Shaped SAEs：扩展机制可解释性的工具箱

# Shaped SAEs: Expanding the Toolkit for Mechanistic Interpretability
Source: [https://arxiv.org/html/2610.27375](https://arxiv.org/html/2610.27375)
Oveys Delafrooz NorooziBhagyashree PuranikAffiliation:University of California, Santa BarbaraEmail:[bpuranik@ucsb\.edu](mailto:)Upamanyu MadhowAffiliation:University of California, Santa BarbaraEmail:[madhow@ucsb\.edu](mailto:)

###### Abstract

Sparse autoencoders \(SAEs\) seek to decompose the dense activations of large language models \(LLMs\) into sparse, potentially interpretable features using a learned overcomplete dictionary\. In this work, we ask whether it is possible to enhance the semantic selectivity of the learned dictionary atoms by linearly*shaping*the dense activations prior to learning an SAE\. We explore rank\-reducing transformations based on the recently introduced J\-space corresponding to the averaged Jacobian from the current layer to the LLM output, projecting activations onto the “live” subspace spanned by eigenmodes of the Jacobian with larger singular values \(which we expect to be more semantically relevant\)\. We also introduce a new approach for training SAEs in which \(a\) an affine shaping transformation is learned along with the dictionary atoms, \(b\) the atom selection mechanism is based on the TopK projections along the one\-dimensional subspace spanned by the atoms, and \(c\) reconstruction is carried out using the same atoms\. We call this approach a Matched Filter SAE \(MF\-SAE\) based on analogies with communication theory\. The promise of shaping is demonstrated via evaluations against dense activations and state\-of\-the\-art SAEs on benchmarks including SAEBench and SAE Probes\.

## 1Introduction

Mechanistic interpretability seeks to develop insight into the internal representations and computations of large language models \(LLMs\)\. A key challenge is that model activations are high\-dimensional dense representations encoding many concepts at once \(i\.e\., they are “polysemantic”\), making it difficult to isolate individual concepts and computational variables, and to relate them to model behavior\. Sparse autoencoders \(SAEs\) have been proposed as a practical approach for replacing dense neural activations with potentially more interpretable feature vectors corresponding to their sparse representation with respect to an overcomplete dictionary\([Cunningham et al\., 2023](https://arxiv.org/html/2610.27375#bib.bib2);[Bricken et al\., 2023](https://arxiv.org/html/2610.27375#bib.bib8);[Templeton et al\., 2024](https://arxiv.org/html/2610.27375#bib.bib3)\)\. Significant effort has been devoted to developing a variety of SAE formulations[Gao et al\. \(2025\)](https://arxiv.org/html/2610.27375#bib.bib9);[Bussmann et al\. \(2024\)](https://arxiv.org/html/2610.27375#bib.bib12);[Rajamanoharan et al\. \(2024a\)](https://arxiv.org/html/2610.27375#bib.bib10);[Rajamanoharan et al\. \(2024b\)](https://arxiv.org/html/2610.27375#bib.bib11);[Bussmann et al\. \(2025\)](https://arxiv.org/html/2610.27375#bib.bib13), along with benchmarks assessing semantic selectivity[Karvonen et al\. \(2025\)](https://arxiv.org/html/2610.27375#bib.bib14);[Kantamneni et al\. \(2025\)](https://arxiv.org/html/2610.27375#bib.bib15)\.

Our broad objective is to investigate whether the toolkit for mechanistic interpretability using SAEs can be expanded by asking a simple question: can we enhance the semantic selectivity of the learned dictionary atoms by allowing ourselves the freedom to*shape*activations prior to learning an SAE? We make two key contributions that answer this question in the affirmative:

\(1\)*J\-space Shaping:*Using the recently introduced Jacobian method[Gurnee et al\. \(2026\)](https://arxiv.org/html/2610.27375#bib.bib5), we shape activations accounting for their impact on*downstream transmission*\. Locally, the Jacobian of a downstream mapping describes how perturbations to a residual\-stream activation propagate through the model\. Its leading right singular vectors therefore identify source\-space directions with high sensitivity under that mapping\. We use the leading singular vectors of a corpus\-averaged Jacobian to divide the residual stream into a high\-transmission*live*subspace and its orthogonal*quiet*complement\. The terminology is deliberately relative: quiet directions are less transmitted by the selected average Jacobian, but they are not necessarily an exact null space, devoid of semantic information, or irrelevant in every context\. However, we do expect the live directions to be more semantically relevant, and therefore investigate the impact of shaping activations by projecting them onto the live subspace\.

\(2\)*Matched Filter SAE \(MF\-SAE\):*We introduce a new SAE formulation in which we apply a learnable affine transformation prior to encoding and decoding with an overcomplete dictionary, with the affine transformation and dictionary atoms learned jointly based on reconstruction loss\. The encoder outputs are TopK rectified projections along the unit\-norm atoms, while the decoder uses the same atoms and projections for reconstruction\. We are inspired by the concept of matched filters from communication theory to learn atoms corresponding to dominant directions in the data, but noting that matched filters are only optimal when the noise is white, we insert a linear transformation up front as a learnable “whitening” transformation\. The bias term is motivated as a learnable centering mechanism that minimizes the energy of the atoms needed to represent the data, similar to the centering routinely used in signal constellations in communication systems\.

We show that our proposed methods lead to improved performance on standard benchmark tasks on semantic feature extraction under a constrained feature budget on SAEBench and SAE Probes\. We make the following high\-level observations regarding performance on SAEBench:

- •Semantically shaped dense activations \(prior to learning an SAE\) perform better than original dense activations\.
- •The MF\-SAE applied to standard dense activations performs better than state\-of\-the\-art SAEs as well as dense activations on SAEBench\.
- •The MF\-SAE applied to semantically shaped activations performs slightly better than the MF\-SAE at very tight feature budgets, but does not appear to improve performance at higher budgets\. Thus, the problem of how best to combine semantic shaping with sparse coding remains open\.

Performance trends on the SAE Probes benchmark are more variable, as we discuss in more detail in our results section \(Section[6](https://arxiv.org/html/2610.27375#S6)\)\. We evaluate on Gemma\-2\-2B at layers 12 and 20, and on GPT\-2 Small at layers 7 and 8 \(Appendix[E](https://arxiv.org/html/2610.27375#A5)\)\. On SAEBench, the MF\-SAEs outperform the available baseline SAEs for both models, while on SAE Probes the gains are marginal\. Probing accuracy is higher on Gemma\-2\-2B than on GPT\-2 Small on both benchmarks, which we attribute to model scale: the tasks of SAE Probes in particular are more complex and benefit from a larger model\.

## 2Related Work

SAEs for language models:SAEs use dictionary learning as a means of decomposing the dense, superposed representations of language models into sparse features that are potentially more interpretable\([Cunningham et al\., 2023](https://arxiv.org/html/2610.27375#bib.bib2);[Bricken et al\., 2023](https://arxiv.org/html/2610.27375#bib.bib8)\)\. A variety of SAE formulations have been proposed, includingL1L\_\{1\}\-regularized SAEs, TopK and BatchTopK SAEs[Gao et al\. \(2025\)](https://arxiv.org/html/2610.27375#bib.bib9);[Bussmann et al\. \(2024\)](https://arxiv.org/html/2610.27375#bib.bib12), Gated SAEs[Rajamanoharan et al\. \(2024a\)](https://arxiv.org/html/2610.27375#bib.bib10), JumpReLU SAEs[Rajamanoharan et al\. \(2024b\)](https://arxiv.org/html/2610.27375#bib.bib11), and Matryoshka SAEs[Bussmann et al\. \(2025\)](https://arxiv.org/html/2610.27375#bib.bib13)\. These methods differ in how they impose sparsity and trade it against reconstruction fidelity\. With respect to this state of the art, our contribution is two\-fold: \(a\) we propose shaping dense activations prior to learning sparse representations, focusing on J\-space[Gurnee et al\. \(2026\)](https://arxiv.org/html/2610.27375#bib.bib5)based transformations that account for downstream information transmission; \(b\) we propose a new SAE formulation including a learned geometric shaping and an atom selection mechanism combining ideas from TopK and matching pursuit\.

