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

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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 WGTW_{\mathrm{GT}} with sparse codes aGT\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 β\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 β\beta . rr and ρ\rho denote the Pearson and Spearman correlation coefficients.
Architecture# AtomsDCSℓ0\ell_{0}MCCInput R2R^{2}
AmortizedRe-optimizedAmortizedRe-optimizedAmortizedRe-optimized
ReLU3K0.98367.463.80.8330.9750.6590.885
6K0.97076.575.60.7800.9620.6330.872
9K0.77762.592.80.5550.6720.4340.585
12K0.36786.970.60.1570.1680.2680.249
TopK3K0.98260.060.00.8620.9470.8120.920
Table 1: Per-sample re-optimization improves coefficient recovery under a fixed dictionary. Under uniform atom-selection frequencies ( β=0\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 ℓ0\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 16×16\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/16DINOv2-B/14Pythia-160MGPT-2 Small
Method@1@5@1@5@1@5@1@5
ReLU0.48 ±\pm 0.010.55 ±\pm 0.000.45 ±\pm 0.000.53 ±\pm 0.000.70 ±\pm 0.000.78 ±\pm 0.000.74 ±\pm 0.000.79 ±\pm 0.00
TopK0.46 ±\pm 0.010.53 ±\pm 0.000.43 ±\pm 0.000.53 ±\pm 0.000.72 ±\pm 0.000.80 ±\pm 0.000.75 ±\pm 0.000.80 ±\pm 0.00
BatchTopK0.45 ±\pm 0.000.52 ±\pm 0.000.43 ±\pm 0.000.53 ±\pm 0.000.73 ±\pm 0.000.80 ±\pm 0.000.75 ±\pm 0.000.80 ±\pm 0.00
Matryoshka0.42 ±\pm 0.010.46 ±\pm 0.000.42 ±\pm 0.000.51 ±\pm 0.000.72 ±\pm 0.000.79 ±\pm 0.000.74 ±\pm 0.000.79 ±\pm 0.00
Table 2: Sparse probing performance at 16×16\times dictionary expansion ( ℓ0≈128\ell_{0}\approx 128 ). Probes use the kk most discriminative latents ( k∈{1,5}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.
BackboneVariantSparsityReconstructionCross-seed ConsistencyDictionary Utilization
λ\lambdaℓ0\ell_{0}R2R^{2}DCSMCCUsable atom frac.# Dup.# Dead
CLIP ViT-B/16Sparse coding1.2116 ±\pm 5.360.87 ±\pm 0.000.84 ±\pm 0.000.72 ±\pm 0.000.25 ±\pm 0.009,270 ±\pm 35.30 ±\pm 0.00
+ Refitting0.8116 ±\pm 1.880.92 ±\pm 0.000.85 ±\pm 0.000.70 ±\pm 0.000.55 ±\pm 0.005,564 ±\pm 74.90 ±\pm 0.00
+ Similarity reg.1.0110 ±\pm 2.560.89 ±\pm 0.000.70 ±\pm 0.000.49 ±\pm 0.001.00 ±\pm 0.0056 ±\pm 1.760 ±\pm 0.00
Full0.75130 ±\pm 2.130.92 ±\pm 0.000.78 ±\pm 0.000.66 ±\pm 0.001.00 ±\pm 0.001 ±\pm 0.900 ±\pm 0.00
Pythia-160MSparse coding3.0113 ±\pm 2.450.69 ±\pm 0.000.29 ±\pm 0.000.17 ±\pm 0.000.03 ±\pm 0.012,144 ±\pm 52.49,767 ±\pm 103
Table 3: Ablation study at 16×16\times expansion and matched sparsity ( ℓ0≈128\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 ℓ0\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 ( 16×16\times , ℓ0≈128\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 ( ∼0.5\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 4×4\times ( ℓ0≈32\ell_{0}\approx 32 ) and 16×16\times ( ℓ0≈128\ell_{0}\approx 128 ) expansion. (A) A ConSDL atom associated with “Golden Retriever” at 4×4\times corresponds to six 16×16\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 ( ∼0.5\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 16×16\times , ℓ0≈128\ell_{0}\approx 128 , colored by top 30 clusters from a reference run. (B) Cluster-size profiles for ConSDL and Matryoshka. ρ\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

Architecture4×4\times8×8\times16×16\times
InferenceTrainingInferenceTrainingInferenceTraining
Pythia-160M
ReLU0.04 (4.7)0.32 (28.3)0.08 (9.4)0.53 (56.6)0.14 (18.9)0.97 (113.2)
TopK0.12 (4.7)0.36 (28.3)0.21 (9.4)0.59 (56.6)0.37 (18.9)1.05 (113.2)
BatchTopK0.07 (4.7)0.48 (28.3)0.13 (9.4)0.75 (56.6)0.24 (18.9)1.40 (113.2)
Matryoshka0.07 (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 μ\mu s with MFLOPs in parentheses, using matched sparsity budgets of ℓ0=32\ell_{0}=32 , 6464 , and 128128 for 4×4\times , 8×8\times , and 16×16\times , respectively.
