On the Limits of Sparse Autoencoders: A Theoretical Framework and Reweighted Remedy
Authors: Jingyi Cui
Qi Zhang
Yifei Wang
Yisen Wang
Venue: ICLR 2026 Poster (2026)
Keywords: sparse autoencoder
SAE
theoretical understanding
Primary area: interpretability and explainable AI

Abstract: Sparse autoencoders (SAEs) have recently emerged as a powerful tool for interpreting the features learned by large language models (LLMs). By reconstructing features with sparsely activated networks, SAEs aim to recover complex superposed polysemantic features into interpretable monosemantic ones. Despite their wide applications, it remains unclear under what conditions SAEs can fully recover the ground truth monosemantic features from the superposed polysemantic ones. In this paper, we provide the first theoretical analysis with a closed-form solution for SAEs, revealing that they generally fail to fully recover the ground truth monosemantic features unless the ground truth features are extremely sparse. To improve the feature recovery of SAEs in general cases, we propose a reweighting strategy targeting at enhancing the reconstruction of the ground truth monosemantic features instead of the observed polysemantic ones. We further establish a theoretical weight selection principle for our proposed weighted SAE (WSAE). Experiments across multiple settings validate our theoretical findings and demonstrate that our WSAE significantly improves feature monosemanticity and interpretability.

Published as a conference paper at ICLR 2026
ON THE LIMITS OF SPARSE AUTOENCODERS: A THEO- RETICAL FRAMEWORK AND REWEIGHTED REMEDY
Jingyi Cui'- Qi Zhang'
Yifei Wang² Yisen Wang
State Key Lab of General AI, School of Intelligence Science and Technology, Peking University
2 Amazon AGI SF Lab IIntitttee for Artificial Intelligence, Peking University
ABSTRACT
Sparse autoencoders (SAEs) have recently emerged as powerful tool for inter- preting the features learned by large language models (LLMs). By reconstructing features with sparsely activated networks, SAEs aim to recover complex super- posed polysemantic features into interpretable monosemantic ones. Despite their wide applications, it remains unclear under what conditions SAEs can fully recover the ground truth monosemantic features from the superposed polysemantic ones. this paper, we provide the first theoretical analysis with a foosed-form solution for SAEs, revealing that they generally fail to fully recover the ground truth monose- mantic features unless the ground truth features are extremely sparse. To improve the feature recovery of SAEs in general cases, we propose a reweighting strategy targeting at enhancing the reconstruction of the ground truth monosemantic fea- tures instead of the observed polysemantic ones. We further establish a theoretical weight selection principle for our proposed weighted SAE (WSAE). Experiments across multiple settings validate theoretical findings and demonstrate that our WSAE significantly improves feature monosemanticity and interpretability.
INTRODUCTION
The "black- box" nature is a long-standing problem plaguing the mechanistic interpretability of deep neural networks. One of the key problems is feature polysemanticity, where an individual neuron is often activated by multiple semantically unrelated features (Elhage et al.) [022b] Hänni al., Scherlis et al., 2022. Zhang et al. 2023 2025). This issue is particularly evident in large language models (LLMs) as neurons rarely correspond distinct and well-defined features. Previous studies have proposed the superposition hypothesis that the polysemantic features are linear combinations of the monosemantic ones, so that models can represent more features than they have dimensions (Elhage et al.) 2022b; Hänni et al. Scherlis et al. 2022).
To disentangle the superposed polysemantic features, sparse autoencoders (SAEs) (Gao al. [025
Makhzani & Frey| 2013) Minegishi et al., (2025) Ng et al. 2011) have demonstrated significant
potential for identifying interpretable monosemantic features. They have been widely used as a
promising approach to enhancing the interpretability of LLMs (Cunningham et al. 2023 Ferrando ct al. 2025 Gao et al. 20255) Lieberum et al. 2024) and VLMs (Daujotas 2024 Lim et al. 2025
Lou et al. 2025; Thasarathan et al. 2025). Typically, an SAE has single wide layered encoder- decoder architecture, with sparse activations such as ReLU (Cunningham et al. 2023). ) JumpReLU (Rajamanoharan et al.] 2024b), TopK (Gao et al.) 2025), BatchTopk (Bussmann et al. 2024), etc.
While previous works mainly focused on studying architecture (Braun et al. 2024 Makhzani & Frey
2013 Rajamanoharan et al. 2024a Tibshirani] 1996, [Bussmann et al. (025) evaluation (Minegishi et al.
2025; Karvonen et al. (2025) of SAES, the theoretical understanding of the identifiability of
SAEs still lacking. Specifically, we wonder: Can SAEs recover the ground truth monosemantic
features from the polysemantic inputs?
To answer the above question, in this paper, we investigate the theoretical conditions of SAE feature recovery. First of all, under the superposition hypothesis, we propose theoretical framework and
"Equal contribution
soorespnn author: Yisen Wang (yisen. @ pku.edu.cn)
1 This work was completed MIT prior Yifei Wang joining Amazon.

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Published as a conference paper at ICLR 2026
derive the closed-form optimal solution to SAEs. Nonetheless, we show that full recovery of the ground truth monosemantic features is not theoretically guaranteed under general conditions, with standard SAEs plagued by feature shrinking and feature vanishing. As possible explanation why SAEs do work well in some empirical cases, we further show that the sparsity of the ground truth features might be the key. Specifically, if the ground truth monosemantic features extremely sparse, the optimal solution to an SAE is unique and precisely recovers the ground truth features. Moreover, considering that the sparsity of the ground truth features is not controllable through training, when the extreme sparsity assumption is not met (SAE feature not fully recoverable), we propose reweighting strategy enhance the reconstruction of the ground truth features. We also propose principle for the weight selection through theoretical analysis. Specifically, we derive the theoretical relationship between the loss of SAE feature reconstruction and that of the ground truth feature reconstruction, and discuss that smaller weights on the more polysemantic dimensions help reduce the negative interferences hindering the reconstruction of ground truth features.
Our contributions are summarized as follows.
We propose theoretical framework for analyzing SAE feature recovery based on the superposition hypothesis and derive a closed-form solution to SAEs. Based on this, we show that SAEs fails to recover the ground truth monosemantic features in general unless the ground truth features are extremely sparse.
In low sparsity conditions where SAE full recovery does not meet, we propose a reweighting strategy improve the reconstruction of ground truth features by SAEs and theoretically discuss the principle of weight selection.
We validate theoretical findings through experiments and show that our reweighting strategy significantly enhances the monosemanticity and interpretability of SAE features.
2 RELATED WORKS
2.1 POLYSEMANTICITY AND SUPERPOSITION HYPOTHESIS.
A widely accepted explanation for feature polysemanticity is the superposition hypothesis (Elhage ctt al., 2022b) Hänni et al.) Scherlis et al. 2022). which regards a polysemantic dimension to be an approximately linear combination of several natural semantic concepts. To investigate the mechanism of polysemanticity. mainstream studies reveal that superposition occurs when a model represent more features than they have dimensions. Specifically, Elhage et al. (20226) introduced a single-layer toy model demonstrating that polysemanticity occurs when ReLU networks are trained on synthetic data with sparse input features. Hänni et al.) proved that superposition actively helpful for efficiently accomplishing the task of emulating circuits via mathematical models of computation. Scherlis et al.) (2022) explained polysemanticity through the lens of feature capacity, suggesting that the optimal capacity allocation tends to polysemantically represent less important features. Aside from the dimension restriction, Lecomte et al. (2024) demonstrated that polysemanticity can also incidentally restriction,LLcoomte occur because of sparse regularization or neural noise.
In this paper, we adopt the superposition hypothesis to simulate the generation of polysemantic features from the ground truth monosemantic inputs.
2.2 SPARSE AUTOENCODERS (SAEs).
Sparse autoencoder (SAEs) automatically learn features from unlabeled data, typically trained with sparsity priors such as ReLU activation and (-1 penalty (Ng et al. (011).. To mitigate the feature suppression caused by 1-1 regularization (Tibshirani] Wright & Sharkey 2024). Makhzani & Frey (2013) proposed k-sparse autoencoders, which replace the 7-1 penalty with Top-k activation function. To address the same problem, Rajamanoharan et al. (2024a) proposed the gated SAE that decouples detection of which features are present from estimating their magnitudes. Besides, to ensure the features learned are functionally important, Braun et al. (2024) propose to optimize the downstream KL divergence instead of reconstruction MSE. Moreover, by using k-sparse autoencoders, Gao et al. (2025) found clean scaling laws with respect to autoencoder size and sparsity. MMnegiishi
al. 20255 proposed suite of evaluations for SAFS analyze the quality of monosemantic features
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Feature Superposition
Sparse Autoencoder (SAE)
SAAK
Input
Wp
W
W
Alm to recover
Figure I: Theoretical framework for sparse autoencoder (SAE) feature recovery. The superposed polysemantic features Zp. composed of ground truth monosemantic features with matrix W p. serve as the input to the SAE. For the SAE, Wm denotes the weight matrix, denotes the sparse activation function, and LSAE denotes the reconstruction loss of Tp- Ideally, we expect the SAE output Tm to fully recover the ground truth monosemantic features through reconstruction of Tp.
by focusing on polysemous words. They showed that compared with alternative activations such as JumpRel.U and Top-k, ReI.U still leads to competitive semantic quality.
