Beyond SAEs: Consistent Concept Discovery with Sparse Coding

Jinho Choi; Hyesu Lim; Jaegul Choo; Steffen Schneider

cs.AIOct 16, 2026

Beyond SAEs: Consistent Concept Discovery with Sparse Coding

Authors: Jinho Choi, Hyesu Lim, Jaegul Choo, Steffen Schneider

Organizations: KAIST · Helmholtz Munich

Abstract

Sparse autoencoders (SAEs) explain foundation model representations by decomposing activations into sparse combinations of learned dictionary atoms. However, the learned dictionaries can vary across training runs, limiting the reliability and reproducibility of SAE-based analyses. We propose Consistent Sparse Dictionary Learning (ConSDL), which replaces the standard linear encoder with per-sample sparse coding. This is motivated by our synthetic analysis showing that coefficients from a single linear encoder pass are suboptimal compared with those obtained by per-sample optimization, suggesting that sparse coding is a reliable mechanism for dictionary learning. Across 16 synthetic settings, ConSDL achieves near-perfect dictionary and coefficient recovery. These gains translate to state-of-the-art cross-seed consistency across four vision and language settings, where ConSDL improves dictionary and coefficient consistency by 17.6% and 35.9%, respectively, over the average of six SAE baselines.

Figures & tables

Figure 1: Method. Left : Consistent Sparse Dictionary Learning (ConSDL) alternates between per-sample sparse coefficient inference with refitting and decoder updates under a reconstruction objective with co-active similarity regularization, holding the other component fixed at each stage. Standard SAEs (inset) instead use a learned encoder trained jointly with the decoder. Stacked panels indicate independent training seeds, across which ConSDL aims to recover consistent codes and dictionaries. Right : Python-style pseudocode.

Figure 2: Synthetic recovery and cross-seed consistency. (A) Activations are generated from a ground-truth dictionary $W_{\mathrm{GT}}$ with sparse codes $\mathrm{a}_{\mathrm{GT}}$ . Insets show the sampling distributions for the active-set size, coefficient magnitudes, and atom selection frequencies. (B) Dictionary ( DCS ; upper-left) and coefficient ( MCC ; lower-right) recovery across dictionary sizes and tail exponents $\beta$ . (C) Cross-seed consistency correlates with ground-truth recovery for both dictionaries (top) and coefficients (bottom). Each point is one method at a given dictionary size and $\beta$ . $r$ and $\rho$ denote the Pearson and Spearman correlation coefficients.

Architecture	# Atoms	DCS	$\ell_{0}$		MCC		Input $R^{2}$
Architecture	# Atoms	DCS	Amortized	Re-optimized	Amortized	Re-optimized	Amortized	Re-optimized
ReLU	3K	0.983	67.4	63.8	0.833	0.975	0.659	0.885
	6K	0.970	76.5	75.6	0.780	0.962	0.633	0.872
	9K	0.777	62.5	92.8	0.555	0.672	0.434	0.585
	12K	0.367	86.9	70.6	0.157	0.168	0.268	0.249
TopK	3K	0.982	60.0	60.0	0.862	0.947	0.812	0.920

Table 1: Per-sample re-optimization improves coefficient recovery under a fixed dictionary. Under uniform atom-selection frequencies (

\beta=0

), we compare amortized coefficients with per-sample FISTA re-optimization using fixed trained decoders. Darker green indicates larger improvements over amortized inference.

Figure 3: Consistency and dictionary utilization of foundation-model activations. We compare seven methods across two vision and two language backbones, three expansion factors, and three sparsity levels

\ell_{0}

. Cross-seed dictionary and coefficient consistency are measured by DCS and MCC , and dictionary utilization by the fraction of atoms that are neither dead nor duplicated. ConSDL achieves consistently higher cross-seed consistency, with the largest gains at

16\times

, while maintaining high dictionary utilization.

Figure 4: Atom-level cross-seed consistency distributions and activation patterns. (A) Distributions for CLIP ViT-B/16 and Pythia-160M. Each anchor atom’s consistency is the mean decoder cosine similarity to its matched atoms in the other two seeds. (B) Selected consistent ConSDL and inconsistent TopK atoms across three seeds, showing mean similarity for anchors and pairwise similarity for matches.

	Vision (AUPRC)				Language (AUROC)
	CLIP ViT-B/16		DINOv2-B/14		Pythia-160M		GPT-2 Small
Method	@1	@5	@1	@5	@1	@5	@1	@5
ReLU	0.48 $\pm$ 0.01	0.55 $\pm$ 0.00	0.45 $\pm$ 0.00	0.53 $\pm$ 0.00	0.70 $\pm$ 0.00	0.78 $\pm$ 0.00	0.74 $\pm$ 0.00	0.79 $\pm$ 0.00
TopK	0.46 $\pm$ 0.01	0.53 $\pm$ 0.00	0.43 $\pm$ 0.00	0.53 $\pm$ 0.00	0.72 $\pm$ 0.00	0.80 $\pm$ 0.00	0.75 $\pm$ 0.00	0.80 $\pm$ 0.00
BatchTopK	0.45 $\pm$ 0.00	0.52 $\pm$ 0.00	0.43 $\pm$ 0.00	0.53 $\pm$ 0.00	0.73 $\pm$ 0.00	0.80 $\pm$ 0.00	0.75 $\pm$ 0.00	0.80 $\pm$ 0.00
Matryoshka	0.42 $\pm$ 0.01	0.46 $\pm$ 0.00	0.42 $\pm$ 0.00	0.51 $\pm$ 0.00	0.72 $\pm$ 0.00	0.79 $\pm$ 0.00	0.74 $\pm$ 0.00	0.79 $\pm$ 0.00

Table 2: Sparse probing performance at

16\times

dictionary expansion (

\ell_{0}\approx 128

). Probes use the

k

most discriminative latents (

k\in\{1,5\}

). Vision reports class-wise AUPRC averaged over six datasets. Language reports AUROC averaged over 113 binary tasks in SAE-Probes. Values are mean

\pm

standard deviation over three seeds. Boldface and underlining indicate the best and second-best mean scores in each column, respectively.