Jacobian\-based representations and computation:Our Jacobian\-based shaping builds on the work in[Gurnee et al\. \(2026\)](https://arxiv.org/html/2610.27375#bib.bib5), which averages the linearized map from an intermediate residual stream to later residuals and uses the resulting token\-associated directions to define and interrogate a sparse “J\-space\.” While we build on the insight that activation variance and downstream information transfer are not directly related, we use the Jacobian in[Gurnee et al\. \(2026\)](https://arxiv.org/html/2610.27375#bib.bib5)in a different way, employing its singular value decomposition to identify important directions for downstream information transfer to shape the activations prior to learning an SAE\-based sparse representation\. Unrelated but worth mentioning are Jacobian Sparse Autoencoders[Farnik et al\. \(2025\)](https://arxiv.org/html/2610.27375#bib.bib6), which jointly seek sparsity in the input features, output features, and the connections between them for an MLP layer\.

Cautionary notes:Recent work has raised important questions about the extent to which SAE features improve mechanistic analysis\. Good reconstruction does not by itself guarantee a faithful or useful feature decomposition\. Feature splitting and absorption can cause apparently coherent latents to omit expected instances[Chanin et al\. \(2025\)](https://arxiv.org/html/2610.27375#bib.bib4)\. It was originally shown in[Kantamneni et al\. \(2025\)](https://arxiv.org/html/2610.27375#bib.bib15)\(the SAE Probes benchmark\) that SAE representations do not consistently outperform dense activations or simple baselines for semantic probing, including under data scarcity, label noise, class imbalance, and distribution shift; however, improved SAEs \(including our own\) have addressed these objections\. It is shown in[Mueller et al\. \(2025\)](https://arxiv.org/html/2610.27375#bib.bib16)\(the MIB benchmark\) that SAE features do not uniformly improve localization of causal variables relative to alternative representations\. The AxBench benchmark provides complementary evidence in the context of concept detection and steering, where simple non\-SAE baselines can be highly competitive\([Wu et al\., 2025](https://arxiv.org/html/2610.27375#bib.bib17)\)\. These results caution against treating sparse, interpretable\-looking features or reconstruction fidelity as ends in themselves, and motivate exploring alternatives to the current SAE paradigm\. As a counterpoint, it is argued in[Peng et al\. \(2026\)](https://arxiv.org/html/2610.27375#bib.bib1)that the primary utility of SAEs is as a tool for concept discovery, despite their limitations in tasks such as steering and circuit analysis\. In this paper, we do not seek to resolve broader questions regarding the utility of SAEs; our focus is to expand the universe of available sparse representations with the goal of enhancing semantic selectivity under stringent feature budgets\.

Benchmarks for mechanistic interpretability:Several recent benchmarks seek to evaluate semantic selectivity of proposed feature representations\.SAEBenchis the most comprehensive SAE\-oriented evaluation, comparing architectures across reconstruction, sparsity, feature quality, probing, and functional metrics\([Karvonen et al\., 2025](https://arxiv.org/html/2610.27375#bib.bib14)\)\.SAE Probesdirectly tests whether SAE coordinates provide a useful sparse basis for recovering semantic information, including comparisons with dense representations\([Kantamneni et al\., 2025](https://arxiv.org/html/2610.27375#bib.bib15)\)\. Our experimental evaluations compare our proposed shaped SAEs to both dense activations and state\-of\-the\-art SAEs using SAEBench and SAE Probes\.

MIBshifts the emphasis toward mechanistic validity, evaluating localization of causal variables and circuits rather than feature interpretability alone\([Mueller et al\., 2025](https://arxiv.org/html/2610.27375#bib.bib16)\)\.AxBenchevaluates representation\-based concept detection and steering, explicitly comparing SAEs with alternative approaches to representation intervention\([Wu et al\., 2025](https://arxiv.org/html/2610.27375#bib.bib17)\)\. For the steering, concept detection, and causal variable localization tasks considered in these benchmarks, a variety of supervised techniques are shown to outperform SAEs\. In this paper, we do not seek to address this gap\. Our focus is on the complementary question of improving the unsupervised, semantically selective, sparse representations provided by SAEs\.

## 3Jacobian\-based Shaping

Letdddenote the model dimension in a decoder\-only language model, and lethℓ,t∈ℝdh\_\{\\ell,t\}\\in\\mathbb\{R\}^\{d\}denote a residual\-stream activation at a specified layerℓ\\elland token positiontt\. Lethfinal,t′∈ℝdh\_\{\\text\{final\},t^\{\\prime\}\}\\in\\mathbb\{R\}^\{d\}denote the final\-layer residual stream at positiont′t^\{\\prime\}\. The averaged Jacobian matrixJJdefined in[Gurnee et al\. \(2026\)](https://arxiv.org/html/2610.27375#bib.bib5)is a first\-order approximation for how perturbations inhℓ,th\_\{\\ell,t\}propagate downstream to the final layer and affecthfinal,t′h\_\{\\text\{final\},t^\{\\prime\}\}fort′≥tt^\{\\prime\}\\geq t, averaged over a corpus𝒟\\mathcal\{D\}of prompts sampled from a pretraining\-like distribution:

Jℓ=𝔼t,t′≥t,prompt∼𝒟​\[∂hfinal,t′∂hℓ,t\]∈ℝd×d\.J\_\{\\ell\}\\;=\\;\\mathbb\{E\}\_\{\\,t,\\ t^\{\\prime\}\\geq t,\\ \\text\{prompt\}\\sim\\mathcal\{D\}\}\\left\[\\frac\{\\partial h\_\{\\text\{final\},t^\{\\prime\}\}\}\{\\partial h\_\{\\ell,t\}\}\\right\]\\;\\in\\;\\mathbb\{R\}^\{d\\times d\}\.\(1\)
The J\-lens defined in[Gurnee et al\. \(2026\)](https://arxiv.org/html/2610.27375#bib.bib5)applies the averaged JacobianJℓJ\_\{\\ell\}to activations at layerℓ\\elland then unembeds them \(after normalization\) to obtain an output token distribution corresponding to that token\. Sorting the output distribution corresponding to an activation at layerℓ\\ellprovides a human\-readable description of what tokens the activation might tend to produce at the output, averaged across contexts\.

We utilize the averaged Jacobian in an entirely different way in this paper\. Prior to developing sparse representations for activations at layerℓ\\ell, we seek to shape them using the JacobianJ=JℓJ=J\_\{\\ell\}to emphasize downstream sensitivity, in order to better use the representation capacity of the SAE\. We suppressℓ\\ellfrom the notation henceforth for simplicity\.

Live and quiet J\-spaces:We quantify downstream sensitivity via the SVD of the Jacobian,J=U​Σ​VTJ=U\\Sigma V^\{T\}, whereUUandVVared×dd\\times dorthonormal matrices containing the left and right singular vectors, andΣ=diag⁡\(σ1,…,σd\)\\Sigma=\\mathrm\{diag\}\(\\sigma\_\{1\},\\ldots,\\sigma\_\{d\}\)is the diagonal matrix of singular values sorted in decreasing order \(σ1≥σ2≥⋯≥σd\\sigma\_\{1\}\\geq\\sigma\_\{2\}\\geq\\cdots\\geq\\sigma\_\{d\}\)\. We define the “live” space as directions corresponding to the largestrrsingular valuesσi\\sigma\_\{i\},i≤ri\\leq r, collected in the diagonal matrixΣlive=diag⁡\(σ1,…,σr\)\\Sigma\_\{\\mathrm\{live\}\}=\\mathrm\{diag\}\(\\sigma\_\{1\},\\ldots,\\sigma\_\{r\}\)\. The corresponding left and right singular vectors are collected intod×rd\\times rmatricesUliveU\_\{\\mathrm\{live\}\}andVliveV\_\{\\mathrm\{live\}\}, respectively\. The value ofrrused to partition into live and quiet spaces is a hyperparameter that we experiment with\.

Shaping via the live space:Under the intuition that the live modes are more semantically informative given their higher downstream gain, we shape the activations by projecting them onto the live space\. That is, for a residual\-stream activationhh, we obtain a shaped activationhliveh\_\{\\mathrm\{live\}\}by projectinghhonto the live space as follows:

hlive=Vlive​VliveT​h\.h\_\{\\mathrm\{live\}\}=V\_\{\\mathrm\{live\}\}V\_\{\\mathrm\{live\}\}^\{T\}h\.\(2\)We may view discarding the projection in the quiet space as a denoising operation\. As we have mentioned, quiet directions are not necessarily devoid of semantic information, but given their lower average downstream gain, we hope to gain in semantic selectivity when operating on a tight feature budget\.