In this paper, we investigate the identifiability of SAEs. Specifically, we are interested in the conditions under which SÄEs can uniquely recover the ground truth monosemantic features from the polysemantic inputs. We consider the reconstruution-baase SAEs with ReLU or Top-k activations.
PRELIMINARIES & MATHEMATICAL FORMULATIONS
(#1...., TT) E R" the ground truth monosemantic feature with dimension n > O, and Tp E Rn* as the superposed polysemantic feature, where we assume Up > O. Moreover, where Tym > For the weight matrices, we denote W, E "R" and Wm. E Im x*p, Throughout denotes the la norm if not otherwise specified. For mathematical conciseness, denote 1 as the all-one vector, and 0 as the all-zero vector. Also, we denote [n]] {l...., n). Furthermore, for matrix W we use WR..) ..]] to denote the i-th row of W, and use Wiai] to denote the j-th column of W. For vector 2, we use to denote the i-th element of z.
Notations. Denote
F' R to denote the feature learned by sparse autoencoder,
this paper,
Sparsity of ground truth features. For the ground truth monosemantic features r (21....,, we assume the x;'3 's are identically and independently distributed, where given the sparse factor [0, 1). we have æ; >> 0 with probability (1 S) and Jı with probability S, for E [n]. The sparsity assumption follows from Elhage et al. (2022b) and has been observed through empirical evidence (Olah et al.) 2020 Elhage et al. 2022a) but ours is weaker since we do not require the uniform distribution assumption of Elhaae et al. 2022b) when J;; 0.
Superposition of features. Superposition has been widely hypothesized in previous works (Arora et al.] 2018 Olah et al.] 2020; Elhage et al.) (2022b). Following the superposition hypothesis, we formulate the superposed polysemantic feature as a linear transformation of the ground truth monosemantic input feature Specifically, assume the polysemantic features to be
Ip 2
(1)
given the ground truth monosemantic features with some weight matrix W Moreover, as discussed in Elhage et al. (2022b), the superposed dimensions usually have negative interferences, i..., W P. Pll < 0 for j. and form geometric sturctures of digons polygons.
Sparse autoencoders (SAEs). We sparse autoencoder recover the ground truth monosemantic feature x from the polysemantic input Tp- SAE is a neural network with a single hidden layer and a
sparse activation function I Rm T Rm such as ReLU (Cunningham et al.) 2023). JumpReLU
(Rajamanobaran et al. 2024b), Top-k (Gao et al. (025), etc. For mathematical simplicity, we omit the bias term and define the encoder and decoder
Jmm a (W mppp))
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I Wm
(3)
It is trained with a reconstruction loss
CSAE (W m: rppp) I., ,, E,, E, W ( W,,)
As not all SAE architectures include the sparsity regularization, e.g. [ao et al. (002), here omit the 4 penalty and focus on the reconstruction loss only. Then under the optimal weight matrix
W arg min CSAB (Wm: Fp)
Wm.
the learned monosemantic feature is I'm a (W pp)))
Feature Recovery. Ideally, we expect the SAE-learned feature Z'm to exactly recover the ground truth i.c., Fm if m Nonetheless, as the ground truth dimension unknown, we believe additional zero-valued dimensions and dimension reordering are also acceptable. Therefore, this paper, say SAE fully recovers the ground truth monosemantic features by reconstructing polysemantic features Tp
I' Fz Im (z. O)T,
if T and nim > n2,
(6)
(7)
where we denote x' if after reordering of the feature indexes, i.e., there exists matrix Is formed by swapping rows of the identity matrix, such that x' = l*x. v. Or equivalently, from the perspective of the loss function, we say an SAE fully recovers the monosemantic features the solution W = arg minw... LSAE(Wmi Zp) recovers the ground truth monosemantic features to index reordering and zero padding.
The overall theoretical framework for SAE feature recovery is illustrated in Figure
4
THEORETICAL RESULTS
In this section, we discuss the provable conditions for SAE feature recovery. First of all, in Section 4.2 by deriving closed-form solution SAEs, we show that in general cases, an SAE may fail to fully recover the ground truth monosemantic features, plagued by problems such as feature shrinking and feature vanishing. Then in Section 4.1 we show that one possible reason why an SAE still works in some real cases is the sparsity of the ground truth monosemantic features. We show that under the extreme sparsity condition, the unique optimal solution to an SAE fully recovers the ground truth features. In order to overcome such limitations of SAEs, in Section 4.3 propose reweighted version of SAE to achieve a better reconstruction of the ground truth monosemantic features, and theoretically diseuss the weight selection principle. All proofs be found in Appendix A In addition, we present more discussions and extensions of our theoretical framework in Appendix
4.1 SAES FAIL TO RECOVER GROUND TRUTH FEATURES IN GENERAL
By the mathematical formulations introduced in Section we derive the closed-form solution SAEs in Theorem I Note that the assumption of Wp follows from the feature geometry of superposition observed in Elaage et al. (2022b), where the columns of W, within the superposed dimensions form digons or polygons.
Theorem 1 (Closed-Form Solution to SAES). Let LSAR be defined in equation 4 with sparse activation function a. If n' 2 n and the columns of W, within the superposed dimensions form digons/polygons, then we have W (", 0)* arg minw.,. LSAE(Wm (Wmi xp). where I" is formed by swapping row with row * of the identity matrix.
Theorem shows that given the superposition matrix W., its transpose T serves as the closed-fomm solution to SAES in the sense of zero- vero-padding and row-reordering. Specifically, the SAE recovered features are Z.m = a (. xp)) with zero- padding and index-reordering.
However, by such solution, discuss that the recovered features Z.m do not always recover the ground truth monosemantic feature z. Instead, they often deviate from the ground truth because the feature shrinking and feature vanishing phenomena, which illustrate in the following examples.
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Feature shrinking. The SAE-recoverd features from the polysemantic dimensions are shrunk to be smaller compared to their ground truth value. Typically, we say a superposed feature dimension is "more polysemantic" if it has interference with more ground truth monosemantic features. Then the more polysemantic a feature dimension is, the more severe its SAF-recovered monosemantic features shrunk in value. In Example [ the second and third dimensions of more polysemantic in the superposed feature Tp. they shrink severely after recovered by an SAE (from (1.0, 0.8) to (0.2, 0), whereas the first dimension, which is monosemantic, does shrink. This could lead incorrect interpretation of the feature, because the top-1 activated dimension of the ground truth r should be the second dimension (1.0), whereas when using an SAE, the top-1 activated dimension of zm turns out to be the first one (0.5). In other words, SAEs have tendency to better interpret the relatively monosemantic features, whereas overlook the relatively polysemantic ones.
Example 1 (Feature Shrinking). Suppose 7m
3. np = 2, (0.5, 1.0, and
0 0 -1
W =
Then we have Ip- (0.5, 0.2) and
[ 0
0
0.5
00.5
Em = oWW p Wnx) = of
-1 -1
1.0
=
0.2
(8)
0.8
0
Feature Vanishing. When feature shrinking is severe enough, some features in the polysemantic dimensions even vanish and can rarely recovered by an SAE. In Example [ the second and third dimensions of the SAE-recovered feature 2m vanish completely, making I' have even less effective dimensions than Fp-
Example 2 (Feature Vanishing). Suppose n'm = n = 3. np = 2, x = (0.7,0.5,0.3) and
0 v3/2 -V3/2
W
Then we have Tp = Wn = (0.1 (.0l33,..) 0.3) but
1/2
-1/2
-1/2
-1/27
00.7
T0.3
Im oW, Wpx) =
1 1/2
1
-1/2
0.5
0
(9)
/2
- 11/2
00.3
0
4.2 SAE FULL RECOVERY UNDER EXTREME SPARSITY
Despite the disappointing theoretical conclusion in Section 4.2 in this part, we propose one possible explanation for why SAEs do work well in some real-life cases, that is, the sparsity of the ground truth monosemantic features is the key to SAE feature recovery.
Theorem 2 (Optimality under extreme sparsity). Let LSAE be defined in equation 4 For nmm 2
and the columns of WV have non-positive interferences. if 1, then we have W = =7 (W, 0)
arg minw.,, CS pp), where Ix is formed by swapping row i with row j * i of the identity matrix. Accordingly, we have a" (W".pp) x for arbitrary x.
Theorem 2 shows that when the ground truth feature x is extremely sparse and the hidden dimension nm is large enough, WT is the optimal solution to SAFs in the sense of rero-padding and row- reordering. Note that in Theorem 2 the columns of W are only required to have pos-iise interferences, which follows from Elhage et al. (002bb and is much weaker than the digon/polygon geometry condition required in Theorem [| By such solution, under the extreme sparsity condition
+ 1, the SAE-recovered features can fully recover the ground truth monosemantic feature x. Intuitively, under extreme sparsity, becomes 1-sparse with probability nearly 1, and feature shrinking and feature vanishing do not happen to 1001-1 1-sparse features.
Moreover, in Theorem 3 show that when S + 1, the solution derived in Theorem is unique. Theorem 3 (Uniqueness). Let LSAE be defined in equation of W, have non-positive interferences, if S 1, then W IWT is the unique solution to arg minw.,, LSAE(W,. pp).