Backbone	Variant	Sparsity		Reconstruction	Cross-seed Consistency		Dictionary Utilization
Backbone	Variant	$\lambda$	$\ell_{0}$	$R^{2}$	DCS	MCC	Usable atom frac.	# Dup.	# Dead
CLIP ViT-B/16	Sparse coding	1.2	116 $\pm$ 5.36	0.87 $\pm$ 0.00	0.84 $\pm$ 0.00	0.72 $\pm$ 0.00	0.25 $\pm$ 0.00	9,270 $\pm$ 35.3	0 $\pm$ 0.00
	+ Refitting	0.8	116 $\pm$ 1.88	0.92 $\pm$ 0.00	0.85 $\pm$ 0.00	0.70 $\pm$ 0.00	0.55 $\pm$ 0.00	5,564 $\pm$ 74.9	0 $\pm$ 0.00
	+ Similarity reg.	1.0	110 $\pm$ 2.56	0.89 $\pm$ 0.00	0.70 $\pm$ 0.00	0.49 $\pm$ 0.00	1.00 $\pm$ 0.00	56 $\pm$ 1.76	0 $\pm$ 0.00
	Full	0.75	130 $\pm$ 2.13	0.92 $\pm$ 0.00	0.78 $\pm$ 0.00	0.66 $\pm$ 0.00	1.00 $\pm$ 0.00	1 $\pm$ 0.90	0 $\pm$ 0.00
Pythia-160M	Sparse coding	3.0	113 $\pm$ 2.45	0.69 $\pm$ 0.00	0.29 $\pm$ 0.00	0.17 $\pm$ 0.00	0.03 $\pm$ 0.01	2,144 $\pm$ 52.4	9,767 $\pm$ 103

Table 3: Ablation study at

16\times

expansion and matched sparsity (

\ell_{0}\approx 128

). Starting from per-sample sparse coding, we add coefficient refitting or similarity regularization individually, while Full combines both. Since these components affect sparsity, we compare configurations matched by

\ell_{0}

. Values are mean

\pm

standard deviation over three seeds.

Figure 5: ConSDL captures distinct visual features associated with a broad concept. We compare ConSDL and ReLU SAE on CLIP ViT-B/16 ( $16\times$ , $\ell_{0}\approx 128$ ). (A) Five ConSDL atoms have decoder cosine similarities above 0.7 to a ReLU “beer bottle” atom. (B–C) These atoms exhibit moderate pairwise decoder similarity ( $\sim 0.5$ ) and partially overlapping image-level activations (Jaccard 0.36–0.57). (D) UMAP of image-level activation vectors reveals distinct clusters. (E) ConSDL atoms exhibit overlapping but distinct spatial activation patterns, whereas the ReLU atom responds broadly to beer-related objects.

Figure 6: Finer-grained concepts in larger ConSDL dictionaries. We compare CLIP ViT-B/16 dictionaries at $4\times$ ( $\ell_{0}\approx 32$ ) and $16\times$ ( $\ell_{0}\approx 128$ ) expansion. (A) A ConSDL atom associated with “Golden Retriever” at $4\times$ corresponds to six $16\times$ atoms (decoder cosine similarity 0.65–0.75), capturing features such as fur, tails, muzzle shapes, and puppies. (B) ReLU and TopK retain nearly identical retriever-related atoms across dictionary sizes (cosine similarity 0.99 and 0.98). (C–D) The ConSDL atoms exhibit moderate pairwise decoder similarity ( $\sim 0.5$ ) and partial image-level activation overlap (Jaccard 0.32–0.50).

Figure 7: Dictionary cluster consistency across training runs. (A) Joint UMAP of ConSDL decoder atoms at $16\times$ , $\ell_{0}\approx 128$ , colored by top 30 clusters from a reference run. (B) Cluster-size profiles for ConSDL and Matryoshka. $\rho$ denotes the Spearman correlation of cluster sizes across runs. (C) Normalized cluster-centroid agreement across granularities. Lines show the mean over nine CLIP settings, and shading shows the interquartile range.

Appendix figures & tables28 assets

Supplementary material from the paper’s appendix.

Appendix

Pythia-160M
Architecture	$4\times$		$8\times$		$16\times$
Architecture	Inference	Training	Inference	Training	Inference	Training
ReLU	0.04 (4.7)	0.32 (28.3)	0.08 (9.4)	0.53 (56.6)	0.14 (18.9)	0.97 (113.2)
TopK	0.12 (4.7)	0.36 (28.3)	0.21 (9.4)	0.59 (56.6)	0.37 (18.9)	1.05 (113.2)
BatchTopK	0.07 (4.7)	0.48 (28.3)	0.13 (9.4)	0.75 (56.6)	0.24 (18.9)	1.40 (113.2)
Matryoshka	0.07 (4.7)	0.81 (33.0)	0.13 (9.4)	1.20 (66.1)	0.24 (18.9)	2.02 (132.1)

Appendix

Table 4: Per-token inference and training cost across SAE architectures and dictionary expansion factors. We report latency in

\mu

s with MFLOPs in parentheses, using matched sparsity budgets of

\ell_{0}=32

64

, and

128

for

4\times

8\times

, and

16\times

, respectively.

Backbone	Expansion	Sparse coding	Refitting	Similarity reg.	Other update	Total
Pythia-160M	$4\times$	7.7 (62%)	1.7 (14%)	2.3 (19%)	0.6 (5%)	12.3
	$8\times$	14.3 (66%)	3.2 (15%)	3.5 (16%)	0.7 (3%)	21.7
	$16\times$	27.6 (64%)	6.1 (14%)	8.0 (19%)	1.2 (3%)	42.8
CLIP ViT-B/16	$4\times$	42.0 (74%)	9.3 (16%)	3.8 (7%)	1.6 (3%)	56.7
	$8\times$	82.6 (75%)	18.2 (17%)	6.6 (6%)	2.3 (2%)	109.7
	$16\times$	166.2 (75%)	36.6 (17%)	13.9 (6%)	4.4 (2%)	221.1

Appendix

Table 5: Breakdown of ConSDL training-step cost across dictionary expansion factors. Values report latency in milliseconds per minibatch, with the fraction of total training time in parentheses.

Architecture	Decoder cosine (DCS)		Footprint cosine (50k images)		Code Pearson (MCC)
Architecture	$\pi_{\mathrm{dec}}$	$\pi_{\mathrm{fp}}$	$\pi_{\mathrm{dec}}$	$\pi_{\mathrm{fp}}$	$\pi_{\mathrm{dec}}$	$\pi_{\mathrm{fp}}$
ReLU	0.525	0.496	0.595	0.615	0.417	0.416
TopK	0.558	0.525	0.696	0.715	0.499	0.498
BatchTopK	0.480	0.428	0.524	0.575	0.437	0.458
Matryoshka	0.271	0.189	0.243	0.290	0.184	0.199
MP-SAE	0.694	0.644	0.653	0.672	0.276	0.288

Appendix

Table 6: Three cross-seed consistency metrics on CLIP ViT-B/16 (

16\times

\ell_{0}\approx 128

, 3 seeds), each evaluated under two matchings: the absolute decoder cosine Hungarian used throughout the paper (

\pi_{\mathrm{dec}}

), and an independent Hungarian computed from activation-magnitude footprints over the 50,000-image ImageNet validation split (

\pi_{\mathrm{fp}}

). Higher is better and best per column in bold. The ranking is unchanged across the two matchings.