![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/jacobian_spectrum_splits.png)Figure 1:Singular values of the averaged JacobianJJ\(Gemma\-2\-2B, layer 12\)\.Singular values sorted in decreasing order\. The spectrum decays rapidly, so a small number of directions carry most of the downstream gain and the remaining directions have little effect on the output\. Dashed vertical lines mark the live/quiet splitsr∈\{768,1152,1536,1792\}r\\in\\\{768,1152,1536,1792\\\}studied in Section[6\.2](https://arxiv.org/html/2610.27375#S6.SS2); the right singular vectors to the left of each line span the live space\. We user=1536r=1536throughout\.In our benchmarking with SAEBench and SAE Probes, we compare dense activationshhand shaped dense activationshliveh\_\{\\mathrm\{live\}\}against the SAEs that we train, as well as against state\-of\-the\-art SAEs\.

## 4The Matched Filter SAE \(MF\-SAE\)

Letx∈ℝdx\\in\\mathbb\{R\}^\{d\}denote the input to the SAE\. In our setting, we may havex=hx=h, the original dense activations, orx=hlivex=h\_\{\\mathrm\{live\}\}, the projection onto the live J\-space\.

The MF\-SAE consists of an outer affine encoder, a dictionary withnnelements, and an outer affine decoder \(Figure[2](https://arxiv.org/html/2610.27375#S4.F2)\)\. The norms of the dictionary elements are not constrained \(thus simplifying training\), but the atoms used as matched filters for encoding and decoding are unit\-norm versions of the dictionary elements\.

Sparse coding with the same atoms for selection and reconstructionxxOuter encoderz=We​x\+bez=W\_\{e\}x\+b\_\{e\}Matched filterswj=ReLU⁡\(d^j⊤​z\)w\_\{j\}=\\mathrm\{ReLU\}\(\\hat\{d\}\_\{j\}^\{\\top\}z\)TopK selection𝒮=arg​top​\-​kj⁡wj\\mathcal\{S\}=\\operatorname\{arg\\,top\\text\{\-\}k\}\_\{j\}\\,w\_\{j\}Synthesisz^=∑j∈𝒮wj​d^j\\hat\{z\}=\\sum\_\{j\\in\\mathcal\{S\}\}w\_\{j\}\\hat\{d\}\_\{j\}Outer decoderx^=Wd​z^\+bd\\hat\{x\}=W\_\{d\}\\hat\{z\}\+b\_\{d\}x^\\hat\{x\}Shared dictionaryD∈ℝn×dD\\in\\mathbb\{R\}^\{n\\times d\}, unit\-norm atomsd^j=dj/‖dj‖2\\hat\{d\}\_\{j\}=d\_\{j\}/\\\|d\_\{j\}\\\|\_\{2\}analysissynthesisFigure 2:The MF\-SAE architecture\.The inputxxis mapped to a latentzzby the learned outer affine encoder\(We,be\)\(W\_\{e\},b\_\{e\}\)\. The latent is correlated with the unit\-norm atomsd^j\\hat\{d\}\_\{j\}\(matched filters\), and thekklargest rectified projections are kept, giving the sparse codecc\. The same atoms reconstruct the latentz^\\hat\{z\}fromcc, which the learned outer affine decoder\(Wd,bd\)\(W\_\{d\},b\_\{d\}\)maps back to the reconstructionx^\\hat\{x\}\.#### Outer encoder\.

The input is first mapped to a latent

z=We​x\+be,z=W\_\{e\}x\+b\_\{e\},\(3\)withWe∈ℝd×dW\_\{e\}\\in\\mathbb\{R\}^\{d\\times d\}andbe∈ℝdb\_\{e\}\\in\\mathbb\{R\}^\{d\}learned\.

#### Dictionary elements and normalized atoms\.

The dictionary is a matrixD∈ℝn×dD\\in\\mathbb\{R\}^\{n\\times d\}with rowsdjd\_\{j\}, and the atoms are their normalized versionsd^j=dj/‖dj‖2\\hat\{d\}\_\{j\}=d\_\{j\}/\\\|d\_\{j\}\\\|\_\{2\}\. The atoms have no biases, and the same unit vectord^j\\hat\{d\}\_\{j\}is used both to detect atomjjin the latent and to reconstruct it\. The encoder \(WeW\_\{e\},beb\_\{e\}\) therefore shapes the space, and hence the metric, in which matching is performed\.

#### Sparse coding via TopK matched filter selection\.

We select the top\-kkatoms with the largest rectified matched filter outputs\. Specifically, let

wj=max\(d^j⊤z,0\)=ReLU\(d^j⊤z\),j=1,…,nw\_\{j\}=\\max\(\\hat\{d\}\_\{j\}^\{\\top\}z,0\)=\\mathrm\{ReLU\}\(\\hat\{d\}\_\{j\}^\{\\top\}z\),\\quad j=1,\\ldots,n\(4\)denote the rectified projections along the atoms\.

We keep thekkatoms with the largest projections:

𝒮=arg​top​\-​kj⁡wj\.\\mathcal\{S\}=\\operatorname\*\{arg\\,top\\text\{\-\}k\}\_\{j\}\\ w\_\{j\}\.\(5\)\(If fewer thankkrectified projections are positive, the number of selected atoms falls belowkk\.\)

The sparse codec∈ℝnc\\in\\mathbb\{R\}^\{n\}output by the MF\-SAE is thereforecj=wjc\_\{j\}=w\_\{j\}forj∈𝒮j\\in\\mathcal\{S\}andcj=0c\_\{j\}=0otherwise\.

#### Decoding\.

The latent is reconstructed using the selected atoms, and then mapped back to a reconstruction of the input through the outer affine decoder:

z^=∑j=1ncj​d^j=∑j∈𝒮wj​d^j,x^=Wd​z^\+bd,\\hat\{z\}=\\sum\_\{j=1\}^\{n\}c\_\{j\}\\hat\{d\}\_\{j\}=\\sum\_\{j\\in\\mathcal\{S\}\}w\_\{j\}\\hat\{d\}\_\{j\},\\qquad\\hat\{x\}=W\_\{d\}\\hat\{z\}\+b\_\{d\},\(6\)withWd∈ℝd×dW\_\{d\}\\in\\mathbb\{R\}^\{d\\times d\}andbd∈ℝdb\_\{d\}\\in\\mathbb\{R\}^\{d\}learned\.

#### Gradients and cost\.

Selection uses detached scores, and the coefficients of the selected atoms are recomputed differentiably, soWeW\_\{e\},WdW\_\{d\}, and the atoms receive gradients only through𝒮\\mathcal\{S\}\. Scoring all atoms is a single productZ​D^⊤Z\\hat\{D\}^\{\\top\}and decoding is a single dense product, so the MF\-SAE has the same computational cost as a TopK SAE of the same width\.

#### Relation to TopK SAEs\.

Consider a TopK SAE[Gao et al\. \(2025\)](https://arxiv.org/html/2610.27375#bib.bib9)wrapped in the same linear projections,z=Win​h\+binz=W\_\{\\text\{in\}\}h\+b\_\{\\text\{in\}\}andh^=Wout​z^\\hat\{h\}=W\_\{\\text\{out\}\}\\hat\{z\}\. Atomjjhas pre\-activation

pj=ej⊤​\(z−bpre\)\+bj=‖ej‖​‖z−bpre‖​cos⁡∠⁡\(z−bpre,ej\)\+bj,p\_\{j\}=e\_\{j\}^\{\\top\}\(z\-b\_\{\\text\{pre\}\}\)\+b\_\{j\}=\\\|e\_\{j\}\\\|\\,\\\|z\-b\_\{\\text\{pre\}\}\\\|\\cos\\angle\(z\-b\_\{\\text\{pre\}\},e\_\{j\}\)\+b\_\{j\},\(7\)whereeje\_\{j\}is a free encoder row andbjb\_\{j\}a free bias\. Thekklargest pre\-activations are kept with coefficientsReLU⁡\(pj\)\\mathrm\{ReLU\}\(p\_\{j\}\)and decoded with free columnsWdec\[:,j\]W\_\{\\text\{dec\}\}\[:,j\]\. The MF\-SAE differs in three ways\.