For nm = n and the columns
4.3 A REWEIGHTED REMEDY FOR SAE FEATURE RECOVERY UNDER GENERAL SPARSITY
As we show in previous sections, SAEs fail to recover ground truth monosemantic features under general sparsity conditions. In this section, by taking closer look, we show that this is largely
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because the SAE loss is not direct reconstruction of the ground truth monosemantic feature Instead, it reconstructs the polysemantic 2p = Wr because the ground truth x is unknown. In this case, the superposition matrix W, could mistakenly match the reconstructed polysemantic feature p W Jim with Tpp even if the reconstructed monosemantic features Zm do not match the ground truth z. Specifically, in this section, we first identify gap between the SAE loss and the reconstruction loss of ground truth æ, and then we discuss how a reweighting strategy narrows the gap and enhances the ground truth reconstruction.
4.3.1 THE GAP BETWEEN SAE RECONSTTUUCTION AND GROUND TRUTH RECONSTRUCTION
We first of all consider an ideal case, where we are allowed to directly reconstruct the ground truth monosemantic features z. Without loss of generality, we assume 1lm n, and define the ground truth reconstruction loss of æ as
LCT(Wm::)) = E2,[ xm|l²= Ez|l= - (W = EzIla - o ( wr
(10)
Then by comparing LSAE and LGT, we derive the theoretical gap between the two losses.
Theorem 4 (Gap between LSAE and LGT). Let LSAE and LGT be defined in equation and equation 10 respectively. Then when Wm = W.,. we have
CSAE mi xp)
(Gr(Wm:: =f - oW, W,W,Wa w1 (W, Wp - Inxn){ o(W Wp (11)
Theorem 4 shows that the gap between the SAE loss CSAE W Wmixp and the ground truth reconstruc- tion loss LGT(Wn) (Wmi z). Note that according to Theorem we consider the gap when the SAE loss reaches its optimal, i.e., W.. W,T. The gap term depends ground truth and the recovered feature Im = (W, war and the gap between W Wp and the identity matrix I nxn- On the one hand, the gap term goes to zero if the features are perfectly recovered by an SAE. For instance, by Theorem 2 when x is extremely sparse, we have x o(W, p W₁,) and therefore the gap vanishes. On the other hand, in general cases where SAES fail to perfectly recover the ground truth x, i.e., x # o ( p W, the gap term depends largely on W unfortunately cannot be reduced because for SAEs, W is given input rather than learned.
terms, i.e., the gap between
p Wp Inxas which
4.3.2 ADAPTIVELY REWEIGHTED SPARSE AUTOENCODERS (WSAES)
As one possible solution to narrow the gap shown in Theorem 4 we propose the re Weightted Sparse Autoersss (WSAEs) with adaptive weights according the polysemantic level of each dimension. Specifically, given weights %/ 0, E [Mp)) we define the WSAE loss as
LWSAE W mi xp) = Ex, || WT, ReLU(U
(12)
where r diag(1: (n.g.......... )
Theorem 5 (Gap between LWSAE and LCT). Let LWSAE and LoT be defined in equation 72 and
equation 10 respectively. Then when Wm = W.,. we have
(WsAA(Wm:agg - Lor(W (w [x - o (W W [I (W, (WT.TT,, rw, Inxa)(x - ( W w.
(13)
Theorem 5 shows the gap between reweighted SAE loss LSAR m x and the reconstruction loss of the ground truth features (WT, x.)) Compared with Theorem where / W, T Taxm is fixed, when features are not perfectly recovered, i.e., x * (W,, Wp2). the gap in Theorem 5 is adjustable w.r.t. the weight matrix I. That is, given properly chosen r, we can effectively narrow the gap term in equation] 13 and thus gain a better reconstruction of the ground truth features x. As special case, when the weight matrix I' is an identity matrix (uniform weights), the gap of WSAEs in Theorem S degenerates to that of SAEs shown in Theorem
Then we discuss how to select proper weights to narrow the gap term in equation 13 Specifically, we focus on the polysemantic level of each dimension. Observe that
7 WT p.l:11
W p.I.nl
W.T T TTW, Inxn
(14)
W
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Then according to equation 14 for the relatively monosemantic dimensions with almost zero off- diagonal interference terms, we should assign weights near to reduce 7/2 -1,,, whereas for the relatively polysemantic dimensions, diagonal negative interferences. In short, for WSAE reconstruction, we assign larger weights to the relatively monosemantic dimensions and smaller weights to the relatively polysemantie dimensions.
should assign smaller weights primarily reduce the off-
5 VALIDATION EXPERIMENTS
In this section, we conduct numerical experiments to validate our theoretical findings. In Section 5.1 we conduct validation experiments on synthetic data with known ground truth features. We validate that 1) SAEs in general fail to achieve full feature recovery unless the ground truth features extremely sparse (Sections 4.1 and 4.2 and 2) our reweighting strategy can improve ground truth reconstruction when the sparsity level is low (Section 4.3). In Section 5.2] we conduct real-data experiments both pretrained language and vision models to validate the effectiveness of our reweighted strategy in enhancing feature monosemanticity.
5.1 VALIDATION EXPERIMENTS THROUGH SYNTHETIC DATA
Data Generation. We follow the toy model settings in Elhage al. (202bb). Specifically, the polysemantic features are generated by equation] [ described in Section 3 where the superposition matrix W, is learned by reconstructing the ground truth æ with a ReLU output model. We set the ground truth monosemantic feature dimension
200 and input polysemantic feature dimension 7p = 20 to generate polysemantic embeddings and then attempt to extract monosemantic features with SAEs trained frozen polysemantic embeddings. When analyzing the influence of input sparsity, we set E [0. 1] the sparse factor, and let r; = with a probability S, and x; ~ 2<(0, 1] with probability 5, for E [n]]-
Setups. For both SAE and WSAE, we adopt ReLU activation function, and set the hidden dimension n.m
200 (10 times the input dimension). For the weight selection in WSAE, treat W as unknown, and use the per-dimensional variance of Tp as proxy for monosemaniicity. Specifically, let 8; denote the variance of up the i-th dimension. Then given tunable parameter > O, set the weights as = s." As we find that the experimental results are relatively robust against the choice of a, we set a 1 for the experiments.
Validation of SAE feature recovery. We validate the theoret- ical results in Sections 4.1 and [42 that SAEs general fail to achieve full feature recovery unless the ground truth features are extremely sparse. Specifically, under different sparsity lev- els S E [0, 1]. we measure the monosemanticity of SAE latents by the average number ground-truth features activated the same SAE dimension (with more details shown in Appendix If the SAE successfully recovers monosemantic features, each latent should be activated by only small number of input features. As shown in Figure we observe clear decrease in the average number activated features (indicating a signifi- cant improvement in the monosemanticity) with increased input sparsity. This empirical result aligns well with our theoretical findings that SAEs fully recover ground truth monosemantic features only when the sparsity level is extreme.
Average EverageActionn Activated Features
0.6
Sparse Factor
Figure 2: Monosemanticity (mea- sured by the average activated fea- tures) of SAE features increases with increasing sparsity of ground truth monosemantic features.
Validation of WSAE ground truth reconstruction. We validate the theoretical discussions in Section 4.3 that WSAE features have better ground truth reconstruction than SAE features when the sparsity level of ground truth features is low. Specifically, we respectively measure the reconstruction loss of ground truth monosemantic features LGT defined equation I0 the reconstruction loss of polysemantic features LSAE and LWSAE defined in equation 4 and equation [2] and the monoseman- ticity level measured by per-dimensional variance (where larger per-dimensional variance indicates higher monosemanticity). respectively. In Figure B when the sparsity level of the ground truth features is relatively low, we show that the WSAA-recoveree features have better reconstruction the ground truth monosemantic features compared with SAF-recovered features, whereas maintaining
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5A2 W3.6
SAE
0.00
SAC
0.30
04 10 00 0. 06
4.
3.8
34
(a)
(b)
(c)
(d)
Figure 3: Validation experiments of WSAE ground truth reconstruction on synthetic data, (a) Ground truth reconstruction error LGT, where WSAE has lower error compared with SAE when the sparsity level S is low. (b) Reconstruction error on the non non-sparse dimensions of the ground truth monoseman- tic features, showing a greater error gap between WSAE and SAE. (c) The reconstruction error of the polysemantic features Tp, where the errors of the two methods are comparable. (d) Monosemanticity measured by per-dimensional variance, where WSAE features are more monosemantic compared with SAE features when the sparsity level is low.
a comparable reconstruction of the polysemantic features Tp. Specifically, WSAE has a lower ground truth reconstruction error compared to SAE (Figure 3(a)), J. and if evaluate only the non-sparse dimensions of the ground truth x. the advantage of WSAE over SAE under low sparsity becomes more significant (Figure bbb... These empirical findings well verify the theoretical discussions in Section 4.3 that proper weight selection can narrow the gap between WSAE reconstruction error and ground truth reconstruction error. In addition, FigureB(c) we show that the SAE and WSAE reconstruction errors of the polysemantic features Tp comparable, indicating that WSAEs do not lie far away from the pareto-frrotier of sparsity and reconstruction. In Figure (3d) we show that when sparsity is relatively low, WSAE features have better monosemanticity than SAE features, indicating better recovery of the ground truth monosemantic features.
5.2 EMPIRICAL EVALUATION OF REWEIGHTED SAES THROUGH REAL DATA
In this section, evaluate the effectiveness of proposed strategies on real-world datasets. We evaluate on both pretrained language models and vision models.