Architecture	$\beta$	# Atoms	DCS	$\ell_{0}$		MCC		Input $R^{2}$
Architecture	$\beta$	# Atoms	DCS	Amortized	Re-optimized	Amortized	Re-optimized	Amortized	Re-optimized
ReLU	0.0	3K	0.983	67.4	63.8	0.833	0.975	0.659	0.885
	0.0	6K	0.970	76.5	75.6	0.780	0.962	0.633	0.872
	0.0	9K	0.777	62.5	92.8	0.555	0.672	0.434	0.585
	0.0	12K	0.367	86.9	70.6	0.157	0.168	0.268	0.249
	0.5	3K	0.967	113	69.6	0.832	0.954	0.755	0.869

Appendix

Table 7: Per-sample re-optimization improves coefficient recovery under a fixed dictionary. For each trained decoder, we compare amortized coefficients with per-sample FISTA re-optimization. For dictionaries with DCS

\geq 0.5

, re-optimized cells are shaded by the improvement over amortized inference, with darker green indicating larger gains up to

+0.2

. The gains are larger when dictionary recovery is stronger.

Figure 9: Atom-level consistency distributions for DINOv2-B/14 and GPT-2 Small. Most ConSDL atoms exhibit high consistency in both backbones, whereas several baselines show broader distributions with a larger proportion of low-consistency atoms, particularly in DINOv2-B/14.

Figure 10: Number of consistent atoms across dictionary sizes and sparsity levels. We report the number of atoms with cross-seed consistency above 0.7 across four backbones, three dictionary expansions ( $4\times$ , $8\times$ , and $16\times$ ), and three sparsity levels ( $\ell_{0}\approx 32,64,128$ ). ConSDL consistently learns more consistent atoms as dictionary size increases, whereas most baselines exhibit slower growth or saturation, particularly in vision models. Higher sparsity levels generally yield smaller increases.

Figure 11: Finer-grained language concepts identified by ConSDL on Pythia-160M. (A) Top-activating text samples and token activation patterns for a ReLU “baseball” atom and seven matched ConSDL atoms capturing more specific concepts, including team names, pitching, and field positions. Concept labels were assigned by Claude Opus 5.5 based on top-activating samples. (B) The matched ConSDL atoms have high decoder cosine similarity to the ReLU atom (0.73–0.79) but substantially lower pairwise similarity. (C) The ConSDL atoms exhibit limited pairwise token-level activation overlap (0.06–0.28).

Figure 12: Finer-grained language concepts in larger ConSDL dictionaries. We compare Pythia-160M dictionaries at $4\times$ ( $\ell_{0}\approx 32$ ) and $16\times$ ( $\ell_{0}\approx 128$ ) expansion. (A) A ConSDL atom associated with “plants” at $4\times$ matches four $16\times$ atoms capturing more specific concepts, including plant species, crops, and flowers. Concept labels were assigned by Claude Opus 5.5 based on top-activating samples. (B) ReLU retains highly similar plant-related atoms across dictionary sizes (cosine similarity 0.95), whereas the matched ConSDL atoms exhibit low pairwise decoder similarity. (C) The ConSDL $16\times$ atoms exhibit limited token-level activation overlap (0.11–0.21).

Figure 13: One-to-many atom correspondence across methods. Each cell shows the percentage of source atoms with at least two counterparts in the target dictionary. We restrict the analysis to atoms with cross-seed consistency $\geq 0.7$ and identify counterparts using decoder cosine similarity $\geq 0.7$ . The bottom row reports the mean across the six other source methods.

Figure 14: Distribution of cross-method atom correspondence. Percentage of source atoms with at least 2, 3, 5, or 8 counterparts in the target dictionary, comparing ConSDL with ReLU and TopK across four backbones. Only atoms with cross-seed consistency

\geq 0.7

are considered, and counterparts are identified using decoder cosine similarity

\geq 0.7

. The vertical axis uses a logarithmic scale.

Figure 15: One-to-many atom correspondence across dictionary sizes. Percentage of source atoms with at least 2, 3, 5, or 8 counterparts when matching

4\times

(

\ell_{0}\approx 32

) and

16\times

(

\ell_{0}\approx 128

) dictionaries in both directions. The analysis includes only atoms with cross-seed consistency

\geq 0.7

, using decoder cosine similarity

\geq 0.7

to identify counterparts. The vertical axis uses a logarithmic scale.

Figure 16: Effect of dictionary redundancy on cross-seed consistency. Duplicate atom fraction and DCS after deduplication across cosine similarity thresholds for CLIP ViT-B/16 and Pythia-160M. Dotted horizontal lines indicate DCS before deduplication without similarity regularization.

Figure 17: Hyperparameter sensitivity of ConSDL. Results on CLIP ViT-B/16 (

4\times

\ell_{0}\approx 32

) across sparsity penalties

\lambda

and refitting iterations

T_{\mathrm{refit}}

. The left panel shows measured sparsity, while the remaining panels compare performance. Dark markers and gray shading indicate the target sparsity range, red highlights mark the selected configuration, and dashed lines show the best baseline.

Backbone	Metric	ReLU	TopK	BatchTopK	Matryoshka	MP-SAE	iSAE	ConSDL	vs. Avg.	vs. Best
CLIP ViT-B/16	DCS	0.61 $\pm$ 0.15	0.67 $\pm$ 0.10	0.56 $\pm$ 0.10	0.37 $\pm$ 0.08	0.75 $\pm$ 0.04	0.75 $\pm$ 0.06	0.78 $\pm$ 0.07	+26.7%	+4.8%
	MCC	0.53 $\pm$ 0.17	0.63 $\pm$ 0.13	0.52 $\pm$ 0.11	0.31 $\pm$ 0.10	0.45 $\pm$ 0.15	0.40 $\pm$ 0.06	0.73 $\pm$ 0.09	+54.7%	+16.8%
	Usable atom frac.	0.99 $\pm$ 0.00	1.00 $\pm$ 0.00	1.00 $\pm$ 0.00	1.00 $\pm$ 0.00	0.85 $\pm$ 0.05	0.96 $\pm$ 0.10	1.00 $\pm$ 0.00	+3.5%	+0.0%
DINOv2-B/14	DCS	0.68 $\pm$ 0.16	0.73 $\pm$ 0.13	0.65 $\pm$ 0.15	0.50 $\pm$ 0.12	0.83 $\pm$ 0.04	0.77 $\pm$ 0.10	0.84 $\pm$ 0.05	+21.4%	+1.6%
	MCC	0.61 $\pm$ 0.19	0.69 $\pm$ 0.17	0.61 $\pm$ 0.19	0.47 $\pm$ 0.16	0.49 $\pm$ 0.16	0.44 $\pm$ 0.07	0.78 $\pm$ 0.06	+42.3%	+14.1%
	Usable atom frac.	0.98 $\pm$ 0.01	0.98 $\pm$ 0.04	1.00 $\pm$ 0.00	0.99 $\pm$ 0.01	0.75 $\pm$ 0.11	0.99 $\pm$ 0.01	0.99 $\pm$ 0.02	+4.5%	-0.6%

Appendix

Table 8: Cross-seed consistency and dictionary utilization across foundation-model backbones. For each backbone and method, we report the mean and standard deviation across nine combinations of dictionary expansion (

4\times

8\times

16\times

) and sparsity (

\ell_{0}\approx 32

64

128

). The variation therefore reflects differences across these settings rather than training seeds. The last two columns report the relative improvement of ConSDL over the average of the six SAE baselines and the strongest baseline for each backbone and metric. The final three rows report averages over the four backbones with equal weighting.