BackboneExpansionSparse codingRefittingSimilarity reg.Other updateTotal
Pythia-160M4×4\times7.7 (62%)1.7 (14%)2.3 (19%)0.6 (5%)12.3
8×8\times14.3 (66%)3.2 (15%)3.5 (16%)0.7 (3%)21.7
16×16\times27.6 (64%)6.1 (14%)8.0 (19%)1.2 (3%)42.8
CLIP ViT-B/164×4\times42.0 (74%)9.3 (16%)3.8 (7%)1.6 (3%)56.7
8×8\times82.6 (75%)18.2 (17%)6.6 (6%)2.3 (2%)109.7
16×16\times166.2 (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.
ArchitectureDecoder cosine (DCS)Footprint cosine (50k images)Code Pearson (MCC)
πdec\pi_{\mathrm{dec}}πfp\pi_{\mathrm{fp}}πdec\pi_{\mathrm{dec}}πfp\pi_{\mathrm{fp}}πdec\pi_{\mathrm{dec}}πfp\pi_{\mathrm{fp}}
ReLU0.5250.4960.5950.6150.4170.416
TopK0.5580.5250.6960.7150.4990.498
BatchTopK0.4800.4280.5240.5750.4370.458
Matryoshka0.2710.1890.2430.2900.1840.199
MP-SAE0.6940.6440.6530.6720.2760.288
Appendix
Table 6: Three cross-seed consistency metrics on CLIP ViT-B/16 ( 16×16\times , ℓ0≈128\ell_{0}\approx 128 , 3 seeds), each evaluated under two matchings: the absolute decoder cosine Hungarian used throughout the paper ( πdec\pi_{\mathrm{dec}} ), and an independent Hungarian computed from activation-magnitude footprints over the 50,000-image ImageNet validation split ( πfp\pi_{\mathrm{fp}} ). Higher is better and best per column in bold. The ranking is unchanged across the two matchings.
Architectureβ\beta# AtomsDCSℓ0\ell_{0}MCCInput R2R^{2}
AmortizedRe-optimizedAmortizedRe-optimizedAmortizedRe-optimized
ReLU0.03K0.98367.463.80.8330.9750.6590.885
0.06K0.97076.575.60.7800.9620.6330.872
0.09K0.77762.592.80.5550.6720.4340.585
0.012K0.36786.970.60.1570.1680.2680.249
0.53K0.96711369.60.8320.9540.7550.869
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 ≥0.5\geq 0.5 , re-optimized cells are shaded by the improvement over amortized inference, with darker green indicating larger gains up to +0.2+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 ( 4×4\times , 8×8\times , and 16×16\times ), and three sparsity levels ( ℓ0≈32,64,128\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 4×4\times ( ℓ0≈32\ell_{0}\approx 32 ) and 16×16\times ( ℓ0≈128\ell_{0}\approx 128 ) expansion. (A) A ConSDL atom associated with “plants” at 4×4\times matches four 16×16\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 16×16\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 ≥0.7\geq 0.7 and identify counterparts using decoder cosine similarity ≥0.7\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 ≥0.7\geq 0.7 are considered, and counterparts are identified using decoder cosine similarity ≥0.7\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 4×4\times ( ℓ0≈32\ell_{0}\approx 32 ) and 16×16\times ( ℓ0≈128\ell_{0}\approx 128 ) dictionaries in both directions. The analysis includes only atoms with cross-seed consistency ≥0.7\geq 0.7 , using decoder cosine similarity ≥0.7\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 ( 4×4\times , ℓ0≈32\ell_{0}\approx 32 ) across sparsity penalties λ\lambda and refitting iterations TrefitT_{\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.