5.2.1 EXPERIMENTS ON LANGUAGE MODELS
Setups. We use Pythia- 160M (Biderman et al. 2023) as the frozen language model backbone and train sparse autoencoders (SAEs) on its internal activations. The latent dimension of the SAE is set to 32 times the input dimension, and we employ Top-k activation (A: = 32) in the latent layer of SAEs. Guided by the theoretical analysis, compare two training objectives: standard reconstruction loss with uniform weights, and a weighted reconstruction loss that emphasizes monosemantic features.
In real-world settings, cannot directly access the ground-truth monosemanticity of features. Nonetheless, several surrogate metrics have been proposed to evaluate monosemanticityy For instance, Wang et al. (0244) uses variance within each feature dimension (2024) introduces the auto-interpretabiliiy score, which leverages large language models to summarize whether the features activated the same dimension are semantically similar. In our experiments, we use the variance as the metric when assigning weights, due to its computational simplicity. As discussed in Wang et (2024a), monosemantic neurons have high deviation of activation value and low frequency of activation, so higher variance indicates stronger monosemanticity. Specifically, we let s; denote the variance of activations in the SAE latent dimension i. We set the weights as % sf for E [Mp]. where > 0 is a tunable parameter. When increases, the reconstruction loss assigns more weights to the reconstruction of monosemantic features.
surrogate while Paulo al.
On the other hand, for the evaluation of SAE latents, we use the auto-interpretabilit score (Paulo ct al. 2024) as it is more accurate to assess monosemanticity. To compute the score for given SAE latent dimension, we first identify the top-activated samples for that dimension. A large language model (e.g., Llama3.1-82 Touvron et al. (2023) is then used to generate a natural language summary describing the shared characteristics of these samples. In the next step, another language model is prompted with the generated summary to predict whether additional samples in the dataset would activate the given SAE dimension. The prediction accuracy serves as the auto-interpretability
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Table Aut--interpretabiiity scores (%) of SABs trained following different layers (0-11) of Pythia- 160M with original SAE and weighted SAE loss. SAEs trained with weighted SAE loss obtain higher auto-interpretabiiiy scores (i.e., stronger monosemanticity) across different situations.
2
3
4
5
6
7
8
9
10
11
Original SAE Weighted SAE (a=0.5)
74.7
74.1
76.7
77.8
78.5
79.5
79.3
77.8
74.6
75.6
71.6
72.5
75.4
77.6
76.4
77.9
79.3
79.6
79.8
77.4
78.6
79.3
76.1
72.6
Gains
+0.7
+3.5
-0.
+0.1
+0.8
+0.1
+0.5
-0.4
+4.0
43.7
+4.5
+0.1
Weighted SAE (a=1)
77.2
78.9
81.3
84.6
83.9
83.3
83.9
79.6
81.5
77.6
72.4
73.5
Gains
+2.5
+4.8
+4.6
+6.8
+5.4
+3.8
+4.6
+1.8
+6.9
+2.0
+0.8
+1.0
score: higher accuracy indicates that the activated samples in the dimension are semantically similar, reflecting stronger monosemanticity. More details can be found in Appendix
Results. As shown in Table ! assigning greater weight to the reconstruction of monosemantic features leads to significant improvement in the monosemanticity of SAE latents, yielding average 3.8% gain in auto-interpretability score when a = 1. The improvements are consistent across SAEs trained on different layers of backbone models. We also note that the improvement is more significant when the baseline SAE already exhibits relatively strong monosemanticity, These empirical findings further support our theoretical insight in Section 4.3 that reweighting SAE with monosemanticity level can enhance the recovery of the monosemantic features. As supplement, additional experiments with an alternative Top-k realization, evaluations the Llama model, and sensitivity analysis are shown in Appendix
5.2.2 EXPERIMENTS ON VISION MODELS
In addition to language models, we also evaluate our strat- egy on SAEs trained following pretrained vision back- bones. As noted by Wang al. (2024b), features learned by current vision pertaining paradigıss, such as contrastive learning, often highly polysemantic, making difficult to select monosemantic dimensions. Consequently, we adopt Non-negative Contrastive Learning (NCL) (Wang) et al. 2024b) to pretrain the backbone, as it can simulta- neously obtain monosemantic and polysemantic features.
43.0
Semantic Consistency
42.5
42.2
42.0
41.5
41.0
40.7
40.5
40.0
Original SAE Weighted SAE Weighted SAE
(a 0..)
Specifically, we pretrain a ResNet-18 with NCL on ImageNet-100, and subsequently train SAEs on the learned representations. During the training process of SAEs, we set the latent dimension to 16384 and use the topK sparse autoencoder with & 16. When evaluating monosemanticity, we follow Wang et al. (2024b) and use semantic consistency the metric, which calculates the proportion of top-activated samples that belong to their most frequent class along each dimension. As the higher semantic consistency indicates stronger monosemanticity, uniform weights and more weights on the dimension with higher semantic consistency. Specifically, denote 3 as the semantic consistency of the i-th dimension, and set the weights ni = 3 for i E [ppp]: where a > O a tunable parameter. In Figure we observe that assigning greater weight to the reconstruction monosemantic features leads a notable improvement semantic consistency (monosemanticitt) of SAE latents. These results further validate both our theoretical analysis and the effectiveness of the proposed weighting strategy.
Figure 4: Semantic consistency (%) of SAEs trained on the embeddings of ResNet-18 with original SAE and weighted SAE loss.
respectively train SAEs with
CONCLUSION
In this paper, we establish a theoretical framework to reveal the inherent limits of SAEs to recover ground-truth monosemantic features. Specifically, based closed-form theoretical solution, show that standard SAEs inevitably suffer from feature shrinking and vanishing, preventing full recovery except when the true features extremely sparse. While many interpretability studies

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assume that increasing sparsity width of SAEs improves feature separation indefinitely, this work shows that such improvement plateaus due to intrinsic representational interference, meaning full disentanglement is mathematically impossible under realistic sparsity. Consequently, SAE-based interpretability should be regarded as an approximation tool, not as a faithful feature recovery mechanism. This reframes SAE-derived neurons approximate projections of overlapping features, rather than direct encodings of ground-truth concepts.
Furthermore, to enhance feature recovery in the low sparsity situations, we introduced a simple yet effective strategy called re reWeighted Sparse Autoencoder (WSAE) and propose a theoretical weight-selection rule. Validation experiments on both synthetic and real data confirm our theory and demonstrate that WSAEs can enhance monosemanticity and interpretability of features learned. We note that the proposed WSAE serves as just one exemplar remedy, and our theoretical framework has the potential to motivate further methodological advances (e.g. alternative matrix designs to close the loss gap) aimed overcoming the fundamental feature-recovery limits of SAEs.
ACKNOWLEDGEMENT
Yisen Wang was supported by National Natural Science Foundation China (92370129, 62376010), Beijing Major Science and Technology Project under Contract no. Z25 100008425006, Beijing Nova Program (20230484344, 20240484642), and State Key Laboratory of General Artificial Intelligence.
ETHICS STATEMENT
This work makes use of publicly available datasets and models. No private or sensitive data are involved, and no harmful content is included. Therefore, we believe this paper does not raise any ethical concerns.
REFERENCES
Sanjeev Arora, Yuanzhi Li, Yingyu Liang, Tengyu Ma, and Andrej Risteski. Linear algebraic structure of word senses, with applications to polysemy. Transactions of the Association for Computational Linguistics, 6:483-495, 2018.
Stella Biderman, Hailey Schoelkopf, Quentin Gregory Anthony, Herbie Bradley, Kyle O'Brien, Eric Hallahan, Mohammad Aflah Khan, Shivanshu Purohit, USVSN Sai Prashanth, Edward Raff, et al. Pythia: A suite for analyzing large language models across training and scaling. In ICML, 2023.
Dan Braun, Jordan Taylor, Nicholas Goldowsky-DD -Dill, and Lee Sharkey. Identifying functionally important features with end-to-end sparse dictionary learning. In NeurlPS, 2024.
Bart Bussmann, Patrick Leask, and Neel Nanda. Batchtopk: A simple improvement for topksaes. In Al Alignment Forum, 2024.
Bart Bussmann, Noa Nabeshima, Adam Karvonen, and Neel Nanda. Learning multi-level features with matryoshka sparse autoencoders. arXiv preprint arXiv:2503.17547 2025.
Hoagy Cunningham, Aidan Ewart, Logan Riggs, Robert Huben, and Lee Sharkey. Sparse autoen- coders find highly interpretable features in language models. arXiv preprint urXiv:2309.08600 2023.
Gytis Daujotas. Interpreting and steering features in images, 2024. URL https://www. lesswrong. -nddr--end--ddnnddee fentrrinnnfeaturr---iiiinne 2024.
Nelson Elhage, Tristan Hume, Catherine Olsson, Neel Nanda, Tom Henighan, Scott Johnston, Sheer ElShowk, Nicholas Joseph, Nova DasSarma, Ben Mann, et al. Softmax linear units. Transformer Circuits Thread, 2022a.
Nelson Elhage, Tristan Hume, Catherine Olsson, Nicholas Schiefer, Tom Henighan, Shauna Kravec, Zac Hatfield-Dodds, Robert Lasenby, Dawn Drain, Carol Chen, et al. Toy models of superposition. arXiv preprint 2209 200.:2662..00 2022b.