1. \(i\)*Scale\.*In equation[7](https://arxiv.org/html/2610.27375#S4.E7)the norm‖ej‖\\\|e\_\{j\}\\\|and the biasbjb\_\{j\}can make an atom win the selection independently of how well it matches, and the coefficient is the pre\-activation itself\. In the MF\-SAE the score depends only on the alignment and the coefficient is the least\-squares fit of a unit atom\.
2. \(ii\)*Bias\.*The only per\-atom offset in the MF\-SAE isd^j⊤​be\\hat\{d\}\_\{j\}^\{\\top\}b\_\{e\}, which is fixed by the direction of the atom and one shared vector\. TopK[Gao et al\. \(2025\)](https://arxiv.org/html/2610.27375#bib.bib9), BatchTopK[Bussmann et al\. \(2024\)](https://arxiv.org/html/2610.27375#bib.bib12), Matryoshka[Bussmann et al\. \(2025\)](https://arxiv.org/html/2610.27375#bib.bib13), and JumpReLU[Rajamanoharan et al\. \(2024b\)](https://arxiv.org/html/2610.27375#bib.bib11)SAEs all learn a free bias per latent, and JumpReLU additionally learns a per\-latent threshold\.
3. \(iii\)*Tying\.*The effective encoder direction of atomjjin the activation space isWe⊤​d^jW\_\{e\}^\{\\top\}\\hat\{d\}\_\{j\}and its decoder atom isWd​d^jW\_\{d\}\\hat\{d\}\_\{j\}\. WhenWdW\_\{d\}is invertible, We⊤​d^j=G⁡\(Wd​d^j\),G=We⊤​Wd−1,W\_\{e\}^\{\\top\}\\hat\{d\}\_\{j\}=G\\,\(W\_\{d\}\\hat\{d\}\_\{j\}\),\\qquad G=W\_\{e\}^\{\\top\}W\_\{d\}^\{\-1\},\(8\)withGGshared by all atoms, so encoder and decoder are tied up to one learned linear map\. In a TopK SAE,eje\_\{j\}andWdec\[:,j\]W\_\{\\text\{dec\}\}\[:,j\]are independent\. Consequently the linear projections are redundant for a TopK SAE, sinceWinW\_\{\\text\{in\}\}andWoutW\_\{\\text\{out\}\}can be absorbed intoeje\_\{j\}andWdecW\_\{\\text\{dec\}\}, but not for the MF\-SAE, where they are the only mechanism that lets the encoder differ from the decoder\.

Relation to matching pursuit SAEs:Subject to the non\-negativity constraint in equation[4](https://arxiv.org/html/2610.27375#S4.E4),wj​d^j⊤w\_\{j\}\\hat\{d\}\_\{j\}^\{\\top\}is the least\-squares estimate ofzzwhen we use only thejj\-th atom\. Thus, we may view this as the first step of non\-negative matching pursuit[Mallat and Zhang \(1993\)](https://arxiv.org/html/2610.27375#bib.bib7), but applied to all atoms at once: every atom is scored against the same latent, with no residual update between selections\. This is unlike MP\-SAE[Costa et al\. \(2026\)](https://arxiv.org/html/2610.27375#bib.bib18), which selects atoms sequentially against a residual\. Sequential selection improves reconstruction on LLM activations but has not improved sparse probing or feature absorption[Chanin and Garriga\-Alonso \(2026\)](https://arxiv.org/html/2610.27375#bib.bib19), and it requireskkserial encoder passes\.

### 4\.1Training objective

For a batch ofBBtokens with MF\-SAE inputsxbx\_\{b\}and reconstructionsx^b\\hat\{x\}\_\{b\}, the training loss is

ℒ=1B​m​∑b=1B‖x^b−xb‖22\+λ​c¯,c¯=1B​k​∑b=1B∑j∈𝒮bcb​j,\\mathcal\{L\}=\\frac\{1\}\{B\\,m\}\\sum\_\{b=1\}^\{B\}\\big\\\|\\hat\{x\}\_\{b\}\-x\_\{b\}\\big\\\|\_\{2\}^\{2\}\+\\lambda\\,\\bar\{c\},\\qquad\\bar\{c\}=\\frac\{1\}\{B\\,k\}\\sum\_\{b=1\}^\{B\}\\sum\_\{j\\in\\mathcal\{S\}\_\{b\}\}c\_\{bj\},\(9\)where𝒮b\\mathcal\{S\}\_\{b\}andcb​jc\_\{bj\}are the selected atoms and coefficients of tokenbbfrom equation[5](https://arxiv.org/html/2610.27375#S4.E5)and equation[6](https://arxiv.org/html/2610.27375#S4.E6), andmmis the dimension of the space in which the input lives\. On raw activations,xb=hbx\_\{b\}=h\_\{b\}andm=dm=d\. On the live space,xb=hlive,bx\_\{b\}=h\_\{\\text\{live\},b\}, the reconstruction is projected back onto the live space, andm=rm=r, so the error is measured only on the live directions and normalized by their number\. Since the coefficients are non\-negative,c¯\\bar\{c\}is the mean coefficient magnitude of the active atoms\. Sparsity is enforced by the top\-kkselection, so the auxiliary term only regularizes the scale of the coefficients and must be kept small: we useλ=0\.01\\lambda=0\.01in all experiments, as larger values collapse the dictionary to a small number of alive atoms\.

## 5Experimental Setup

### 5\.1Models, Data, and Training

#### Models\.

We study Gemma\-2\-2B[Gemma Team \(2024\)](https://arxiv.org/html/2610.27375#bib.bib20)\(d=2304d=2304, 26 layers\) at the residual stream after blocks 12 and 20\. Results on GPT\-2 Small[Radford et al\. \(2019\)](https://arxiv.org/html/2610.27375#bib.bib21)are reported in Appendix[E](https://arxiv.org/html/2610.27375#A5)\.

#### Jacobian\.

For each layer, the averaged Jacobian equation[1](https://arxiv.org/html/2610.27375#S3.E1)is estimated once by reverse\-mode automatic differentiation over10001000prompts truncated to6464tokens, with the final block as the target, and is shared by every SAE trained at that layer\. Unless stated otherwise, the live space usesr=1536r=1536directions \(768768quiet\)\.

#### Training\.

SAEs are trained on text from the Pile[Gao et al\. \(2020\)](https://arxiv.org/html/2610.27375#bib.bib22)\(uncopyrighted subset\) in sequences of128128tokens\. Activations from20482048sequences are pooled and shuffled, and each step uses a minibatch of40964096tokens drawn from the pool, so that a minibatch mixes tokens from many contexts\. We train for70007000steps \(about28\.728\.7M tokens\) with AdamW[Loshchilov and Hutter \(2019\)](https://arxiv.org/html/2610.27375#bib.bib23), learning rate2×10−42\\times 10^\{\-4\}, no weight decay, andλ=0\.01\\lambda=0\.01\. Evaluation uses a held\-out partition of the same corpus\.

#### Dictionary and sparsity\.

At each layer we train both the raw and the live MF\-SAE withn∈\{16384,32768\}n\\in\\\{16384,32768\\\}atoms \(16K and 32K\) andk∈\{40,80,160,320,640\}k\\in\\\{40,80,160,320,640\\\}active atoms per token\.

### 5\.2Baselines

#### Dense activations\.

As SAE\-free references, we probe the coordinates of the raw activationhhand of the shaped activationhliveh\_\{\\text\{live\}\}directly\. The latter isolates the effect of shaping from that of sparse coding\.

#### SAEBench TopK SAEs\.

SAEBench[Karvonen et al\. \(2025\)](https://arxiv.org/html/2610.27375#bib.bib14)provides TopK SAEs[Gao et al\. \(2025\)](https://arxiv.org/html/2610.27375#bib.bib9)on Gemma\-2\-2B with1616K latents atk∈\{40,80,160,320,640\}k\\in\\\{40,80,160,320,640\\\}, matching our sparsity grid\. We use two checkpoints of each: one after about3030M training tokens, which matches the28\.728\.7M tokens seen by our SAEs, and the final one after300300M tokens\.

#### Gemma Scope\.

We compare against the residual\-stream JumpReLU SAEs[Rajamanoharan et al\. \(2024b\)](https://arxiv.org/html/2610.27375#bib.bib11)of Gemma Scope[Lieberum et al\. \(2024\)](https://arxiv.org/html/2610.27375#bib.bib24)at widths 16K and 32K\. Their sparsity is set by a learned threshold rather than a fixedkk, so we pair each with the MF\-SAE of nearest average L0 \(Table[2](https://arxiv.org/html/2610.27375#A1.T2)\)\.

#### Matryoshka SAE\.

We include the Matryoshka BatchTopK SAE[Bussmann et al\. \(2025\)](https://arxiv.org/html/2610.27375#bib.bib13)with3232K latents and L0=40=40, the only configuration available for Gemma\-2\-2B\.