BackboneMetricReLUTopKBatchTopKMatryoshkaMP-SAEiSAEConSDLvs. Avg.vs. Best
CLIP ViT-B/16DCS0.61 ±\pm 0.150.67 ±\pm 0.100.56 ±\pm 0.100.37 ±\pm 0.080.75 ±\pm 0.040.75 ±\pm 0.060.78 ±\pm 0.07+26.7%+4.8%
MCC0.53 ±\pm 0.170.63 ±\pm 0.130.52 ±\pm 0.110.31 ±\pm 0.100.45 ±\pm 0.150.40 ±\pm 0.060.73 ±\pm 0.09+54.7%+16.8%
Usable atom frac.0.99 ±\pm 0.001.00 ±\pm 0.001.00 ±\pm 0.001.00 ±\pm 0.000.85 ±\pm 0.050.96 ±\pm 0.101.00 ±\pm 0.00+3.5%+0.0%
DINOv2-B/14DCS0.68 ±\pm 0.160.73 ±\pm 0.130.65 ±\pm 0.150.50 ±\pm 0.120.83 ±\pm 0.040.77 ±\pm 0.100.84 ±\pm 0.05+21.4%+1.6%
MCC0.61 ±\pm 0.190.69 ±\pm 0.170.61 ±\pm 0.190.47 ±\pm 0.160.49 ±\pm 0.160.44 ±\pm 0.070.78 ±\pm 0.06+42.3%+14.1%
Usable atom frac.0.98 ±\pm 0.010.98 ±\pm 0.041.00 ±\pm 0.000.99 ±\pm 0.010.75 ±\pm 0.110.99 ±\pm 0.010.99 ±\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 ( 4×4\times , 8×8\times , 16×16\times ) and sparsity ( ℓ0≈32\ell_{0}\approx 32 , 6464 , 128128 ). 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.
BackboneCellMethodImageNet ObjectSUN397 SceneFood101 FoodDTD TextureCaltech256 ObjectRAF-DB EmotionAvg.
@1@5@1@5@1@5@1@5@1@5@1@5@1@5
CLIP ViT-B/164×4\times ℓ0≈32\ell_{0}\approx 32ReLU0.33 ±\pm 0.000.39 ±\pm 0.000.35 ±\pm 0.000.43 ±\pm 0.000.42 ±\pm 0.010.46 ±\pm 0.000.51 ±\pm 0.010.61 ±\pm 0.010.70 ±\pm 0.000.77 ±\pm 0.000.45 ±\pm 0.040.51 ±\pm 0.010.46 ±\pm 0.010.53 ±\pm 0.00
TopK0.28 ±\pm 0.000.36 ±\pm 0.000.34 ±\pm 0.000.45 ±\pm 0.000.42 ±\pm 0.000.50 ±\pm 0.000.49 ±\pm 0.010.61 ±\pm 0.000.67 ±\pm 0.000.76 ±\pm 0.000.48 ±\pm 0.000.52 ±\pm 0.010.45 ±\pm 0.000.54 ±\pm 0.00
BatchTopK0.28 ±\pm 0.000.36 ±\pm 0.000.34 ±\pm 0.000.45 ±\pm 0.000.42 ±\pm 0.010.50 ±\pm 0.000.49 ±\pm 0.010.61 ±\pm 0.000.67 ±\pm 0.000.76 ±\pm 0.000.46 ±\pm 0.040.51 ±\pm 0.010.44 ±\pm 0.010.53 ±\pm 0.00
Matryoshka0.26 ±\pm 0.000.33 ±\pm 0.000.31 ±\pm 0.000.42 ±\pm 0.000.39 ±\pm 0.010.46 ±\pm 0.010.48 ±\pm 0.010.59 ±\pm 0.000.61 ±\pm 0.010.71 ±\pm 0.010.42 ±\pm 0.040.49 ±\pm 0.000.41 ±\pm 0.010.50 ±\pm 0.00