Backbone	Cell	Method	ImageNet Object		SUN397 Scene		Food101 Food		DTD Texture		Caltech256 Object		RAF-DB Emotion		Avg.
Backbone	Cell	Method	@1	@5	@1	@5	@1	@5	@1	@5	@1	@5	@1	@5	@1	@5
CLIP ViT-B/16	$4\times$ $\ell_{0}\approx 32$	ReLU	0.33 $\pm$ 0.00	0.39 $\pm$ 0.00	0.35 $\pm$ 0.00	0.43 $\pm$ 0.00	0.42 $\pm$ 0.01	0.46 $\pm$ 0.00	0.51 $\pm$ 0.01	0.61 $\pm$ 0.01	0.70 $\pm$ 0.00	0.77 $\pm$ 0.00	0.45 $\pm$ 0.04	0.51 $\pm$ 0.01	0.46 $\pm$ 0.01	0.53 $\pm$ 0.00
		TopK	0.28 $\pm$ 0.00	0.36 $\pm$ 0.00	0.34 $\pm$ 0.00	0.45 $\pm$ 0.00	0.42 $\pm$ 0.00	0.50 $\pm$ 0.00	0.49 $\pm$ 0.01	0.61 $\pm$ 0.00	0.67 $\pm$ 0.00	0.76 $\pm$ 0.00	0.48 $\pm$ 0.00	0.52 $\pm$ 0.01	0.45 $\pm$ 0.00	0.54 $\pm$ 0.00
		BatchTopK	0.28 $\pm$ 0.00	0.36 $\pm$ 0.00	0.34 $\pm$ 0.00	0.45 $\pm$ 0.00	0.42 $\pm$ 0.01	0.50 $\pm$ 0.00	0.49 $\pm$ 0.01	0.61 $\pm$ 0.00	0.67 $\pm$ 0.00	0.76 $\pm$ 0.00	0.46 $\pm$ 0.04	0.51 $\pm$ 0.01	0.44 $\pm$ 0.01	0.53 $\pm$ 0.00
		Matryoshka	0.26 $\pm$ 0.00	0.33 $\pm$ 0.00	0.31 $\pm$ 0.00	0.42 $\pm$ 0.00	0.39 $\pm$ 0.01	0.46 $\pm$ 0.01	0.48 $\pm$ 0.01	0.59 $\pm$ 0.00	0.61 $\pm$ 0.01	0.71 $\pm$ 0.01	0.42 $\pm$ 0.04	0.49 $\pm$ 0.00	0.41 $\pm$ 0.01	0.50 $\pm$ 0.00
		MP-SAE	0.28 $\pm$ 0.00	0.36 $\pm$ 0.00	0.34 $\pm$ 0.00	0.45 $\pm$ 0.00	0.42 $\pm$ 0.00	0.50 $\pm$ 0.01	0.51 $\pm$ 0.01	0.60 $\pm$ 0.01	0.67 $\pm$ 0.00	0.76 $\pm$ 0.00	0.42 $\pm$ 0.03	0.50 $\pm$ 0.02	0.44 $\pm$ 0.00	0.53 $\pm$ 0.00

Appendix

Table 9: Sparse probing of vision dictionaries. AUPRC from the

k

most discriminative dictionary atoms per class,

k\in\{1,5\}

, at each expansion and its matched sparsity (

4\times

\ell_{0}\approx 32

8\times

\ell_{0}\approx 64

16\times

\ell_{0}\approx 128

). Avg. is the unweighted mean over the six evaluation datasets.

\pm

is the standard deviation across training seeds of the reported score.

Backbone	Method	$\mathbf{4\times}$ , $\ell_{0}\approx 32$		$\mathbf{8\times}$ , $\ell_{0}\approx 64$		$\mathbf{16\times}$ , $\ell_{0}\approx 128$
Backbone	Method	@1	@5	@1	@5	@1	@5
Pythia-160M	ReLU	0.69 $\pm$ 0.00	0.75 $\pm$ 0.00	0.70 $\pm$ 0.00	0.77 $\pm$ 0.00	0.70 $\pm$ 0.00	0.78 $\pm$ 0.00
	TopK	0.70 $\pm$ 0.00	0.77 $\pm$ 0.00	0.71 $\pm$ 0.00	0.78 $\pm$ 0.00	0.72 $\pm$ 0.00	0.80 $\pm$ 0.00
	BatchTopK	0.70 $\pm$ 0.00	0.77 $\pm$ 0.00	0.72 $\pm$ 0.01	0.79 $\pm$ 0.00	0.73 $\pm$ 0.00	0.80 $\pm$ 0.00
	Matryoshka	0.71 $\pm$ 0.00	0.77 $\pm$ 0.00	0.72 $\pm$ 0.01	0.79 $\pm$ 0.00	0.72 $\pm$ 0.00	0.79 $\pm$ 0.00
	MP-SAE	0.70 $\pm$ 0.00	0.78 $\pm$ 0.00	0.68 $\pm$ 0.00	0.76 $\pm$ 0.00	0.67 $\pm$ 0.00	0.74 $\pm$ 0.00

Appendix

Table 10: Sparse probing performance of language-model dictionaries. We report AUROC using the

k

most discriminative dictionary atoms per probe for

k\in\{1,5\}

across three matched expansion and sparsity settings:

4\times

with

\ell_{0}\approx 32

8\times

with

\ell_{0}\approx 64

, and

16\times

with

\ell_{0}\approx 128

. Scores are averaged over the 113 binary probes in SAE-Probes.

\pm

is the standard deviation across training seeds of the reported score.