### 5\.3Evaluation

#### SAEBench sparse probing\.

Following SAEBench[Karvonen et al\. \(2025\)](https://arxiv.org/html/2610.27375#bib.bib14), we evaluate on 36 binary classification tasks from 8 datasets \(Bias in Bios, Amazon reviews and sentiment, GitHub code, AG News, and Europarl\), with 4000 training and 1000 test examples per task and a context length of 128 tokens\. SAE features are mean\-pooled over tokens, thekprobek\_\{\\text\{probe\}\}features with the largest difference in class means are selected, and a logistic\-regression probe is trained on them\. We report test accuracy averaged over tasks forkprobe∈\{1,2,4,8,16,32\}k\_\{\\text\{probe\}\}\\in\\\{1,2,4,8,16,32\\\}, and refer to it as SB\-Acc\.

#### SAE Probes\.

To test probing on a broader set of tasks, we use the benchmark of[Kantamneni et al\. \(2025\)](https://arxiv.org/html/2610.27375#bib.bib15), with 113 binary datasets in its standard data setting, and report test accuracy averaged over datasets for probes on the top\-kprobek\_\{\\text\{probe\}\}SAE features\.

#### Reconstruction and downstream loss\.

We report the fraction of variance explained \(PVE\) of the MF\-SAE inputxx, the fraction of alive atoms \(atoms that are active on at least one evaluation token\), and the cross\-entropy loss recovered when the activation is replaced by its reconstruction,

CE recovered=CEzero−CErecCEzero−CEclean,\\text\{CE recovered\}=\\frac\{\\mathrm\{CE\}\_\{\\text\{zero\}\}\-\\mathrm\{CE\}\_\{\\text\{rec\}\}\}\{\\mathrm\{CE\}\_\{\\text\{zero\}\}\-\\mathrm\{CE\}\_\{\\text\{clean\}\}\},\(10\)whereCEzero\\mathrm\{CE\}\_\{\\text\{zero\}\}is obtained by zero\-ablating the activation andCEclean\\mathrm\{CE\}\_\{\\text\{clean\}\}is the loss of the unmodified model\. For the live MF\-SAE we patch the model in three ways:*live only*, replacinghhwithx^\\hat\{x\};*live\+quiet*, replacinghhwithx^\+\(h−hlive\)\\hat\{x\}\+\(h\-h\_\{\\text\{live\}\}\), which restores the exact quiet component; and*projection only*, replacinghhwithhliveh\_\{\\text\{live\}\}without any SAE, which measures the cost of dropping the quiet component and upper\-bounds the live\-only value\. Since the live PVE is computed onhliveh\_\{\\text\{live\}\}rather thanhh, it is not directly comparable to the raw PVE\.

#### Automated interpretability\.

For each feature we collect its top\-activating windows on60006000sequences of128128tokens from the Pile\. Qwen2\.5\-7B\-Instruct[Qwen Team \(2025\)](https://arxiv.org/html/2610.27375#bib.bib25)is shown1212activating windows and writes a one\-line explanation\. Given only the explanation, the same model then classifies88held\-out activating windows against88top windows of randomly chosen other features, and we report the balanced accuracy as the detection score\. Features with fewer than2020activating windows are not scored\. Figure[3](https://arxiv.org/html/2610.27375#S5.F3)shows example dashboards of features from layer 20\. Dashboards for all features with detection score above0\.70\.7, including layer 12 and other models, are available at[https://huggingface\.co/spaces/user1378916/MF\-SAE/tree/main](https://huggingface.co/spaces/user1378916/MF-SAE/tree/main)\.

![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/Feature_Examples/Gemma_Layer20/Raw/gemma2_l20_raw_feature2283_little_bit.png)\(a\)Raw, feature 2283: “a little bit”
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/Feature_Examples/Gemma_Layer20/Raw/gemma2_l20_raw_feature4548_reason_behind.png)\(b\)Raw, feature 4548: “the reason behind”
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/Feature_Examples/Gemma_Layer20/Live/gemma2_l20_jlive_feature2703_later.png)\(c\)Live, feature 2703: “later”
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/Feature_Examples/Gemma_Layer20/Live/gemma2_l20_jlive_feature9055_example.png)\(d\)Live, feature 9055: “example”

Figure 3:Example feature dashboards \(Gemma\-2\-2B, layer 20, 16K dictionary\)\.\(a, b\)Features of the raw MF\-SAE\.\(c, d\)Features of the live MF\-SAE\. Each dashboard shows the feature’s top\-activating contexts together with its LLM\-generated explanation\. Dashboards for all features with detection score above0\.70\.7, for layer 12 and other models, are available at the link in Section[5\.3](https://arxiv.org/html/2610.27375#S5.SS3)\.

## 6Results and Analysis

### 6\.1Shaping without an SAE

Before training any SAE, we test whether shaping preserves the information that probes rely on\. We apply both probing benchmarks directly to the coordinates of the shaped activationhliveh\_\{\\text\{live\}\}forr∈\{768,1152,1536,1792\}r\\in\\\{768,1152,1536,1792\\\}and compare with the raw activationhh\(Figure[4](https://arxiv.org/html/2610.27375#S6.F4)\)\.

On SAEBench,hliveh\_\{\\text\{live\}\}outperformshhat both layers and for every live dimensionrr\(Figures[4\(a\)](https://arxiv.org/html/2610.27375#S6.F4.sf1)and[4\(b\)](https://arxiv.org/html/2610.27375#S6.F4.sf2)\)\. Removing the quiet directions therefore does not discard the information these tasks need; it makes that information easier to find for a probe restricted to a few coordinates\. On SAE Probes,hliveh\_\{\\text\{live\}\}is comparable tohhat layer 12 and slightly better at layer 20 \(Figures[4\(c\)](https://arxiv.org/html/2610.27375#S6.F4.sf3)and[4\(d\)](https://arxiv.org/html/2610.27375#S6.F4.sf4)\)\. Across both benchmarks, shaping never hurts probing, and the result is not sensitive to the choice ofrr\. We user=1536r=1536throughout\.

![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Bench/Gemma_Layer12/gemma2_l12_live_space_no_sae_saebench.png)\(a\)SAEBench, layer 12
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Bench/Gemma_Layer20/gemma2_l20_live_space_no_sae_saebench.png)\(b\)SAEBench, layer 20
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Probes/Gemma_Layer12/gemma2_l12_live_space_no_sae_saeprobes.png)\(c\)SAE Probes, layer 12
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Probes/Gemma_Layer20/gemma2_l20_live_space_no_sae_saeprobes.png)\(d\)SAE Probes, layer 20

Figure 4:Probing shaped activations without an SAE \(Gemma\-2\-2B\)\.Probe accuracy againstkprobek\_\{\\text\{probe\}\}for the raw activationhhand for the shaped activationhliveh\_\{\\text\{live\}\}withr∈\{768,1152,1536,1792\}r\\in\\\{768,1152,1536,1792\\\}live directions\.\(a, b\)On SAEBench,hliveh\_\{\\text\{live\}\}outperformshhat both layers and for everyrr\.\(c, d\)On SAE Probes,hliveh\_\{\\text\{live\}\}is comparable tohhat layer 12 and slightly better at layer 20\.![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Bench/Gemma_Layer12/16K/gemma2_l12_16k_k40_saebench.png)\(a\)Layer 12,k=40k=40
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Bench/Gemma_Layer12/16K/gemma2_l12_16k_k80_saebench.png)\(b\)Layer 12,k=80k=80
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Bench/Gemma_Layer20/16K/gemma2_l20_16k_k40_saebench.png)\(c\)Layer 20,k=40k=40
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Bench/Gemma_Layer20/16K/gemma2_l20_16k_k80_saebench.png)\(d\)Layer 20,k=80k=80

Figure 5:SAEBench sparse probing \(Gemma\-2\-2B, 16K dictionary\)\.SB\-Acc againstkprobek\_\{\\text\{probe\}\}for the raw and live MF\-SAEs and the available baseline SAEs\. Rows are layers 12 and 20; columns are sparsityk=40k=40andk=80k=80\. Gemma Scope has no fixed sparsity, and the SAE with the closest average L0 is shown\. Both MF\-SAEs outperform the baselines in every panel\. Raw and live are similar except in \(b\), where the live MF\-SAE is consistently better\.![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Probes/Gemma_Layer12/16K/gemma2_l12_16k_k40_saeprobes.png)\(a\)Layer 12,k=40k=40
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Probes/Gemma_Layer12/16K/gemma2_l12_16k_k80_saeprobes.png)\(b\)Layer 12,k=80k=80
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Probes/Gemma_Layer20/16K/gemma2_l20_16k_k40_saeprobes.png)\(c\)Layer 20,k=40k=40
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Probes/Gemma_Layer20/16K/gemma2_l20_16k_k80_saeprobes.png)\(d\)Layer 20,k=80k=80

Figure 6:SAE Probes \(Gemma\-2\-2B, 16K dictionary\)\.Mean test accuracy over the 113 datasets of[Kantamneni et al\. \(2025\)](https://arxiv.org/html/2610.27375#bib.bib15)againstkprobek\_\{\\text\{probe\}\}, with the same layout and methods as Figure[5](https://arxiv.org/html/2610.27375#S6.F5)\. The MF\-SAEs improve over the baselines only marginally, and raw and live perform similarly\.
### 6\.2Shaping without an SAE

Before training any SAE, we test whether shaping preserves the information that probes rely on\. We apply both probing benchmarks directly to the coordinates of the shaped activationhliveh\_\{\\text\{live\}\}forr∈\{768,1152,1536,1792\}r\\in\\\{768,1152,1536,1792\\\}and compare with the raw activationhh\(Figure[4](https://arxiv.org/html/2610.27375#S6.F4)\)\.