MP-SAE0.28 ±\pm 0.000.36 ±\pm 0.000.34 ±\pm 0.000.45 ±\pm 0.000.42 ±\pm 0.000.50 ±\pm 0.010.51 ±\pm 0.010.60 ±\pm 0.010.67 ±\pm 0.000.76 ±\pm 0.000.42 ±\pm 0.030.50 ±\pm 0.020.44 ±\pm 0.000.53 ±\pm 0.00
Appendix
Table 9: Sparse probing of vision dictionaries. AUPRC from the kk most discriminative dictionary atoms per class, k∈{1,5}k\in\{1,5\} , at each expansion and its matched sparsity ( 4×4\times at ℓ0≈32\ell_{0}\approx 32 , 8×8\times at ℓ0≈64\ell_{0}\approx 64 , 16×16\times at ℓ0≈128\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.
BackboneMethod4×\mathbf{4\times} , ℓ0≈32\ell_{0}\approx 328×\mathbf{8\times} , ℓ0≈64\ell_{0}\approx 6416×\mathbf{16\times} , ℓ0≈128\ell_{0}\approx 128
@1@5@1@5@1@5
Pythia-160MReLU0.69 ±\pm 0.000.75 ±\pm 0.000.70 ±\pm 0.000.77 ±\pm 0.000.70 ±\pm 0.000.78 ±\pm 0.00
TopK0.70 ±\pm 0.000.77 ±\pm 0.000.71 ±\pm 0.000.78 ±\pm 0.000.72 ±\pm 0.000.80 ±\pm 0.00
BatchTopK0.70 ±\pm 0.000.77 ±\pm 0.000.72 ±\pm 0.010.79 ±\pm 0.000.73 ±\pm 0.000.80 ±\pm 0.00
Matryoshka0.71 ±\pm 0.000.77 ±\pm 0.000.72 ±\pm 0.010.79 ±\pm 0.000.72 ±\pm 0.000.79 ±\pm 0.00
MP-SAE0.70 ±\pm 0.000.78 ±\pm 0.000.68 ±\pm 0.000.76 ±\pm 0.000.67 ±\pm 0.000.74 ±\pm 0.00
Appendix
Table 10: Sparse probing performance of language-model dictionaries. We report AUROC using the kk most discriminative dictionary atoms per probe for k∈{1,5}k\in\{1,5\} across three matched expansion and sparsity settings: 4×4\times with ℓ0≈32\ell_{0}\approx 32 , 8×8\times with ℓ0≈64\ell_{0}\approx 64 , and 16×16\times with ℓ0≈128\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. 16×16\times dictionary ( m=12288m=12288 ), ℓ0≈128\ell_{0}\approx 128 , three seeds. (A) Each 14×1414\times 14 patch grid is independently clustered into k=10k=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\cos=0.75 ) and #889 ( cos⁡=0.79\cos=0.79 ). Bottom: top-activating images at ranks 1,21,2 , 50,5150,51 , and 99,10099,100 , 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\cos=0.90 ) and #10300 ( cos⁡=0.70\cos=0.70 ), showing a heart/romance motif.