Figure 18: Region decomposition and cross-seed latent analysis in CLIP ViT-B/16. $16\times$ dictionary ( $m=12288$ ), $\ell_{0}\approx 128$ , three seeds. (A) Each $14\times 14$ patch grid is independently clustered into $k=10$ regions from its coefficient vectors. Hue encodes region position, and headers report active latents and their mean per patch. (B) One seed-0 patch is selected (red outline). Top: activation map and shared UMAP, fitted on seed 0 with seeds 1–2 projected into the same space. Black rings mark latents active at the patch, red the selected latent, and gray its nearest matches. Latent #10694 matches #8244 ( $\cos=0.75$ ) and #889 ( $\cos=0.79$ ). Bottom: top-activating images at ranks $1,2$ , $50,51$ , and $99,100$ , shown unmasked and activation-masked, reveal animals in close contact. (C) Same analysis for a second latent at the patch. #3107 matches #7465 ( $\cos=0.90$ ) and #10300 ( $\cos=0.70$ ), showing a heart/romance motif.

Figure 19: Region decomposition and cross-seed latent analysis in Pythia-160M. $16\times$ dictionary ( $m=12288$ ), three seeds, using a 96-token window with approximately 112 active latents per token. (A) Tokens are independently clustered into $k=10$ regions for each seed. Hue encodes position in the text rather than content. (B) One seed-0 token is selected (red outline). For each seed, we show the passage with latent activations highlighted, a shared UMAP fitted on seed 0 with seeds 1–2 projected into the same space, and top-activating contexts at ranks $1,2$ , $50,51$ , and $99,100$ . Black rings mark latents active at the token, red the selected latent, and gray its nearest matches. Latent #12079 matches #12005 ( $\cos=0.97$ ) and #7854 ( $\cos=0.97$ ), capturing animal nouns. (C) Same analysis for a second latent. #5107 matches #6110 ( $\cos=0.65$ ) and #10394 ( $\cos=0.79$ ), capturing sentence-final punctuation and the preceding token.

Setting	Architecture	Sparsity	Reconstruction	Ground-truth Recovery		Cross-seed Consistency		Dictionary Utilization
Setting	Architecture	$\ell_{0}$	$R^{2}$	DCS	MCC	DCS	MCC	# Dup	# Dead	Usable atom frac.
$\beta=0.0$	ReLU	67.0 $\pm$ 0.1	0.67 $\pm$ 0.00	0.98 $\pm$ 0.00	0.83 $\pm$ 0.00	0.97 $\pm$ 0.00	0.97 $\pm$ 0.00	10 $\pm$ 2	0 $\pm$ 0	1.00 $\pm$ 0.00
	TopK	60.0 $\pm$ 0.0	0.82 $\pm$ 0.00	0.98 $\pm$ 0.00	0.86 $\pm$ 0.00	0.97 $\pm$ 0.00	0.93 $\pm$ 0.00	1 $\pm$ 1	0 $\pm$ 0	1.00 $\pm$ 0.00
	BatchTopK	61.0 $\pm$ 0.0	0.81 $\pm$ 0.00	0.98 $\pm$ 0.00	0.86 $\pm$ 0.00	0.96 $\pm$ 0.00	0.93 $\pm$ 0.00	1 $\pm$ 1	0 $\pm$ 0	1.00 $\pm$ 0.00
	Matryoshka	64.0 $\pm$ 0.0	0.69 $\pm$ 0.00	0.82 $\pm$ 0.00	0.67 $\pm$ 0.00	0.69 $\pm$ 0.00	0.58 $\pm$ 0.00	1 $\pm$ 1	0 $\pm$ 0	1.00 $\pm$ 0.00
	MP-SAE	57.7 $\pm$ 0.0	0.95 $\pm$ 0.00	0.99 $\pm$ 0.00	0.89 $\pm$ 0.00	0.99 $\pm$ 0.00	0.95 $\pm$ 0.00	20 $\pm$ 4	0 $\pm$ 0	1.00 $\pm$ 0.00

Appendix

Table 11: Full synthetic results with a 3K-atom ground-truth dictionary across tail exponents

\beta

. We evaluate all methods across

\beta\in\{0,0.5,0.7,1.0\}

. We report sparsity

\ell_{0}

, reconstruction (

R^{2}

), ground-truth recovery ( DCS , MCC ), cross-seed consistency, and dictionary utilization measured by the usable atom fraction. The numbers of dead and duplicated atoms are also reported. Metrics are averaged across training seeds, while cross-seed DCS and MCC are averaged across seed pairs.

\pm

denotes the corresponding standard deviation.

Setting	Architecture	Sparsity	Reconstruction	Ground-truth Recovery		Cross-seed Consistency		Dictionary Utilization
Setting	Architecture	$\ell_{0}$	$R^{2}$	DCS	MCC	DCS	MCC	# Dup	# Dead	Usable atom frac.
$\beta=0.0$	ReLU	76.0 $\pm$ 0.3	0.64 $\pm$ 0.00	0.97 $\pm$ 0.00	0.78 $\pm$ 0.00	0.94 $\pm$ 0.00	0.94 $\pm$ 0.00	44 $\pm$ 5	0 $\pm$ 0	1.00 $\pm$ 0.00
	TopK	60.0 $\pm$ 0.0	0.65 $\pm$ 0.00	0.81 $\pm$ 0.00	0.61 $\pm$ 0.00	0.67 $\pm$ 0.00	0.53 $\pm$ 0.00	2 $\pm$ 1	0 $\pm$ 0	1.00 $\pm$ 0.00
	BatchTopK	71.0 $\pm$ 0.0	0.63 $\pm$ 0.00	0.76 $\pm$ 0.00	0.55 $\pm$ 0.00	0.59 $\pm$ 0.00	0.44 $\pm$ 0.00	1 $\pm$ 1	0 $\pm$ 0	1.00 $\pm$ 0.00
	Matryoshka	75.0 $\pm$ 0.0	0.48 $\pm$ 0.00	0.35 $\pm$ 0.00	0.17 $\pm$ 0.00	0.20 $\pm$ 0.00	0.05 $\pm$ 0.00	0 $\pm$ 0	0 $\pm$ 0	1.00 $\pm$ 0.00
	MP-SAE	58.0 $\pm$ 0.0	0.95 $\pm$ 0.00	0.99 $\pm$ 0.00	0.98 $\pm$ 0.00	0.98 $\pm$ 0.00	0.98 $\pm$ 0.00	51 $\pm$ 12	0 $\pm$ 0	1.00 $\pm$ 0.00

Appendix

Table 12: Full synthetic results with a 6K-atom ground-truth dictionary across tail exponents

\beta

. We evaluate all methods across

\beta\in\{0,0.5,0.7,1.0\}

. We report sparsity

\ell_{0}

, reconstruction (

R^{2}

\pm

denotes the corresponding standard deviation.

