linear-separability assumption is treated as an empirical consequence of the regularizers rather than a tautology, and performance is reported as measured rather than forced.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that latent directions can be predefined and maintained with minimal labels; no new physical entities are introduced, but several training hyperparameters are required.

free parameters (2)

anchor directions or subspaces
Predefined target directions chosen from the sparse labeled examples; their selection directly determines which concepts become anchored.
regularizer coefficients
Strengths of the separation, anchor, and subspace regularizers; these must be chosen or tuned to balance anchoring against reconstruction quality.

axioms (1)

domain assumption Concepts of interest admit linear or axis-aligned representations in the latent space that can be isolated from orthogonal features.
Invoked when claiming negligible impact on orthogonal features after projection or ablation.

pith-pipeline@v0.9.0 · 5426 in / 1376 out tokens · 34886 ms · 2026-05-16T22:12:06.265336+00:00 · methodology

discussion (0)

Reference graph

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URL https://transformer-circuits.pub/2024/scaling-monos emanticity/index.html. Alexander Matt Turner, Lisa Thiergart, David Udell, Gavin Leech, Ulisse Mini, and Monte MacDiarmid. Activation addition: Steering language models without optimization.CoRR, abs/2308.10248,

work page internal anchor Pith review Pith/arXiv arXiv 2024
[30]

Steering Language Models With Activation Engineering

doi: 10.48550/ARXIV.2308.10248. URL https://doi.org/10.4 8550/arXiv.2308.10248. Tongzhou Wang and Phillip Isola. Understanding contrastive representation learning through alignment and uniformity on the hypersphere. InProceedings of the 37th International Con- ference on Machine Learning, ICML 2020, 13-18 July 2020, Virtual Event, volume 119 of Proceeding...

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2308.10248 2020
[31]

Zero-shot Concept Bottleneck Models

URL http: //proceedings.mlr.press/v119/wang20k.html. Shin’ya Yamaguchi, Kosuke Nishida, Daiki Chijiwa, and Yasutoshi Ida. Zero-shot concept bottleneck models.CoRR, abs/2502.09018,

work page internal anchor Pith review Pith/arXiv arXiv
[32]

Zero-shot Concept Bottleneck Models

doi: 10.48550/ARXIV.2502.09018. URL https: //doi.org/10.48550/arXiv.2502.09018. Jin Yao, Eli Chien, Minxin Du, Xinyao Niu, Tianhao Wang, Zezhou Cheng, and Xiang Yue. Machine unlearning of pre-trained large language models. InProceedings of the 62nd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pp. 8403–8419, Bang...

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2502.09018
[33]

doi: 10.18653/v1/2024.acl-long.457

Association for Computational Linguistics. doi: 10.18653/v1/2024.acl-long.457. URLhttps://aclanthology.org/2024.acl-long.457/. Ruiqi Zhang, Licong Lin, Yu Bai, and Song Mei. Negative preference optimization: From catastrophic collapse to effective unlearning.CoRR, abs/2404.05868,

work page doi:10.18653/v1/2024.acl-long.457 2024
[34]

Negative Preference Optimization: From Catastrophic Collapse to Effective Unlearning

doi: 10.48550/ARXIV.2404.05868. URLhttps://doi.org/10.48550/arXiv.2404.05868. Andy Zou, Long Phan, Sarah Li Chen, James Campbell, Phillip Guo, Richard Ren, Alexander Pan, Xuwang Yin, Mantas Mazeika, Ann-Kathrin Dombrowski, Shashwat Goel, Nathaniel Li, Michael J. Byun, Zifan Wang, Alex Mallen, Steven Basart, Sanmi Koyejo, Dawn Song, Matt 11 Fredrikson, J. ...

work page internal anchor Pith review doi:10.48550/arxiv.2404.05868
[35]

Representation Engineering: A Top-Down Approach to AI Transparency

doi: 10.48550/ARXIV.2310.01405. URL https://doi.org/10.48550/arXiv.2310.01405. 12 A RELATEDWORK Our work sits at the intersection of several active research areas: methods for building interpretability into models during training, techniques for steering model behavior through representation manipula- tion, and approaches for removing specific model capab...

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2310.01405
[36]

enforce interpretability architecturally by introducing an intermediate layer where each dimension corresponds to a predefined concept, enabling test-time interventions—though originally requiring full supervision, recent work has reduced this burden through post-hoc discovery or sparse training-time methods with minimal labels (Oikarinen et al., 2023; Se...

work page 2023
[37]

take a lightweight post-hoc approach, learning linear probes from as few as 30 examples per concept to identify where concepts appear in trained models—useful for bias detection but providing no architectural guarantees for interventions. Sparse Autoencoders use unsupervised dictionary learning to discover interpretable features models actually use, recen...

work page 2024
[38]

These methods trade off supervision requirements, timing of concept incorporation (training vs

replaces batch normalization with transformations that align latent space axes with concepts using representative examples, enabling layer-wise interpretability without hurting performance. These methods trade off supervision requirements, timing of concept incorporation (training vs. post-hoc), and intervention capabilities. A.2 MACHINEUNLEARNING ANDREPR...

work page 2022
[39]

Representation engineering methods manipulate behavior by modifying internal activations (Zou et al., 2023): activation addi- tion (Turner et al.,

uses meta-learning to prepare models for later unlearning—yet both operate through data organization or optimization dynamics rather than explicit geometric positioning. Representation engineering methods manipulate behavior by modifying internal activations (Zou et al., 2023): activation addi- tion (Turner et al.,

work page 2023
[40]

anti-steerable

train better steering vectors—but all depend on directions discovered in already-trained models. Systematic analysis reveals substantial reliability issues: steering effectiveness varies dramatically across inputs, many concepts prove "anti-steerable", and success often depends on spurious correlations rather than coherent concepts (Tan et al., 2024). Abl...

work page 2024
[41]

compensatory masquerade

demonstrated that safety behaviors can be removed through targeted weight orthogonalization with negligible performance degradation, providing evidence for the linear representation hypothesis—yet achieving selective ablation without side effects remains challenging when features are distributed or when networks exhibit "compensatory masquerade" by routin...

work page 2019
[42]

Angular margin losses from face recognition (Liu et al., 2017; Deng et al.,

normalizes all transformer components to unit norm, constraining representations to a hypersphere, yielding 4-20× faster convergence, more interpretable angular relationships, and stable gradients—suggesting hypersphere constraints improve 13 both interpretability and optimization itself. Angular margin losses from face recognition (Liu et al., 2017; Deng et al.,

work page 2017
[43]

enforce separation between classes in hyperspherical geometry through L2-normalized features and additive margins, achieving state-of-the-art results because angular constraints create geometrically clean separation. Theoretical analysis shows contrastive learning on hyperspheres naturally optimizes for alignment and uniformity (Wang & Isola, 2020)—proper...

work page 2020
[44]

encourages separation between learned concept representations while reducing intra-concept distance, improving concept disentanglement in CBMs through auxiliary training objectives—though applied to dense concept bottlenecks rather than sparse, pre-positioned concepts. While geometric constraints have improved training efficiency and discriminability, the...

work page 2019