Figure 19: Region decomposition and cross-seed latent analysis in Pythia-160M. 16×16\times dictionary ( m=12288m=12288 ), three seeds, using a 96-token window with approximately 112 active latents per token. (A) Tokens are independently clustered into k=10k=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 1,21,2 , 50,5150,51 , and 99,10099,100 . Black rings mark latents active at the token, red the selected latent, and gray its nearest matches. Latent #12079 matches #12005 ( cos⁡=0.97\cos=0.97 ) and #7854 ( cos⁡=0.97\cos=0.97 ), capturing animal nouns. (C) Same analysis for a second latent. #5107 matches #6110 ( cos⁡=0.65\cos=0.65 ) and #10394 ( cos⁡=0.79\cos=0.79 ), capturing sentence-final punctuation and the preceding token.
SettingArchitectureSparsityReconstructionGround-truth RecoveryCross-seed ConsistencyDictionary Utilization
ℓ0\ell_{0}R2R^{2}DCSMCCDCSMCC# Dup# DeadUsable atom frac.
β=0.0\beta=0.0ReLU67.0 ±\pm 0.10.67 ±\pm 0.000.98 ±\pm 0.000.83 ±\pm 0.000.97 ±\pm 0.000.97 ±\pm 0.0010 ±\pm 20 ±\pm 01.00 ±\pm 0.00
TopK60.0 ±\pm 0.00.82 ±\pm 0.000.98 ±\pm 0.000.86 ±\pm 0.000.97 ±\pm 0.000.93 ±\pm 0.001 ±\pm 10 ±\pm 01.00 ±\pm 0.00
BatchTopK61.0 ±\pm 0.00.81 ±\pm 0.000.98 ±\pm 0.000.86 ±\pm 0.000.96 ±\pm 0.000.93 ±\pm 0.001 ±\pm 10 ±\pm 01.00 ±\pm 0.00
Matryoshka64.0 ±\pm 0.00.69 ±\pm 0.000.82 ±\pm 0.000.67 ±\pm 0.000.69 ±\pm 0.000.58 ±\pm 0.001 ±\pm 10 ±\pm 01.00 ±\pm 0.00
MP-SAE57.7 ±\pm 0.00.95 ±\pm 0.000.99 ±\pm 0.000.89 ±\pm 0.000.99 ±\pm 0.000.95 ±\pm 0.0020 ±\pm 40 ±\pm 01.00 ±\pm 0.00
Appendix
Table 11: Full synthetic results with a 3K-atom ground-truth dictionary across tail exponents β\beta . We evaluate all methods across β∈{0,0.5,0.7,1.0}\beta\in\{0,0.5,0.7,1.0\} . We report sparsity ℓ0\ell_{0} , reconstruction ( R2R^{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.
SettingArchitectureSparsityReconstructionGround-truth RecoveryCross-seed ConsistencyDictionary Utilization
ℓ0\ell_{0}R2R^{2}DCSMCCDCSMCC# Dup# DeadUsable atom frac.
β=0.0\beta=0.0ReLU76.0 ±\pm 0.30.64 ±\pm 0.000.97 ±\pm 0.000.78 ±\pm 0.000.94 ±\pm 0.000.94 ±\pm 0.0044 ±\pm 50 ±\pm 01.00 ±\pm 0.00
TopK60.0 ±\pm 0.00.65 ±\pm 0.000.81 ±\pm 0.000.61 ±\pm 0.000.67 ±\pm 0.000.53 ±\pm 0.002 ±\pm 10 ±\pm 01.00 ±\pm 0.00
BatchTopK71.0 ±\pm 0.00.63 ±\pm 0.000.76 ±\pm 0.000.55 ±\pm 0.000.59 ±\pm 0.000.44 ±\pm 0.001 ±\pm 10 ±\pm 01.00 ±\pm 0.00
Matryoshka75.0 ±\pm 0.00.48 ±\pm 0.000.35 ±\pm 0.000.17 ±\pm 0.000.20 ±\pm 0.000.05 ±\pm 0.000 ±\pm 00 ±\pm 01.00 ±\pm 0.00
MP-SAE58.0 ±\pm 0.00.95 ±\pm 0.000.99 ±\pm 0.000.98 ±\pm 0.000.98 ±\pm 0.000.98 ±\pm 0.0051 ±\pm 120 ±\pm 01.00 ±\pm 0.00
Appendix
Table 12: Full synthetic results with a 6K-atom ground-truth dictionary across tail exponents β\beta . We evaluate all methods across β∈{0,0.5,0.7,1.0}\beta\in\{0,0.5,0.7,1.0\} . We report sparsity ℓ0\ell_{0} , reconstruction ( R2R^{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.