10

---

Published as a conference paper at ICLR 2026
Javier Ferrando, Oscar Obeso, Senthooran Rajamanoharan, and Neel Nanda. Do i know this entity? knowledge awareness and hallucinations in language models. In ICLR, 2025.
Leo Gao, Tom Dupré Tour, Henk Tillman, Gabriel Goh, Rajan Troll, Alec Radford, llya Sutskever, Jan Leike, and Jeffrey Wu. Scaling and evaluating sparse autoencoders. In ICLR, 2025.
Kaarel Hänni, Jake Mendel, Dmitry Vaintrob, and Lawrence Chan. Mathematical models of compu- tation in superposition. In ICML 2024 Workshop on Mechanistic Interpretability
Adam Karvonen, Can Rager, Johnny Lin, Curt Tigges, Joseph Bloom, David Chanin, Yeu-Tong Lau, Eoin Farrell, Callum McDougall, Kola Ayonrinde, al. Saebench: A comprehensive benchmark for sparse autoencoders in language model interpretability. ICML, 2025.
Victor Lecomte, Kushal Thaman, Rylan Schaeffer, Naomi Bashkansky, Trevor Chow, and Sanmi Koyejo. What causes polysemanticity? an alternative origin story of mixed selectivity from incidental causes. In ICLR 2024 Workshop on Representational Aligument, 2024.
Tom Lieberum, Senthooran Rajamanoharan, Arthur Conmy, Lewis Smith, Nicolas Sonnerat, Vikrant Varma, János Kramár, Anca Dragan, Rohin Shah, and Neel Nanda. Gemma scope: Open sparse autoencoders everywhere all at once on gemma 2. In Proceedings of the 7th BlackboxNLP Workshop: Analyzing and Interpreting Neural Networks for NLP, 2024.
Hyesu Lim, Jinho Choi, Jaegul Choo, and Steffen Schneider. Sparse autoencoders reveal selective remapping of visual concepts during adaptation. In ICLR, 2025.
Hantao Lou, Changye Li, Jiaming Ji, and Yaodong Yang. Sae-v: Interpreting multimodal models for enhanced alignment. ICML, 2025.
Alireza Makhzani and Brendan Frey. K-sparse autoencoders. arXiv preprint arXiv:1312.5663. 2013.
Gouki Minegishi, Hiroki Furuta, Yusuke Iwasawa, and Yutaka Matsuo. Rethinking evaluation of sparse autoencoders through the representation of polysemous words. In ICLR, 2025.
Andrew Ng al. Sparse autoencoder. CS294A Lecture notes, 72(201 1):1-19, 2011.
Chris Olah, Nick Cammarata, Ludwig Schubert, Gabriel Goh, Michael Petrov, and Shan Carter. Zoom in: An introduction to circuits. Distill, 5(3):c00024-0000 2020.
Gonçalo Paulo, Alex Mallen, Caden Juang, and Nora Belrose. Automatically interpreting millions of features in large language models. arXiv preprint arXiv:2410.13922 2024.
Senthooran Rajamanoharan, Arthur Commy, Lewis Smith, Tom Lieberum, Vikrant Varma, János Kramár, Rohin Shah, and Neel Nanda. Improving dictionary learning with gated sparse autoen- coders. arXiv preprint arXiv:2404.100 16014, 2024a.
Senthooran Rajamanoharan, Tom Lieberum, Nicolas Sonnerat, Arthur Conmy, Vikrant Varma, János Kramár, and Neel Nanda. Jumping ahead: Improving reconstruction fidelity with jumprelu sparse autoencoders. arXiv preprint arXiv:2407. 02044074.45,, /4435, 2024b.
Adam Scherlis, Kshitij Sachan, Adam Jermyn, Joe Benton, and Buck Shlegeris. Polysemanticity and capacity in neural networks. arXiv preprint arXiv:2210.01892 2022.
Harrish Thasarathan, Julian Forsyth, Thomas Fel, Matthew Kowal, and Konstantinos Derpanis. Universal sparse autoencoders: Interpretable cross-model concept alignment. ICML, 2025.
Robert Tibshirani. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology, 58( 1)226777-8 1996.
Hugo Touvron, Thibaut Lavril, Gautier Izacard, Xavier Martinet, Marie-Amn Lachaux, Timothée Lacroix, Baptiste Rozière, Naman Goyal, Eric Hambro, Faisal Azhar, et al. Llama: Open and efficient foundation language models. arXiv preprint arXiv:2302.13977, 2023,
Jiachuan Wang, Shimin Di, Lei Chen, and Charles Wang Wai Ng. Learning from emergence: A study on proactively inhibiting the monosemantic neurons of artificial neural networks. In SIGKDD, 2024a.
11

---

Published as a conference paper at ICLR 2026
Yifei Wang, Qi Zhang, Yaoyu Guo, and Yisen Wang. Non-negative contrastive learning. In ICLR, 2024b.
Benjamin Wright and Lee Sharkey. Addressing feature suppression in saes. In Al Alignment Forum, volume 6, 2024.
Qi Zhang, Yifei Wang, and Yisen Wang. Identifiable contrastive learning with automatic feature importance discovery. In NeurlPS, 2023.
Qi Zhang, Yifei Wang. Jingyi Cui, Xiang Pan, Qi Lei, Stefanie Jegelka, and Yisen Wang. Beyond interpretability: The gains feature monosemanticity on model robustness, In ICLR, 2025.
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A PROOFS
Without loss of generality. we consider the ReLU sparse activation function for proofs, i.e. (xx) =
ReLU(x) x1 [ > 0]. Note that for other activation functions such TopK or Jump-ReLU, the theoretical results are similar because they are based on the ReLU function form. Specifically, we have Lump-ReLUg c) ReLU(< c) and To)-k(x)) ReLU(x 2(K+
I I(k+1) 1), [ 2 denotes the Heaviside function, -((--)) denotes the (A: + 1) largest
where (x) element of x, and denotes the all-one vector. The sparse activation functions have a general form
(x) ReLU( ( + and the general gradient form @o((=) H(r c).
Also note that when 1 is formed by swapping solution to the SAE loss minimization, then I WTT I' p
with row of the identity matrix if W
is also a solution because
CSAE(["W :kpp W p.= ( W W,z))l W. - T (W W
-E,lWpT WW W,')
-SAE :
(15)
where the second last equation holds because (I*) T = I. In the following, we will not repetitively
prove this point the proofs of Theorem 2 and 3
A.1
PROOFS RELATED TO SECTION
Proof of Theorem 7 By definition, we have
CSAB(Wm:
Zp)
= Ex
Σ
[W p.
W (W W
iE/np]
Er
Σ
Σ kEim]
WT m a Σ
Wm. W [.).)))))
(16)
iE(mg]
jE[ny]
For u [n],, V [mpll we have
0LSAE
Ex {
p.[..... WT o (W WWW Wpw))o[Wmm.lu.. Wnz)
OW m./u.u]
Σ
[W W.A4F WT W...gg (W. W,>)) mm, H (W., ( mm.jaa (17)
sE/mp]
where H() denotes the Heaviside function. Observe that for non-zero F, aw m.m.....
BCEAR if and only
Ex (i.j
W m. W =
(18)
When W.. = WT, because the columns of W. form digons/polygons, we have W. W, being block diagonal matrix, with block B₁ Imxm for the (0 I m S np) monosemantic dimensions and blocks
-1/(pj - 1)
- 11/(p 1)
B, -1/(p; 1) - 11/(p, -
(19)
1
for the polysemantic dimensions, where P5 is the number of ground truth monosemantic features entangled in the j-th block. Then we have
Imxm
B2
0
σ a(Bg9(m+
oWW (WWWWpt wyz) =
=
(20)
0
B
o (Bbit (Bo07(npottimn))
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Denote P, + Eierg Pt and note that
BBo(EE,-11 EIO w-ー P 21)
15-1 -1 E - 2i)
(B ++1:3))=
(21)
For E [py]. we have
2P
Σ
11
Pj -
PJ- P
1
Σ
1
a) P3 1tisis
Σ
=
PP
21 PPJ-iti
Pj
Pj Pj
P PP(M, iti
1
Σ
Fi
Pj
P
PP Because is's are i.i.d., we have 1 = 14p p = Pí- Denote l = Ey
(22)
1)T (1.....1)
as the all-one vector, and 0 = (0....,0) 0) as the all-zero vector. Then we have
plm m U2
plm U20pp
ul
E, W (B, (B, - W
0 m
Lhh pp
(23)
where the second last equation holds because the columns of W, form digons/polygons.
Therefore, when W.. WT,, have equation 18 holds because
k1m M 0n
1m
p1m m
WFE W W W,x) 1,W,
= 0,
(24)
On-m
0 On-m
ie., W. argmin.... CSAB(Wm:s mi D.))