On SAEBench,hliveh\_\{\\text\{live\}\}outperformshhat both layers and for every live dimensionrr\(Figures[4\(a\)](https://arxiv.org/html/2610.27375#S6.F4.sf1)and[4\(b\)](https://arxiv.org/html/2610.27375#S6.F4.sf2)\)\. Removing the quiet directions therefore does not discard the information these tasks need; it makes that information easier to find for a probe restricted to a few coordinates\. On SAE Probes,hliveh\_\{\\text\{live\}\}is comparable tohhat layer 12 and slightly better at layer 20 \(Figures[4\(c\)](https://arxiv.org/html/2610.27375#S6.F4.sf3)and[4\(d\)](https://arxiv.org/html/2610.27375#S6.F4.sf4)\)\. Across both benchmarks, shaping never hurts probing, and the result is not sensitive to the choice ofrr\. We user=1536r=1536throughout\.

### 6\.3Sparse Probing

We compare the raw and live MF\-SAEs with the baseline SAEs of Section[5\.2](https://arxiv.org/html/2610.27375#S5.SS2)at sparsityk=40k=40andk=80k=80with a 16K dictionary, at layers 12 and 20\. For Gemma Scope, sparsity is not fixed; we use the SAE whose average L0 is closest tokk, so this comparison is approximate\.

#### SAEBench\.

Both MF\-SAEs outperform the available baselines at both layers and both sparsity levels \(Figure[5](https://arxiv.org/html/2610.27375#S6.F5)\)\. The raw and live MF\-SAEs perform similarly in most settings\. The exception is layer 12 atk=80k=80, where the live MF\-SAE is consistently better than the raw one acrosskprobek\_\{\\text\{probe\}\}\(Figure[5\(b\)](https://arxiv.org/html/2610.27375#S6.F5.sf2)\)\. Most of the improvement over the baselines therefore comes from the MF\-SAE itself, with shaping adding a further gain in some settings\. The benefit of shaping without an SAE \(Section[6\.2](https://arxiv.org/html/2610.27375#S6.SS2)\) thus carries over only partially once sparse coding is applied\.

#### SAE Probes\.

On the 113 datasets of[Kantamneni et al\. \(2025\)](https://arxiv.org/html/2610.27375#bib.bib15), the MF\-SAEs improve over the baselines only marginally, and the raw and live MF\-SAEs are again similar \(Figure[6](https://arxiv.org/html/2610.27375#S6.F6)\)\. This is consistent with the finding of[Kantamneni et al\. \(2025\)](https://arxiv.org/html/2610.27375#bib.bib15)that SAE features offer little advantage on this broader benchmark, and indicates that the gains on SAEBench do not transfer fully to it\.

### 6\.4Reconstruction and Downstream Loss

Table[1](https://arxiv.org/html/2610.27375#S6.T1)reports the reconstruction metrics of Section[5\.3](https://arxiv.org/html/2610.27375#S5.SS3)for the raw and live MF\-SAEs with a 16K dictionary\. Since the live PVE is computed onhliveh\_\{\\text\{live\}\}rather thanhh, PVE is comparable only within each of the raw and live groups\. For the live MF\-SAE, the gap between*live only*and*live\+quiet*measures the contribution of the discarded quiet component, while*projection only*upper\-bounds the live\-only value\. The raw MF\-SAE recovers 97–99% of the CE loss\. The live MF\-SAE recovers 89–94% when patched alone, close to the 92–95% bound set by dropping the quiet component without any SAE \(projection only\), so most of this gap comes from the projection rather than from sparse coding\. Restoring the quiet component recovers 99% in every setting\. The live MF\-SAE also uses more of its dictionary, with 81–99% of atoms alive compared with 62–89% for the raw MF\-SAE, and the fraction of alive atoms increases withkkfor both\.

Table 1:Reconstruction and downstream loss \(Gemma\-2\-2B, 16K dictionary\)\.PVE of the MF\-SAE input, fraction of alive atoms, and CE loss recovered equation[10](https://arxiv.org/html/2610.27375#S5.E10)\. For the live MF\-SAE, CE recovered is reported for the three patching modes of Section[5\.3](https://arxiv.org/html/2610.27375#S5.SS3): live only \(L\), live\+quiet \(L\+Q\), and projection only \(P\)\.

## 7Discussion and Limitations

Our results illustrate the potential gains from shaping activations to enhance semantic selectivity under tight feature budgets\. We have chosen to illustrate our ideas through transformations for which we can offer intuitive interpretations\. Projecting onto the live J\-space selects directions with larger downstream gain, while the matched filter SAE is inspired by concepts of whitening and matched filtering from communication theory\. Apart from performing well on benchmarks, the MF\-SAE formulation results in a large fraction of useful atoms without requiring additional mechanisms to address dead atoms\. We hope our work stimulates further research into the concept of shaping via linear or nonlinear transformations\.

#### Limitations\.

*Shaping mechanisms\.*Our current work shows that, on SAEBench, semantic shaping performs better than dense activations, and MF\-SAE performs better than state\-of\-the\-art SAEs\. However, combining semantic shaping with the MF\-SAE yields gains only at tight feature budgets\. Enhancing performance by combining shaping mechanisms remains an important topic for future work\.

*Jacobian analysis\.*While we have empirically shown the benefits of projecting onto the live J\-space, a more detailed analysis of the live and quiet components is an important topic for future work, complementary to the directions pursued in[Gurnee et al\. \(2026\)](https://arxiv.org/html/2610.27375#bib.bib5)\. The quiet space is of particular interest: the Jacobian is a linearization averaged over contexts and target positions, so a quiet direction with small average output\-layer gain may matter in specific contexts or at intermediate layers\.

*Scope\.*Our main results are on Gemma\-2\-2B at two layers, with GPT\-2 Small in the appendix; we have not tested larger models\.

*Baselines\.*Gemma Scope and the Matryoshka SAE were trained on different data and token budgets, and their sparsity is only approximately matched to ours\. SAE Probes results are available for these baselines at a single sparsity level\.

*Automated interpretability\.*Explanations and detection scores come from a single 7B model, and negatives are drawn from other features’ activating windows rather than from non\-activating text\.

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## Appendix AAvailable baseline SAEs

Tables[2](https://arxiv.org/html/2610.27375#A1.T2)and[3](https://arxiv.org/html/2610.27375#A1.T3)list the open\-source SAEs we compare against at each model and layer\. JumpReLU SAEs have no fixed sparsity; their L0 is the average number of active latents per token, and we pair each MF\-SAE with the JumpReLU SAE of closest average L0\. The Matryoshka SAE exists only with a 32K dictionary at L0=40=40, so it appears only in the 32K comparisons of Appendix[C](https://arxiv.org/html/2610.27375#A3)and not in the main text\.