Setting	Architecture	Sparsity	Reconstruction	Ground-truth Recovery		Cross-seed Consistency		Dictionary Utilization
Setting	Architecture	$\ell_{0}$	$R^{2}$	DCS	MCC	DCS	MCC	# Dup	# Dead	Usable atom frac.
$\beta=0.0$	ReLU	63.0 $\pm$ 0.0	0.44 $\pm$ 0.00	0.78 $\pm$ 0.00	0.55 $\pm$ 0.00	0.62 $\pm$ 0.00	0.53 $\pm$ 0.00	66 $\pm$ 5	0 $\pm$ 0	1.00 $\pm$ 0.00
	TopK	60.0 $\pm$ 0.0	0.41 $\pm$ 0.00	0.23 $\pm$ 0.00	0.05 $\pm$ 0.00	0.15 $\pm$ 0.00	0.01 $\pm$ 0.00	0 $\pm$ 0	0 $\pm$ 0	1.00 $\pm$ 0.00
	BatchTopK	89.0 $\pm$ 0.0	0.45 $\pm$ 0.00	0.22 $\pm$ 0.00	0.05 $\pm$ 0.00	0.15 $\pm$ 0.00	0.02 $\pm$ 0.00	0 $\pm$ 0	0 $\pm$ 0	1.00 $\pm$ 0.00
	Matryoshka	80.0 $\pm$ 0.4	0.44 $\pm$ 0.00	0.19 $\pm$ 0.00	0.03 $\pm$ 0.00	0.14 $\pm$ 0.00	0.01 $\pm$ 0.00	0 $\pm$ 0	0 $\pm$ 0	1.00 $\pm$ 0.00
	MP-SAE	60.0 $\pm$ 0.0	0.47 $\pm$ 0.00	0.17 $\pm$ 0.00	0.01 $\pm$ 0.00	0.18 $\pm$ 0.00	0.00 $\pm$ 0.00	0 $\pm$ 0	0 $\pm$ 0	1.00 $\pm$ 0.00

Appendix

Table 13: Full synthetic results with a 9K-atom ground-truth dictionary across tail exponents

\beta

. We evaluate all methods across

\beta\in\{0,0.5,0.7,1.0\}

. We report sparsity

\ell_{0}

, reconstruction (

R^{2}

\pm

denotes the corresponding standard deviation.

Setting	Architecture	Sparsity	Reconstruction	Ground-truth Recovery		Cross-seed Consistency		Dictionary Utilization
Setting	Architecture	$\ell_{0}$	$R^{2}$	DCS	MCC	DCS	MCC	# Dup	# Dead	Usable atom frac.
$\beta=0.0$	ReLU	87.0 $\pm$ 0.0	0.27 $\pm$ 0.00	0.37 $\pm$ 0.00	0.16 $\pm$ 0.00	0.22 $\pm$ 0.00	0.05 $\pm$ 0.00	1 $\pm$ 1	0 $\pm$ 0	1.00 $\pm$ 0.00
	TopK	60.0 $\pm$ 0.0	0.40 $\pm$ 0.00	0.18 $\pm$ 0.00	0.02 $\pm$ 0.00	0.15 $\pm$ 0.00	0.01 $\pm$ 0.00	0 $\pm$ 0	0 $\pm$ 0	1.00 $\pm$ 0.00
	BatchTopK	88.0 $\pm$ 0.6	0.45 $\pm$ 0.00	0.17 $\pm$ 0.00	0.02 $\pm$ 0.00	0.15 $\pm$ 0.00	0.02 $\pm$ 0.00	0 $\pm$ 0	0 $\pm$ 0	1.00 $\pm$ 0.00
	Matryoshka	85.0 $\pm$ 0.0	0.44 $\pm$ 0.00	0.17 $\pm$ 0.00	0.02 $\pm$ 0.00	0.14 $\pm$ 0.00	0.01 $\pm$ 0.00	0 $\pm$ 0	0 $\pm$ 0	1.00 $\pm$ 0.00
	MP-SAE	60.0 $\pm$ 0.0	0.57 $\pm$ 0.00	0.18 $\pm$ 0.00	0.02 $\pm$ 0.00	0.15 $\pm$ 0.00	0.01 $\pm$ 0.00	0 $\pm$ 0	0 $\pm$ 0	1.00 $\pm$ 0.00

Appendix

Table 14: Full synthetic results with a 12K-atom ground-truth dictionary across tail exponents

\beta

. We evaluate all methods across

\beta\in\{0,0.5,0.7,1.0\}

. We report sparsity

\ell_{0}

, reconstruction (

R^{2}

\pm

denotes the corresponding standard deviation.

Expansion	$\ell_{0}$ band	Method	Sparsity	Reconstruction	Consistency		Dictionary Utilization
Expansion	$\ell_{0}$ band	Method	$\ell_{0}$	$R^{2}$	DCS	MCC	Usable atom frac.	# Dup.	# Dead
$4\times$	$\ell_{0}\approx 32$	ReLU	34 $\pm$ 0.4	0.89 $\pm$ 0.00	0.74 $\pm$ 0.00	0.68 $\pm$ 0.00	0.99 $\pm$ 0.00	36 $\pm$ 4.1	0 $\pm$ 0.00
		TopK	32 $\pm$ 0.0	0.93 $\pm$ 0.00	0.80 $\pm$ 0.00	0.77 $\pm$ 0.00	1.00 $\pm$ 0.00	1 $\pm$ 0.58	0 $\pm$ 0.00
		BatchTopK	32 $\pm$ 0.0	0.93 $\pm$ 0.00	0.72 $\pm$ 0.00	0.70 $\pm$ 0.00	1.00 $\pm$ 0.00	1 $\pm$ 0.29	0 $\pm$ 0.00
		Matryoshka	32 $\pm$ 0.0	0.92 $\pm$ 0.00	0.51 $\pm$ 0.00	0.47 $\pm$ 0.01	1.00 $\pm$ 0.00	6 $\pm$ 0.87	0 $\pm$ 0.00
		MP-SAE	32 $\pm$ 0.0	0.93 $\pm$ 0.00	0.79 $\pm$ 0.01	0.63 $\pm$ 0.00	0.91 $\pm$ 0.00	275 $\pm$ 9	0 $\pm$ 0.00

Appendix

Table 15: Full results on CLIP ViT-B/16 across dictionary widths and sparsity levels. We evaluate three expansion factors (

4\times

8\times

16\times

) and three target sparsity levels (

\ell_{0}\approx 32

64

128

). We report achieved

\ell_{0}

, reconstruction (

R^{2}

), cross-seed consistency ( DCS , MCC ), and dictionary utilization, including usable atom fraction and numbers of dead and duplicated atoms. Metrics are averaged across training seeds, while DCS and MCC are averaged across seed pairs.

\pm

denotes the corresponding standard deviation.