A.2 PROOFS RELATED TO SECTION
Proof of Theorem
When Wm = (W,, T) we have
W LSAE W mi xp) BrIlW,r (W 0)
=B,lWWrr WT OW, W 11
(25)
When S 1, we have
CSAF. pp)
S"h L(W mm Wp. = 0) Σ
Er, c SS) Wp x xnGa) + o(1 - S)
==1
0 nn (1 S) Σ Er, L(Wm: Wp. x Iyea) + o(1 S)
= =nsa -1--5 Σ
Ex, L(W mi x= x.ew)
u=1
= n mn (1 5) Σ E,, I m./aw) - WT (W, W Wo.).:
(26)
u=1
When the superposed dimensions have negative interferences as discussed in Elhage et al. (022bb).
ie., W W W [..) I 0 for + j, we have LSAE (W m x.p) = 0 = minw., LSAE W mi Tp).
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Also note that for TopK JumpReLU activations, feature recovery under potential positive in- terference is also possible. For example, for top-k activation, when 1 < S R'm p, where p [n) Kj [n]: p.l.. W p.(-j) 0} we have Top-k(W, W W.sssa)) Eu. and correspond- ingly LSAE(W (SAee mp) = 0 minw,, (sAe(Wmiap))
Proof of Theorem Then by definition, we have
CSAB (w mi xp)
= - pI- - W m (W W 2
= =E,[Wpr - W.T. o oW m W p2)] [W W WWW
= -E,x Wp Wpx 2x T W W W oW wa)
+ +o Wn 2²) W,, W m ( W,
ap am
Bp
Eg T WT W p
W
p.laaa Wm Wmilioo
W Wm.l W p.(..))))
p
W m. W W. m.jj
W Wm.le.
W Wm,. W p.l...)).
(27)
S 1, by equation 26 have
CsAe(Wmizp)) n1 S)
Σ
Er, Co Wp zu),
(28)
==1
where for [n] and vu > O, by equation 27
we denote
Lu Cu(W mi W F.)
Up Th
W [:( W Wo
2x
W Wa Wm.Li. o
W (i W p.tt))
Thm
By
+
x
W [m W p.(t.n)) W Wm, [iiol
W [m.
[m.t W W..lt.)).
(29)
j=1 q=1
Then for is E [nmm] and 7 [mp]. by equation 26 we have
WLn (Wmi mi W pP:
aW mWm.le.d
Rp
--2n W..tng
Wm.,ke [R.4 W W..li.)))
=
up
Up
2x 1
W p.lq.u W Wm. o H
Wm, W "e.lt.))
Wp.t.a) p.l.a]
1=
W Wm.kk. W p. W. W m.
W Wm.le.g
W Wm.ls.1 Wp W..le.))
Up
up
+ r
Wm. Wm.[k.)) [k.1]
W p((r))
W Wm.
W Wm.[s.)) W Wp.(e.u))
W m..i W (.
W Wm. W W..te.n))
ng
Up
W Wi W W mi W Wm, mm.kk.l H
W Wm. W (..lt.i) p.l.aal pl.t.al
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U1p
2x
W m. (k.i) 1) W p.li,u) W W p.lt.a] W Wm..kkel
q=1
i=1
W [a
W M.. W p.leea))
W Wo.le..u)
+ 2x 0
W I W p.ll,
Σ W Ia.go
W Wm. We.t,,) W P'..B.u)
(30)
Observe that
(WW:
= 0 for k E [rem] and / E [np] holds if and only if
OW m...e
nl
Σ
W Wm
Wm, [m, W Wnlton) = W.ls
(31)
for l E [mp]. or equivalently
W p/ba0]] W m a (W,,, m W W.lls.u)
(32)
W = WT a (W.. Wp)
(33)
When Am and the superposed dimensions have negative interferences discussed in Elhage et al. 2022b), i.e., W p.l... W p1111 S 0 for i + j, for arbitrary W, E IR"x",, the unique solution is W W (see the following for details).
For special example, if W, = 12x2, equation 33 becomes = W WlTooWWmm))) and we observe that Wm = Wp = I is a solution. Then Wm + 12x2, we can have a decomposition that
Wm = I E E2 D,
(34)
where E1 and Es are off-diagonal with non- non-negative elements on mutually exclusive positions, and D is a diagonal matrix. Then equation 33 becomes
I =1 (T+ E1 Eg + T) (I E1 Es + D)
=(T + F1 - Fg + D) T a(T + E1 + D)
= =T ET F. D + E + E Fi Fgg F + D D
+ +
EgJ D + DJ D
= II+ (2D D E F (E)
Fn + E1).
(35)
where the last equation holds because E.J E DJ E = ESJ D 0. Note that 2D + D2 + F, E
is diagonal and ET ET + E, is off diagonal, so there has to hold that 2D + 2 + E, E =
0 E with E1, 2 Then there has to hold Eg = E1 + +
and Eg = E1 + ET. Let E = E2
0
E1
52 0
which contradicts with that Eg and Eg have non-negative elements on mutually
E1
exclusive positions if or 82 0. Therefore, it has be E, E2 0, and accordingly D 0. That is, we prove that Wm, / is the unique solution to I = W W (W.)) and that W... WT is the unique solution to equation 33 for all Wp.
A.3 PROOFS RELATED TO SECTION
4.3
Proof of Theorem
By definition, we have
SSAE(Wmm) wimpp)
EzIlW,, ,, W WT W
EE,lWW, (W (W + (W, Wo W,r)
= =EollWafr- - (WW, w,,) )1 + Il - w,)) Ww
(W W,,)) (W W2 WWr oW (.,)),)) W,
(36)
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The first term of equation
36 can be further decomposed into
[ oW W 2
= [ a(Wm W px))
W, W plæ - oWWm W Wbx)]
[a: oW W p.)z)
[ o oW.. w,
+ [x oWWm m W w,,)
(W, p W Inxa) @W War
= | σ (Wm W )1
+ [ oW Wrr)] (W, W Iuxn](r a o HiWWwapt ppa)]
(37)
Combining equation 36
equation
37 and that Wm WT finishes the proof.
Proof of Theorem 5
By definition, we have
LWSAE Wm: wimppp))
= E, Wr nt - W W m W.W.WWp ?
= E, o (W W r) + r(w, W W (W W.a))
| E,,, o (w,, IIVV
W,,) oW W,,)ll
+ 2W w2x) (Wn. - ( o(W WwmWax2)
(38)
The first term of equation
38 can be further decomposed into
[T (w,, )11
= [x o (Wm W ,x)) )]
[x oW W 2.22) [z oW W,,)]
+ [a (W., W,,))
rwp (W TW, Inxn) o(W Wr
= || 0 (W, W.
+ [x oW,,, w (W rwp - Inxn (W,,, Wp
(39)
Combining equation
38
equation
39 and that Wm. WT finishes the proof.
B DISCUSSIONS OF THE THEORETICAL FRAMEWORK
In this section, we discuss the significance and potential extensions of our theoretical results.
Relationship to Sparse dictionary learning (SDL) and identifiability. Sparse dictionary learning (SDL) and SAES are closely related in that both aim to represent data using sparse latent codes, but they differ fundamentally how this objective is achieved. SDL formulated as a strictly linear model in which the data approximated as a linear combination of dictionary atoms, whereas SAEs replace this sparse-coding step with a nonlinear encoder, typically using ReLU-based activations. this nonlinear extension of classical dictionary identifiabilityy
sense, our theorems can be interpreted results. Specifically, from the perspective of identifiability. we can say an SAE is identifiable if the reconstruction of polysemantic features Tp leads to the perfect reconstruction of ground truth feature + Im = I to index reordering and zero padding. Then the observed phenomena
x. i.c., I of feature shrinking and feature vanishing correspond forms of partial non-identifiabilit that arise when the sparsity level is low and the nonlinear encoder cannot perfectly disentangle the ground truth features. On the contrary, by Theorems 2 and 3 we prove the identifiability of SAEs under the extreme sparsity condition. Moreover, Theorem shows that the ground truth reconstrction error of SAEs, which can also be viewed as a measure identifiability, depends on the interference term (W, p W, I), which has similar forms the coherence condition of SDL. In addition, by Theorem we show that introducing additional weights to the interference terms can enhance SAE identifiability
Extension to hierarchical feature structures. From empirical observations of the related works, the general features can be viewed as a combination of the specialized ones. For example, the activation of the general feature "blue" is a combination of activations of specialized features "blue circle", "blue square", and "blue triangle" with positive weights, because ideally, if any of the three specialized feature fires, the general "blue" should fire. Therefore, possible extension
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of our theoretical framework, specialized features x assumed in our submission, i... 2/ = Wi,.. Then the polysemantic SAE inputs I, = 2, W2, 2 W, nr still coincide with theoretical formulations. Note that under this formulation, for any concatenations of general and specialized features Fconeat, there exists a weight matrix W such that pp D Frencat. Then according to theorems, if ceconett features are non-overlapping and extremely sparse, they can be fully recovered by SAEs. This can partly explain the feature absorption phenomenon that the general features are absorbed by the overlapping specialized ones, because non-overlapping sparse features are more recoverable.
view the general features Σy as linear combinations of the
Extension non-linear feature combinations. We consider the attention attention-weighted weighted combinations of features. We discuss that as softmax(-) ~ max(-), the attention attention-weighter weighted combinations can be approximately reduced to a linear form about certain features, so the theoretical results in our paper still hold. Specifically, for an iitention-weighee combinations of features, we denote the polysemantic feature as
(War))TWaan W (W (Wizz,)
E p.t
(W,w.) " (Waa,) where Wg. We, and W. denote query, key, and value matrices. Note that (WW,))) (Waz,, softmax( (x, W. Wazy)eell) = 1[i arg max, x. W, Waazy, so we have
Σ 1li arg max W. Wia,WWWWWo W, W. max, on W. Wamy"
Ep.