Table 2:Baseline SAEs on Gemma\-2\-2B\.For JumpReLU SAEs, L0 is the average number of active latents per token\.LayerSAEArchitectureWidthSparsity12SAEBench[Karvonen et al\. \[2025\]](https://arxiv.org/html/2610.27375#bib.bib14)Top\-K16Kk=40,80,160,320k=40,80,160,320Gemma Scope[Lieberum et al\. \[2024\]](https://arxiv.org/html/2610.27375#bib.bib24)JumpReLU16KL0=22,41,82,176,445=22,41,82,176,445Gemma Scope[Lieberum et al\. \[2024\]](https://arxiv.org/html/2610.27375#bib.bib24)JumpReLU32KL0=12,22,40,76,155,360=12,22,40,76,155,360Matryoshka[Bussmann et al\. \[2025\]](https://arxiv.org/html/2610.27375#bib.bib13)BatchTopK32KL0=40=4020Gemma Scope[Lieberum et al\. \[2024\]](https://arxiv.org/html/2610.27375#bib.bib24)JumpReLU16KL0=22,38,71,139,294=22,38,71,139,294Matryoshka[Bussmann et al\. \[2025\]](https://arxiv.org/html/2610.27375#bib.bib13)BatchTopK32KL0=40=40Table 3:Baseline SAEs on GPT\-2 Small\.Our MF\-SAEs are trained at the same hook point, with the same dictionary size and matched sparsity\.
## Appendix BSparsity level ablation, 16K dictionary

This section extends Section[6\.3](https://arxiv.org/html/2610.27375#S6.SS3)to all sparsity levelsk∈\{40,80,160,320\}k\\in\\\{40,80,160,320\\\}with a 16K dictionary\. Across sparsity levels,k=320k=320gives the best MF\-SAE on SAE Probes in every setting, while on SAEBench it is among the best sparsity levels but not always the best \(Figures[7](https://arxiv.org/html/2610.27375#A2.F7)and[8](https://arxiv.org/html/2610.27375#A2.F8)\)\. At layer 12 the baselines are the SAEBench Top\-K and Gemma Scope 16K SAEs; at layer 20, only Gemma Scope 16K is available \(Table[2](https://arxiv.org/html/2610.27375#A1.T2)\)\. The comparisons with these baselines are shown in Figures[9](https://arxiv.org/html/2610.27375#A2.F9)and[10](https://arxiv.org/html/2610.27375#A2.F10)for layer 12 and Figures[11](https://arxiv.org/html/2610.27375#A2.F11)and[12](https://arxiv.org/html/2610.27375#A2.F12)for layer 20\.

![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Bench/Gemma_Layer12/16K/gemma2_l12_16k_mfsae_raw_sparsity_saebench.png)\(a\)Raw, layer 12
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Bench/Gemma_Layer12/16K/gemma2_l12_16k_mfsae_live_sparsity_saebench.png)\(b\)Live, layer 12
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Bench/Gemma_Layer20/16K/gemma2_l20_16k_mfsae_raw_sparsity_saebench.png)\(c\)Raw, layer 20
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Bench/Gemma_Layer20/16K/gemma2_l20_16k_mfsae_live_sparsity_saebench.png)\(d\)Live, layer 20

Figure 7:SAEBench, MF\-SAE across sparsity levels \(Gemma\-2\-2B, 16K dictionary\)\.SB\-Acc againstkprobek\_\{\\text\{probe\}\}for the MF\-SAE atk∈\{40,80,160,320\}k\\in\\\{40,80,160,320\\\}\. Columns are the raw and live MF\-SAE; rows are layers 12 and 20\.k=320k=320is among the best sparsity levels but not the best in every panel\.![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Probes/Gemma_Layer12/16K/gemma2_l12_16k_mfsae_raw_sparsity_saeprobes.png)\(a\)Raw, layer 12
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Probes/Gemma_Layer12/16K/gemma2_l12_16k_mfsae_live_sparsity_saeprobes.png)\(b\)Live, layer 12
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Probes/Gemma_Layer20/16K/gemma2_l20_16k_mfsae_raw_sparsity_saeprobes.png)\(c\)Raw, layer 20
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Probes/Gemma_Layer20/16K/gemma2_l20_16k_mfsae_live_sparsity_saeprobes.png)\(d\)Live, layer 20

Figure 8:SAE Probes, MF\-SAE across sparsity levels \(Gemma\-2\-2B, 16K dictionary\)\.Mean test accuracy againstkprobek\_\{\\text\{probe\}\}for the MF\-SAE atk∈\{40,80,160,320\}k\\in\\\{40,80,160,320\\\}, with the same layout as Figure[7](https://arxiv.org/html/2610.27375#A2.F7)\.k=320k=320is consistently the best sparsity level\.![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Bench/Gemma_Layer12/16K/gemma2_l12_16k_k40_saebench.png)\(a\)k=40k=40
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Bench/Gemma_Layer12/16K/gemma2_l12_16k_k80_saebench.png)\(b\)k=80k=80
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Bench/Gemma_Layer12/16K/gemma2_l12_16k_k160_saebench.png)\(c\)k=160k=160
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Bench/Gemma_Layer12/16K/gemma2_l12_16k_k320_saebench.png)\(d\)k=320k=320

Figure 9:SAEBench, layer 12, 16K dictionary\.SB\-Acc againstkprobek\_\{\\text\{probe\}\}for the raw and live MF\-SAEs, the SAEBench Top\-K SAE at the samekk, and the Gemma Scope 16K SAE of closest average L0\.![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Probes/Gemma_Layer12/16K/gemma2_l12_16k_k40_saeprobes.png)\(a\)k=40k=40
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Probes/Gemma_Layer12/16K/gemma2_l12_16k_k80_saeprobes.png)\(b\)k=80k=80
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Probes/Gemma_Layer12/16K/gemma2_l12_16k_k160_saeprobes.png)\(c\)k=160k=160
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Probes/Gemma_Layer12/16K/gemma2_l12_16k_k320_saeprobes.png)\(d\)k=320k=320

Figure 10:SAE Probes, layer 12, 16K dictionary\.Mean test accuracy againstkprobek\_\{\\text\{probe\}\}, with the same methods and layout as Figure[9](https://arxiv.org/html/2610.27375#A2.F9)\.![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Bench/Gemma_Layer20/16K/gemma2_l20_16k_k40_saebench.png)\(a\)k=40k=40
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Bench/Gemma_Layer20/16K/gemma2_l20_16k_k80_saebench.png)\(b\)k=80k=80
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Bench/Gemma_Layer20/16K/gemma2_l20_16k_k160_saebench.png)\(c\)k=160k=160
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Bench/Gemma_Layer20/16K/gemma2_l20_16k_k320_saebench.png)\(d\)k=320k=320

Figure 11:SAEBench, layer 20, 16K dictionary\.SB\-Acc againstkprobek\_\{\\text\{probe\}\}for the raw and live MF\-SAEs and the Gemma Scope 16K SAE of closest average L0 \(38, 71, 139 and 294 fork=40,80,160,320k=40,80,160,320\)\. No SAEBench SAEs are available at this layer\.![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Probes/Gemma_Layer20/16K/gemma2_l20_16k_k40_saeprobes.png)\(a\)k=40k=40
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Probes/Gemma_Layer20/16K/gemma2_l20_16k_k80_saeprobes.png)\(b\)k=80k=80
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Probes/Gemma_Layer20/16K/gemma2_l20_16k_k160_saeprobes.png)\(c\)k=160k=160
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Probes/Gemma_Layer20/16K/gemma2_l20_16k_k320_saeprobes.png)\(d\)k=320k=320

Figure 12:SAE Probes, layer 20, 16K dictionary\.Mean test accuracy againstkprobek\_\{\\text\{probe\}\}, with the same methods and layout as Figure[11](https://arxiv.org/html/2610.27375#A2.F11)\.
## Appendix CDictionary size ablation, 32K dictionary

This section repeats Appendix[B](https://arxiv.org/html/2610.27375#A2)with a 32K dictionary\. As with 16K,k=320k=320gives the best MF\-SAE on SAE Probes, while on SAEBench it is among the best sparsity levels but not always the best \(Figures[13](https://arxiv.org/html/2610.27375#A3.F13)and[14](https://arxiv.org/html/2610.27375#A3.F14)\)\. SAEBench does not release 32K SAEs\. At layer 12 the baselines are Gemma Scope 32K and the Matryoshka SAE at L0=40=40; at layer 20, Gemma Scope has no 32K SAEs and only the Matryoshka SAE is available\. Since the Matryoshka SAE exists only with a 32K dictionary, it appears only in this section\. On SAE Probes atk=40k=40, the Matryoshka SAE is comparable to the MF\-SAEs at layer 12 and outperforms them at layer 20\. The comparisons with these baselines are shown in Figures[15](https://arxiv.org/html/2610.27375#A3.F15)and[16](https://arxiv.org/html/2610.27375#A3.F16)for layer 12 and Figures[17](https://arxiv.org/html/2610.27375#A3.F17)and[18](https://arxiv.org/html/2610.27375#A3.F18)for layer 20\.