Expansion	$\ell_{0}$ band	Method	Sparsity	Reconstruction	Consistency		Dictionary Utilization
Expansion	$\ell_{0}$ band	Method	$\ell_{0}$	$R^{2}$	DCS	MCC	Usable atom frac.	# Dup.	# Dead
$4\times$	$\ell_{0}\approx 32$	ReLU	29 $\pm$ 0.0	0.93 $\pm$ 0.00	0.79 $\pm$ 0.00	0.75 $\pm$ 0.00	0.99 $\pm$ 0.00	26 $\pm$ 2.6	0 $\pm$ 0.00
		TopK	32 $\pm$ 0.0	0.96 $\pm$ 0.00	0.90 $\pm$ 0.00	0.88 $\pm$ 0.00	1.00 $\pm$ 0.00	3 $\pm$ 2.02	0 $\pm$ 0.00
		BatchTopK	32 $\pm$ 0.0	0.96 $\pm$ 0.00	0.83 $\pm$ 0.00	0.81 $\pm$ 0.00	1.00 $\pm$ 0.00	0 $\pm$ 0.29	0 $\pm$ 0.00
		Matryoshka	32 $\pm$ 0.0	0.95 $\pm$ 0.00	0.67 $\pm$ 0.00	0.67 $\pm$ 0.00	1.00 $\pm$ 0.00	9 $\pm$ 2.84	0 $\pm$ 0.00
		MP-SAE	32 $\pm$ 0.0	0.96 $\pm$ 0.00	0.87 $\pm$ 0.02	0.69 $\pm$ 0.01	0.89 $\pm$ 0.01	352 $\pm$ 28	0 $\pm$ 0.00

Appendix

Table 16: Full results on DINOv2-B/14 across dictionary widths and sparsity levels. We evaluate three expansion factors (

4\times

8\times

16\times

) and three target sparsity levels (

\ell_{0}\approx 32

64

128

). We report achieved

\ell_{0}

, reconstruction (

R^{2}

\pm

denotes the corresponding standard deviation.

Expansion	$\ell_{0}$ band	Method	Sparsity	Reconstruction	Consistency		Dictionary Utilization
Expansion	$\ell_{0}$ band	Method	$\ell_{0}$	$R^{2}$	DCS	MCC	Usable atom frac.	# Dup.	# Dead
$4\times$	$\ell_{0}\approx 32$	ReLU	31 $\pm$ 0.2	0.87 $\pm$ 0.02	0.91 $\pm$ 0.00	0.86 $\pm$ 0.00	0.95 $\pm$ 0.00	161 $\pm$ 8	6 $\pm$ 3.00
		TopK	32 $\pm$ 0.0	0.89 $\pm$ 0.00	0.83 $\pm$ 0.00	0.79 $\pm$ 0.00	0.99 $\pm$ 0.00	27 $\pm$ 3.4	0 $\pm$ 0.00
		BatchTopK	32 $\pm$ 0.1	0.90 $\pm$ 0.00	0.83 $\pm$ 0.00	0.79 $\pm$ 0.00	0.99 $\pm$ 0.00	17 $\pm$ 3.6	0 $\pm$ 0.00
		Matryoshka	32 $\pm$ 0.1	0.88 $\pm$ 0.00	0.71 $\pm$ 0.00	0.69 $\pm$ 0.00	1.00 $\pm$ 0.00	0 $\pm$ 0.00	0 $\pm$ 0.00
		MP-SAE	32 $\pm$ 0.0	0.98 $\pm$ 0.00	0.87 $\pm$ 0.00	0.64 $\pm$ 0.01	0.83 $\pm$ 0.00	517 $\pm$ 13	0 $\pm$ 0.00

Appendix

Table 17: Full results on GPT-2 Small across dictionary widths and sparsity levels. We evaluate three expansion factors (

4\times

8\times

16\times

) and three target sparsity levels (

\ell_{0}\approx 32

64

128

). We report achieved

\ell_{0}

, reconstruction (

R^{2}

\pm

denotes the corresponding standard deviation.

Expansion	$\ell_{0}$ band	Method	Sparsity	Reconstruction	Consistency		Dictionary Utilization
Expansion	$\ell_{0}$ band	Method	$\ell_{0}$	$R^{2}$	DCS	MCC	Usable atom frac.	# Dup.	# Dead
$4\times$	$\ell_{0}\approx 32$	ReLU	31 $\pm$ 0.3	0.84 $\pm$ 0.00	0.90 $\pm$ 0.00	0.82 $\pm$ 0.01	0.92 $\pm$ 0.00	247 $\pm$ 5	1 $\pm$ 1.15
		TopK	32 $\pm$ 0.0	0.90 $\pm$ 0.00	0.84 $\pm$ 0.00	0.78 $\pm$ 0.00	1.00 $\pm$ 0.00	15 $\pm$ 1.8	0 $\pm$ 0.00
		BatchTopK	32 $\pm$ 0.1	0.91 $\pm$ 0.02	0.83 $\pm$ 0.00	0.78 $\pm$ 0.00	1.00 $\pm$ 0.00	13 $\pm$ 1.0	0 $\pm$ 0.00
		Matryoshka	32 $\pm$ 0.0	0.90 $\pm$ 0.02	0.71 $\pm$ 0.00	0.67 $\pm$ 0.00	1.00 $\pm$ 0.00	0 $\pm$ 0.29	0 $\pm$ 0.00
		MP-SAE	32 $\pm$ 0.0	0.96 $\pm$ 0.00	0.84 $\pm$ 0.01	0.51 $\pm$ 0.00	0.72 $\pm$ 0.00	874 $\pm$ 7	0 $\pm$ 0.00

Appendix

Table 18: Full results on Pythia-160M across dictionary widths and sparsity levels. We evaluate three expansion factors (

4\times

8\times

16\times

) and three target sparsity levels (

\ell_{0}\approx 32

64

128

). We report achieved

\ell_{0}

, reconstruction (

R^{2}

\pm

denotes the corresponding standard deviation.