Then we have the SAE risk
CSAE
Em
app. W (WW
= Ex
W aa max, a. W. WLa, W (W W arg max, W, Wea, Il
(40)
By similar proofs of Theorems 3. if W. forms certain geometry or has non-positive interferences, then for / [ni], have
W = W = arg min E. 1| WW pp.e W..
and accordingly W = W = arg minw.,, CSAE. With this closed-form solution, we can show that the insights of SAE feature recovery still hold. Moreover, by similar proofs, the insights of Theorems 4-5 also still hold, only replacing with Farg max, w. W Waa, in the mathematic forms.
Extensions of loss design beyond reweighting, As Theorem 5 based on very general theoretical framework, any design the matrix that shrinks the gap term W. I IWW, Juxa could improve the SAE feature recovery. In submission, we propose the most simple and intuitive reweighting strategy for an example to demonstrate this. Nonetheless, alternative designs are also foresecable. For example, although W, is unknown, according the closed-form solutions derived in Theorems 1-3, in turn estimate using WT. Then we can deliberately estimate triangular matrix to minimize ||W, rw, Taxall |l- A possible risk of this method would be the estimation error of W, can be large during the early training stages. Another possible alternative direction would be to include additional regularization term ||trace[W]] W,r n/). Specifically, note that trace ( T SeTTTrT,,, = trace(W, I, Then we can design a triangular matrix I' such that trace T r) trace(Inxn) = where the cross-feature interference WT W, can be directly estimated from data.
C EXPERIMENTS DETAILS
C.I ADDITIONAL EXPERIMENTS
Alternative realization for Top-k. We conduct additional validation experiments for Section 5.2. with an alternative realization of the Top-k activation. Specifically, the Top-k dimensions selected
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by decomposing the dimensions into k groups and choosing the Top-1 activated dimensions in each group. This realization could largely enhance the computational efficiency by allowing parallel computing. The results shown in Table 2 The results show that compared with SAE features, WSAR features obtain higher auto-interpretabiliiy scores (i.e., stronger monosemanticity) across different situations.
Table 2: Auto-interpretabiliiy scores (%) of SABs trained following different layers (9-11) of Pythia- 160M with original SAE and weighted SAE loss.
9
10
Original SAE Weighted SAE (a 1)
69.0
61.8
56.3
71.5
63.7
59.0
Gains
+2.5
+1.9
+2.7
Table 3: Auto-interpretabiliy scores (%) of Llama-3-88 with original SAE and weighted SAE loss.
Layer
16
24
SAE
62.0
67.0
66.7
61.4
WSAE (a= 1)
66.5
80.2
70.5
63.6
Gain
+4.5
+2.8
+3.8
+2.2
Evaluation on Llama. We conduct additional experiments on Llama-3-8B and present the last-layer results in Table The results show that compared with SAE features, WSAEs show consistent improvement in interpretability compared with the original SAE across different layers, which demonstrates the effectiveness of our proposed WSAE. This result coincides with our result on the smaller Pythia-160M.
Somantic Consistency
WSAE SAE
per-dimenson
2000
Consaterryy
0.2
2000
200
S4Ele WSAEls
(a) Sensitivity against
(b) Sensitivity against proxy.
(c) SAE reconstruction error.
Figure 5: (a) Semantic consistency of WSAEs under different (b) Semantic consistency of SAE and WSAEs with different monosemanticity proxies, including semantic consistency and per-dimensional variance. (c) SAE reconstruction error of WSAEs under different
Sensitivity analysis regarding a. We run experiments with alpha E {0, 0.5, 1.0, 2.0, and report the semantic consistency. The other settings are kept exactly the same with that of Figure 4 in our submission. The results are shown in Figure ((a) We show that in wide selection range of a, our WSAE show consistent advantage over SAE (a 0) with respect to interpretability, which demonstrates the robustness of WSAE against a.
Sensitivity analysis regarding monosemanticity proxy. We conduct an empirical comparison of different proxies (s;) (variance vs. semantic metrics) to test transferability. Specifically, as an extension Figure 3 we replace the semantic consistency with per-dimensional variance. The results reported in Figure5(()) We show that WSAEs with the two proxies both show consistent advantage over SAE (a = 0) with respect to interpretability. and the variation trend against a is also similar. This demonstrates the robustness of WSAE against proxy selection of monosemanticity.
Influence reweighting on the Pareto frontier. Figure 5(c) we show the SAE reconstruction error of zp of WSAEs under different a' 's the synthetic dataset, where the r-axis is the sparsity level of the ground truth features. We keep the other settings the same as Figure We show that the SAE reconstruction errors of WSAEs approximately the same those of the original SAE, and
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the results remain robust across relatively wide range of selections. This indicates that WSAEs still roughly on the Pareto frontier.
C.2 COMPUTE RESOURCES
For the experiments on toy models in Section conduct experiments on an NVIDIA 3090 GPU with 24GB of memory. The training and evaluation of toy model takes around 30 minutes. For the experiments
language models in Section we conduct experiments on an NVIDIA A100 GPU with 40GB of memory. The training of sparse autoencoder takes around 24 hours, and the evaluation needs 10 minutes. While for experiments on vision models Section 5.2 we conduct experiments on two NVIDIA 3090 GPUs with 24GB of memory. The training of a sparse autoencoder takes around 12 hours, and the evaluation needs minutes.
C.3 AVERAGE ACTIVATED FEATURES
For sample I E I2" we perform n encodings, each time isolating a single input feature. Specifically, for the i-th encoding, we construct x' = (0.... -....., 0), where only the i-th feature is retained. This yields n SAE latent representations [h(x")}; each corresponding to the activation induced by single input feature. We then calculate the total activation values across different samples in each SAE latent dimension. For example, for j-th dimension in SAE latents, we define Mj = E., (x²)
representing the cumulative activation from the i-th input feature. the value Mj exceeds a threshold, consider the j-th SAE latent to be activated by the i-th input. Finally, number of input features that activate each SAE latent dimension. We note that if the SAE successfully recovers monosemantic features, each latent should be activated by only small number of input features.
compute the average
C.4 AN AUTOINTERPPRETTTON PROTOCOL
The autointerpretabiliiy process consists of five steps and yields both an interpretation and an autointerpretability score:
1. On each of the first 50,000 lines of Open Web'Tet, take 64-token centence-fragmeet and measure the feature's activation on each token of this fragment. Feature activations are rescaled to integer values between 0 and 10.
2. Take the 20 fragments with the top activation scores and pass 5 of these to GPT-4, along with the rescaled per-token activations. Instruct GPT-4 suggest an explanation for when the feature (or neuron) fires, resulting in an interpretation.
3. Use GPT-3.56 to simulate the feature across another highly activating fragments and randomly selected fragments (with non-zero variation) by asking it to provide the per-token activations. While the process described in Bills et al. (2023) uses GPT-4 for the simulation step, we use GPT-3.5. This is because the simulation protocol requires the model's logprobs for scoring, and OpenAl's public API for GPT-3.5 (but not GPT-4) supports returning logprobs.
4. Compute the correlation of the simulated activations and the actual activations. This correlation is the autointerpretability score of the feature. The texts chosen for scoring feature can be random text fragments. fragments chosen for containing particularly high activation of that feature, or even mixture of the two. We use mixture of the two unless otherwise noted, also called 'top-random' scoring.
5. If, amongst the 50,000 fragments, there are fewer than 20 which contain non-zero variation in activation, then the feature is skipped entirely.