![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Bench/Gemma_Layer12/32K/gemma2_l12_32k_mfsae_raw_sparsity_saebench.png)\(a\)Raw, layer 12
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Bench/Gemma_Layer12/32K/gemma2_l12_32k_mfsae_live_sparsity_saebench.png)\(b\)Live, layer 12
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Bench/Gemma_Layer20/32K/gemma2_l20_32k_mfsae_raw_sparsity_saebench.png)\(c\)Raw, layer 20
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Bench/Gemma_Layer20/32K/gemma2_l20_32k_mfsae_live_sparsity_saebench.png)\(d\)Live, layer 20

Figure 13:SAEBench, MF\-SAE across sparsity levels \(Gemma\-2\-2B, 32K dictionary\)\.SB\-Acc againstkprobek\_\{\\text\{probe\}\}for the MF\-SAE atk∈\{40,80,160,320\}k\\in\\\{40,80,160,320\\\}\. Columns are the raw and live MF\-SAE; rows are layers 12 and 20\.k=320k=320is among the best sparsity levels but not the best in every panel\.![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Probes/Gemma_Layer12/32K/gemma2_l12_32k_mfsae_raw_sparsity_saeprobes.png)\(a\)Raw, layer 12
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Probes/Gemma_Layer12/32K/gemma2_l12_32k_mfsae_live_sparsity_saeprobes.png)\(b\)Live, layer 12
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Probes/Gemma_Layer20/32K/gemma2_l20_32k_mfsae_raw_sparsity_saeprobes.png)\(c\)Raw, layer 20
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Probes/Gemma_Layer20/32K/gemma2_l20_32k_mfsae_live_sparsity_saeprobes.png)\(d\)Live, layer 20

Figure 14:SAE Probes, MF\-SAE across sparsity levels \(Gemma\-2\-2B, 32K dictionary\)\.Mean test accuracy againstkprobek\_\{\\text\{probe\}\}for the MF\-SAE atk∈\{40,80,160,320\}k\\in\\\{40,80,160,320\\\}, with the same layout as Figure[13](https://arxiv.org/html/2610.27375#A3.F13)\.k=320k=320is consistently the best sparsity level\.![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Bench/Gemma_Layer12/32K/gemma2_l12_32k_k40_saebench.png)\(a\)k=40k=40
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Bench/Gemma_Layer12/32K/gemma2_l12_32k_k80_saebench.png)\(b\)k=80k=80
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Bench/Gemma_Layer12/32K/gemma2_l12_32k_k160_saebench.png)\(c\)k=160k=160
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Bench/Gemma_Layer12/32K/gemma2_l12_32k_k320_saebench.png)\(d\)k=320k=320

Figure 15:SAEBench, layer 12, 32K dictionary\.SB\-Acc againstkprobek\_\{\\text\{probe\}\}for the raw and live MF\-SAEs, the Gemma Scope 32K SAE of closest average L0, and, in \(a\), the Matryoshka SAE\.![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Probes/Gemma_Layer12/32K/gemma2_l12_32k_k40_saeprobes.png)\(a\)k=40k=40
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Probes/Gemma_Layer12/32K/gemma2_l12_32k_k80_saeprobes.png)\(b\)k=80k=80
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Probes/Gemma_Layer12/32K/gemma2_l12_32k_k160_saeprobes.png)\(c\)k=160k=160
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Probes/Gemma_Layer12/32K/gemma2_l12_32k_k320_saeprobes.png)\(d\)k=320k=320

Figure 16:SAE Probes, layer 12, 32K dictionary\.Mean test accuracy againstkprobek\_\{\\text\{probe\}\}, with the same methods and layout as Figure[15](https://arxiv.org/html/2610.27375#A3.F15)\. In \(a\), the Matryoshka SAE is comparable to the MF\-SAEs\.![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Bench/Gemma_Layer20/32K/gemma2_l20_32k_k40_saebench.png)\(a\)k=40k=40
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Bench/Gemma_Layer20/32K/gemma2_l20_32k_k80_saebench.png)\(b\)k=80k=80
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Bench/Gemma_Layer20/32K/gemma2_l20_32k_k160_saebench.png)\(c\)k=160k=160
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Bench/Gemma_Layer20/32K/gemma2_l20_32k_k320_saebench.png)\(d\)k=320k=320

Figure 17:SAEBench, layer 20, 32K dictionary\.SB\-Acc againstkprobek\_\{\\text\{probe\}\}for the raw and live MF\-SAEs and, in \(a\), the Matryoshka SAE\. No SAEBench or Gemma Scope 32K SAEs are available at this layer\.![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Probes/Gemma_Layer20/32K/gemma2_l20_32k_k40_saeprobes.png)\(a\)k=40k=40
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Probes/Gemma_Layer20/32K/gemma2_l20_32k_k80_saeprobes.png)\(b\)k=80k=80
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Probes/Gemma_Layer20/32K/gemma2_l20_32k_k160_saeprobes.png)\(c\)k=160k=160
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Probes/Gemma_Layer20/32K/gemma2_l20_32k_k320_saeprobes.png)\(d\)k=320k=320

Figure 18:SAE Probes, layer 20, 32K dictionary\.Mean test accuracy againstkprobek\_\{\\text\{probe\}\}, with the same methods and layout as Figure[17](https://arxiv.org/html/2610.27375#A3.F17)\. In \(a\), the Matryoshka SAE outperforms the MF\-SAEs\.
## Appendix DLinear projection ablation

We compare the MF\-SAE with and without the linear projectionsWeW\_\{e\}andWdW\_\{d\}\(Section[4](https://arxiv.org/html/2610.27375#S4)\) at layer 12, 16K dictionary,k=40k=40\. The linear encoder and decoder significantly improve probing accuracy on both the raw and the live space \(Figure[19](https://arxiv.org/html/2610.27375#A4.F19)\), consistent with Section[4](https://arxiv.org/html/2610.27375#S4): they are the only mechanism that lets the encoder of the MF\-SAE differ from its decoder\.

![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Bench/Gemma_Layer12/16K/gemma2_l12_16k_k40_proj_vs_noproj_saebench.png)Figure 19:Linear projection ablation \(SAEBench, Gemma\-2\-2B, layer 12, 16K dictionary,k=40k=40\)\.SB\-Acc againstkprobek\_\{\\text\{probe\}\}for the raw and live MF\-SAEs with and without the linear projections\. The projections significantly improve both\.
## Appendix EGPT\-2 results

We train MF\-SAEs on GPT\-2 Small \(d=768d=768\) at layers 7 and 8, matching the hook point, dictionary size and sparsity of an available open\-source SAE at each layer \(Table[3](https://arxiv.org/html/2610.27375#A1.T3)\)\. At layer 7 we train both the raw and the live MF\-SAE, with the Jacobian estimated as in Section[5\.1](https://arxiv.org/html/2610.27375#S5.SS1)andr=512r=512live directions; at layer 8 we train only the raw MF\-SAE\.

#### Layer 7\.

We compare with the residual\-stream SAE of[Bloom \[2024\]](https://arxiv.org/html/2610.27375#bib.bib26)atblocks\.7\.hook\_resid\_pre, a ReLU SAE trained with an L1 penalty on 300M tokens of OpenWebText, with24,57624\{,\}576latents and average L0≈62\\approx 62\. Our MF\-SAEs usen=24,576n=24\{,\}576andk=64k=64\.

#### Layer 8\.

We compare with the Top\-K SAE of[Gao et al\. \[2025\]](https://arxiv.org/html/2610.27375#bib.bib9)atblocks\.8\.hook\_resid\_post, with32,76832\{,\}768latents andk=32k=32\. Our MF\-SAE usesn=32,768n=32\{,\}768andk=32k=32\.

![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Bench/GPT2Small/gpt2small_l7_24k_matched.png)\(a\)SAEBench, layer 7
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Bench/GPT2Small/gpt2small_l8_32k_matched.png)\(b\)SAEBench, layer 8
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Probes/GPT2Small/gpt2small_l7_24k_matched.png)\(c\)SAE Probes, layer 7
![Refer to caption](https://arxiv.org/html/2610.27375v1/Figures/SAE_Probes/GPT2Small/gpt2small_l8_32k_matched.png)\(d\)SAE Probes, layer 8

Figure 20:GPT\-2 Small, matched baselines\.Probe accuracy againstkprobek\_\{\\text\{probe\}\}\. Rows are SAEBench and SAE Probes; columns are layer 7 \(raw and live MF\-SAEs,n=24,576n=24\{,\}576,k=64k=64, against the SAE of[Bloom \[2024\]](https://arxiv.org/html/2610.27375#bib.bib26)\) and layer 8 \(raw MF\-SAE,n=32,768n=32\{,\}768,k=32k=32, against the Top\-K SAE of[Gao et al\. \[2025\]](https://arxiv.org/html/2610.27375#bib.bib9)\)\.