Backbone	Expansion	ReLU			ConSDL
		$\ell_{1}$			$\lambda$			$T_{\mathrm{FISTA}}$			$T_{\mathrm{refit}}$			$\gamma_{\mathrm{sim}}$
		$32$	$64$	$128$	$32$	$64$	$128$	$32$	$64$	$128$	$32$	$64$	$128$	$32$	$64$	$128$
CLIP ViT-B/16	$\times 4$	1.4e-4	1.0e-4	7.5e-5	$0.8$	$0.6$	$0.35$	$30$	$30$	$30$	$10$	$5$	$5$	1e-3	1e-3	1e-3
	$\times 8$	1.5e-4	1.2e-4	8.4e-5	$0.8$	$0.8$	$0.5$	$30$	$30$	$30$	$60$	$10$	$5$	5e-3	1e-3	1e-3
	$\times 16$	1.6e-4	1.3e-4	1.0e-4	$1$	$1.25$	$0.75$	$50$	$30$	$30$	$50$	$10$	$10$	5e-3	5e-3	2e-3
DINOv2-B/14	$\times 4$	1.5e-4	8.5e-5	5.6e-5	$1.76$	$1.08$	$0.74$	$30$	$30$	$30$	$10$	$10$	$10$	5e-3	5e-3	1e-3

Appendix

Table 19: Hyperparameters for sparsity matching and ConSDL training. For each backbone, expansion, and target sparsity level (

\ell_{0}\approx 32,64,128

), we report the ReLU

\ell_{1}

coefficient and ConSDL sparse-coding penalty

\lambda

used for sparsity matching. For ConSDL, we additionally report the number of FISTA iterations

T_{\mathrm{FISTA}}

, coefficient-refitting steps

T_{\mathrm{refit}}

, and the co-active similarity weight

\gamma_{\mathrm{sim}}

$\bm{\beta}$	3K			6K			9K			12K
$\bm{\beta}$	ReLU $\ell_{1}$	Ours $\lambda$	Ours $T_{\mathrm{refit}}$	ReLU $\ell_{1}$	Ours $\lambda$	Ours $T_{\mathrm{refit}}$	ReLU $\ell_{1}$	Ours $\lambda$	Ours $T_{\mathrm{refit}}$	ReLU $\ell_{1}$	Ours $\lambda$	Ours $T_{\mathrm{refit}}$
0.0	1.5e-4	0.3	30	1.4e-4	0.5	10	1.8e-4	0.4	10	1.8e-4	0.5	5
0.5	1e-4	0.3	30	1.2e-4	0.4	10	1.8e-4	0.3	10	1.6e-4	0.4	10
0.7	1.5e-4	0.35	20	2.1e-4	0.25	30	1.8e-4	0.35	30	2e-4	0.3	10
1.0	2.1e-4	0.15	30	2.3e-4	0.15	20	2.1e-4	0.25	30	2.1e-4	0.3	20

Appendix

Table 20: Hyperparameters for the synthetic experiments. For each ground-truth dictionary size and tail exponent

\beta

, we report the ReLU sparsity coefficient

\ell_{1}

and the ConSDL sparse-coding penalty

\lambda

and number of refitting steps

T_{\mathrm{refit}}

. ConSDL uses

T_{\mathrm{FISTA}}=50

, and

\gamma_{\mathrm{sim}}=\texttt{1e-3}

in all settings.

Sparse Autoencoders

Sparse autoencoders decompose neural network activations into sparse features to study, interpret, and control learned representations.

See all 217 papers

May 29, 2026

Toward Identifiable Sparse Autoencoders

Recently, sparse autoencoders (SAEs) have emerged as an attractive tool for interpreting and interacting with representations in practical neural networks. While it is common empirical folklore, we also show theoretically that SAEs are highly unstable: different training runs are likely to produce different concept dictionaries and sparse codes. We characterize the model properties that hinder the stability of real-world SAEs, and address each of these problems through minimal changes to the architecture and training procedure. Together, these changes yield two versions of an identifiable SAE (iSAE), a variant of the standard TopK SAE with lower reconstruction error and improved stability. We explain this improvement theoretically by connecting SAEs with traditional dictionary learning approaches, and show that the dictionaries learned in practice satisfy an approximate restricted isometry condition, rendering the corresponding sparse codes in those models near-identifiable.

Walter Nelson, Theofanis Karaletsos, Francesco Locatello

Institute of Science and Technology Austria · 2Pyramidal Inc. · and Achira Inc., USA.

Parameter Identifiability Sparse Autoencoders Dictionary Learning Sparse Coding

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Jun 1, 2026

How Optimality Structures Sparse Dictionaries: A Theory for Understanding SAE Representations

Sparse Autoencoders (SAEs) have found success parsing neural representations into interpretable concepts, providing a basis for understanding and control. However, what exactly SAEs extract, and, correspondingly, the scientific conclusions we can draw from them, are not obvious. Empirically, the proof is in the pudding: SAEs learn interpretable features. Theoretically, we lack a clear account of what properties a 'concept' must satisfy for an SAE to extract it. There has been extensive identifiability work studying the conditions under which sparse coding recovers ground-truth features; however, these approaches tends to focus on simple data-generating models (e.g. sparse independent features) which poorly approximate the internet-swallowing language-model representations on which SAEs are trained. Here, avoiding data-generating models, we ask simply what properties any dictionary learning optimum must satisfy. Concretely, we extend local optimality analyses (Gribonval & Schnass, 2010) to the nonnegative joint-optimisation problem that vanilla SAEs approximate, and derive constraints relating optimal SAE features to their distributions. We use these constraints to explain a range of observed SAE behaviours - hierarchical splitting & absorption, the structure of residuals, and dense antipodal features - each reflecting how L1+nonnegativity interact with data to structure optimal dictionaries. Finally, we construct a novel large-dictionary convex problem and explore the wide atom-per-datapoint limit. In sum, we hope to tease model assumptions from unexpected observations, letting us learn more from SAEs' successes and provide principles for designing their successors.

William Dorrell

Kempner Institute · Harvard University

Representation Learning Sparse Autoencoders Mechanistic Interpretability Dictionary Learning Sparse Coding

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Sep 3, 2026

SharedSAE: One Feature Dictionary Across Language Models

Sparse autoencoders (SAEs) are widely used to interpret language model activations, but SAE training and latent labelling are typically repeated for every model. Here, we show that a single shared SAE can replace a collection of dedicated per-model SAEs. Our method, SharedSAE, combines a shared dictionary with model-specific encoder-decoder pairs. Unlike the closest prior method, which discards activation magnitudes and requires all models at inference, SharedSAE instead normalizes only selection scores, preserving magnitudes, and uses model dropout for single-model inference. We train SharedSAE on four 1B-scale base language models spanning distinct families and tokenizers. Despite sharing its latents across models, SharedSAE retains 96.6% of dedicated SAEs' mean explained variance; its latent activations exhibit cross-model correlations 1.8 times as high as separate SAEs aligned post-hoc, and its latent descriptions transfer across models. After the dictionary is frozen, new models can be efficiently adapted to it, achieving near-dedicated-SAE reconstruction quality while reusing the shared latent descriptions.

Daniil Ognev, Célian Vasson, Lijie Hu +2

MBZUAI · Sorbonne Université · Tohoku University +1

LLM Interpretability Sparse Autoencoders

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Abstract

Sparse Autoencoders

Explore similar work

Toward Identifiable Sparse Autoencoders

How Optimality Structures Sparse Dictionaries: A Theory for Understanding SAE Representations

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