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Reviews:
[{"review_id": "otF63XYxsr", "reviewer": "Reviewer_DB94", "summary": "This paper provides a theoretical analysis of SAEs which motivates an alternative SAE design proposal (WSAE). They claim WSAE improves feature mono-semanticity and interpretability.", "questions": "- I'd like to see theoretical analysis which assumes more complex underlying data distributions. Would this change the resulting conclusions or insights? \n- Use of established, if imperfect benchmarks like SAE bench could be used to provide a more comprehensive endorsement of the wSAE approach. For example, do wSAEs demonstrate less feature absorption?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- The paper provides a detailed theoretical analysis. \n- The paper proposes a new re-weighting strategy that may reduce polysemanticity. \n- the paper tests their new strategy on language models.", "weaknesses": "- The paper assumes an incorrect model of the underlying data distribution which is no longer considered valid by many researchers in the field. There is not one true set of non-overlapping features, but rather many kinds of features which overlap with one another. Additionally, many features, such as parts of speech are dense. While much work on SAEs has assumed sparse features in a non-overlapping basis - this was more reasonable to do a few years ago before we'd seen so much object level data. For example, see \"Sparse Autoencoders Do Not Find Canonical Units of Analysis\" or \"A is for Absorption: Studying Feature Splitting and Absorption in Sparse Autoencoders\" and indeed some SAE architectures have been proposed with this in mind \"Learning Multi-Level Features with Matryoshka Sparse Autoencoders\". \n\n- Empirical results on real data are filtered through the lens of an auto-interpretability scoring method which doesn't reflect the broader quality of an SAE. \n\nFigure 4 deliberately scales the x-axis to make minor differences in the semantic consistency metric look very large. It's unclear if this is meaningful or not.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "4w2A49Y5Sg", "reviewer": "Reviewer_TYYe", "summary": "This paper provides a rigorous theoretical analysis of the **feature recovery limits of Sparse Autoencoders (SAEs)**, a method widely used for interpreting polysemantic representations in large models.  \nUnder the **superposition hypothesis**, the authors derive a closed-form optimal solution for SAEs and prove that:\n- In general, SAEs cannot perfectly recover the true monosemantic features due to *feature shrinking* and *feature vanishing* effects.\n- Full recovery is guaranteed only under **extremely sparse ground-truth features** (sparsity factor \\(S \\to 1\\)), i.e., nearly 1-sparse activations.\n- To mitigate this, the paper proposes a **Weighted Sparse Autoencoder (WSAE)**, where per-dimension weights reduce the reconstruction gap between the observed polysemantic and unobserved monosemantic features.  \nTheoretical results (Theorem 4–5) show that proper weighting minimizes this gap, and experiments on synthetic data and pretrained models (Pythia-160M, ResNet-18) demonstrate improved interpretability metrics.", "questions": "1. How do Theorem 1–3 connect formally to existing identifiability results in sparse dictionary learning (e.g., spark or coherence conditions)?  \n2. How sparse are real LLM activations compared to the “extreme sparsity” regime analyzed here?  \n3. Could you include a small α-sweep or sensitivity plot to verify robustness?  \n4. Does reweighting ever degrade reconstruction accuracy, or is the Pareto frontier generally preserved (as suggested by Fig. 3(c))?  \n5. Will the theoretical framework and WSAE implementation be publicly released?  \n6. Do you foresee extensions of this theoretical setup beyond reweighting—for example, alternative matrix designs that directly minimize cross-feature interference?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- **Mathematically grounded analysis.**  \n  The derivation of a closed-form SAE solution and identification of feature shrinking/vanishing phenomena clarify long-standing empirical observations in mechanistic interpretability.  \n  While related to classical sparse coding theory, the explicit analytical form for the ReLU-based SAE and the formal characterization of these degradation modes constitute a novel operational understanding that was not previously formalized.\n\n- **Bridging theory and practice.**  \n  The framework connects abstract sparse recovery theory to interpretability practice in LLMs, offering insight into why SAEs sometimes fail to yield cleanly separable features.\n\n- **Theoretically motivated improvement.**  \n  The proposed WSAE provides a principled approach to reweight reconstruction according to estimated polysemanticity, improving monosemanticity without large losses in reconstruction quality (as evidenced in Fig.3(c)).\n\n- **Transparent proofs and clear assumptions.**  \n  All mathematical steps, assumptions (superposition, sparsity distribution), and limitations are clearly stated.", "weaknesses": "### 1. Relationship to dictionary learning and identifiability\nThe theoretical results closely parallel classical **identifiability conditions in sparse dictionary learning**, where full recovery requires incoherence or extreme sparsity of the underlying basis.  \nWhile this correspondence is intuitive, it is not explicitly discussed in the paper.  \nClarifying this relationship would enhance the theoretical positioning of the work.  \n\nIn particular:\n- Theorem 1–3 can be viewed as a **nonlinear (ReLU-based) extension of dictionary identifiability** results, where interference \\(W_p^\\top W_p - I\\) plays the role of coherence.  \n- The observed feature shrinking/vanishing could be interpreted as manifestations of partial non-identifiability under finite sparsity.  \n\nExplicitly situating the paper in relation to established sparse coding theory (e.g., spark condition, mutual coherence) would help readers understand which parts of the contribution are new—e.g., the inclusion of ReLU nonlinearities and overcomplete encoders—and which are theoretical refinements of existing principles.\n\n---\n\n### 2. Implications for interpretability and practical impact\nThe paper’s findings—that SAEs cannot fully disentangle superposed representations except in extreme sparsity—have significant consequences for current interpretability research, though this point could be emphasized more strongly.\n\nIn particular:\n- Many interpretability studies assume that increasing sparsity or width of SAEs improves feature separation indefinitely.  \n  This work shows that such improvement **plateaus due to intrinsic representational interference**, meaning full disentanglement is mathematically impossible under realistic sparsity.\n- Consequently, **SAE-based interpretability should be regarded as an approximation tool**, not as a faithful feature recovery mechanism.  \n  This reframes SAE-derived neurons as *approximate projections of overlapping features*, rather than direct encodings of ground-truth concepts.\n\nThis reinterpretation could reshape how SAE-based analyses are used in mechanistic interpretability: rather than aiming for perfectly monosemantic neurons, practitioners might instead quantify or visualize residual interference between features.\n\nA related empirical suggestion would be to measure the effective sparsity \\(S\\) of real LLM activations, to contextualize how close such models operate to the theoretical extreme-sparsity regime.\n\n---\n\n### 3. Sensitivity and robustness of the reweighting scheme\nThe WSAE introduces weights \\(\\gamma_i = s_i^{\\alpha}\\), with \\(s_i\\) estimated from variance or semantic consistency and \\(\\alpha\\) controlling emphasis on monosemantic dimensions.  \nThe authors note (p.6) that results are “relatively robust” to α and show examples for α = 0.5 and 1.0 (Fig. 4), which supports this claim.  \nNonetheless, a small sensitivity analysis or ablation could strengthen the argument.  \n\nSuggestions:\n- A sweep of α (e.g., {0, 0.5, 1, 2}) to illustrate stability trends.  \n- A comparison of different proxies \\(s_i\\) (variance vs. semantic metrics) to test transferability.  \n- Discussion of reconstruction trade-offs: Fig.3(c) indicates that \\(x_p\\) reconstruction is maintained (i.e., no major Pareto penalty), but confirming this across datasets would reinforce generality.\n\nThese additions would verify that the reported improvements are not dataset-specific and that α tuning is unnecessary in practice.\n\n---\n\n### 4. Reproducibility and release\nAppendix B.2 clearly reports compute resources and training time (e.g., 24 h on A100 for language models).  \nIt would still be helpful to confirm whether **code and theoretical implementations** will be released upon publication, enabling the community to replicate and extend the analysis.\n\n---\n\n### 5. Potential extensions\nThe theoretical framework appears extensible beyond reweighting—for instance, to alternative loss formulations or nonlinear encoder architectures that directly address the feature interference term \\(W_p^\\top W_p - I\\).  \nA short remark in the final version about such future directions would underscore the broader applicability of this analysis.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Pbm47oCMqk", "reviewer": "Reviewer_hKP4", "summary": "This paper presents the first theoretical framework analyzing the fundamental limits of Sparse Autoencoders (SAEs) for recovering monosemantic features from superposed polysemantic representations. The authors derive closed-form solutions showing that, under general conditions, SAEs cannot perfectly recover ground-truth monosemantic features due to feature shrinking and feature vanishing effects. They identify that full recovery is theoretically guaranteed only under extreme sparsity of the true features. To address this limitation, the paper introduces a Weighted Sparse Autoencoder (WSAE) that reweights the reconstruction loss according to the degree of polysemanticity of each input dimension. Theoretical analysis demonstrates that this weighting narrows the gap between the SAE loss and the ideal ground-truth reconstruction loss, and experiments on both synthetic data and pretrained models confirm that WSAE improves feature monosemanticity and interpretability without sacrificing reconstruction fidelity.", "questions": "1. How does the theoretical framework extend to non-linear or multi-layer autoencoders, especially when feature mixing occurs across layers? Could the authors discuss whether similar recovery limits hold in deeper settings?\n\n2. The analysis assumes linear superposition of features. Have the authors explored how deviations from linearity—such as attention-weighted combinations or nonlinear feature interactions—affect the recovery bounds?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "The paper’s originality lies in providing a formal, mathematical explanation for why SAEs sometimes fail to identify interpretable features, moving beyond the purely empirical understanding that dominates current mechanistic interpretability work. The authors establish clear analytical results—closed-form solutions, necessary and sufficient conditions, and a uniqueness theorem—that rigorously connect feature sparsity with successful monosemantic recovery. This theoretical grounding fills a long-standing gap in the field, where SAE behavior had been empirically impressive but conceptually opaque. The proposed reweighting principle is both simple and theoretically motivated, making it an elegant bridge between formal analysis and practical implementation.\n\nIn terms of quality and clarity, the paper is meticulously written, with precise mathematical derivations and well-structured proofs. The relationship between the SAE loss and the ideal ground-truth loss is clearly articulated and forms a compelling narrative that unifies the theoretical and empirical sections. The significance of the contribution is substantial: by formalizing the limitations of current interpretability techniques, the paper reframes the field’s expectations of what SAEs can and cannot achieve, and introduces a principled path forward through reweighted optimization. The experiments, while modest in scale, convincingly validate the theoretical predictions and demonstrate that the framework generalizes across modalities.", "weaknesses": "The main limitation is the scope of empirical validation. The experiments are primarily performed on small or medium-scale models (Pythia-160M, ResNet-18) and under controlled settings. While these choices are appropriate for validating the theory, it remains unclear how well the findings extend to large modern LLMs or to deeper, multi-layer SAE architectures that are increasingly used in practice. Furthermore, the superposition assumption in the theoretical model treats representations as linear mixtures, which simplifies the nonlinear interactions and attention-based dynamics found in real networks. As a result, the framework captures the core geometry of superposition but may not fully describe the behavior of realistic LLM feature spaces.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)