SettingArchitectureSparsityReconstructionGround-truth RecoveryCross-seed ConsistencyDictionary Utilization
ℓ0\ell_{0}R2R^{2}DCSMCCDCSMCC# Dup# DeadUsable atom frac.
β=0.0\beta=0.0ReLU63.0 ±\pm 0.00.44 ±\pm 0.000.78 ±\pm 0.000.55 ±\pm 0.000.62 ±\pm 0.000.53 ±\pm 0.0066 ±\pm 50 ±\pm 01.00 ±\pm 0.00
TopK60.0 ±\pm 0.00.41 ±\pm 0.000.23 ±\pm 0.000.05 ±\pm 0.000.15 ±\pm 0.000.01 ±\pm 0.000 ±\pm 00 ±\pm 01.00 ±\pm 0.00
BatchTopK89.0 ±\pm 0.00.45 ±\pm 0.000.22 ±\pm 0.000.05 ±\pm 0.000.15 ±\pm 0.000.02 ±\pm 0.000 ±\pm 00 ±\pm 01.00 ±\pm 0.00
Matryoshka80.0 ±\pm 0.40.44 ±\pm 0.000.19 ±\pm 0.000.03 ±\pm 0.000.14 ±\pm 0.000.01 ±\pm 0.000 ±\pm 00 ±\pm 01.00 ±\pm 0.00
MP-SAE60.0 ±\pm 0.00.47 ±\pm 0.000.17 ±\pm 0.000.01 ±\pm 0.000.18 ±\pm 0.000.00 ±\pm 0.000 ±\pm 00 ±\pm 01.00 ±\pm 0.00
Appendix
Table 13: Full synthetic results with a 9K-atom ground-truth dictionary across tail exponents β\beta . We evaluate all methods across β∈{0,0.5,0.7,1.0}\beta\in\{0,0.5,0.7,1.0\} . We report sparsity ℓ0\ell_{0} , reconstruction ( R2R^{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.
SettingArchitectureSparsityReconstructionGround-truth RecoveryCross-seed ConsistencyDictionary Utilization
ℓ0\ell_{0}R2R^{2}DCSMCCDCSMCC# Dup# DeadUsable atom frac.
β=0.0\beta=0.0ReLU87.0 ±\pm 0.00.27 ±\pm 0.000.37 ±\pm 0.000.16 ±\pm 0.000.22 ±\pm 0.000.05 ±\pm 0.001 ±\pm 10 ±\pm 01.00 ±\pm 0.00
TopK60.0 ±\pm 0.00.40 ±\pm 0.000.18 ±\pm 0.000.02 ±\pm 0.000.15 ±\pm 0.000.01 ±\pm 0.000 ±\pm 00 ±\pm 01.00 ±\pm 0.00
BatchTopK88.0 ±\pm 0.60.45 ±\pm 0.000.17 ±\pm 0.000.02 ±\pm 0.000.15 ±\pm 0.000.02 ±\pm 0.000 ±\pm 00 ±\pm 01.00 ±\pm 0.00
Matryoshka85.0 ±\pm 0.00.44 ±\pm 0.000.17 ±\pm 0.000.02 ±\pm 0.000.14 ±\pm 0.000.01 ±\pm 0.000 ±\pm 00 ±\pm 01.00 ±\pm 0.00
MP-SAE60.0 ±\pm 0.00.57 ±\pm 0.000.18 ±\pm 0.000.02 ±\pm 0.000.15 ±\pm 0.000.01 ±\pm 0.000 ±\pm 00 ±\pm 01.00 ±\pm 0.00
Appendix
Table 14: Full synthetic results with a 12K-atom ground-truth dictionary across tail exponents β\beta . We evaluate all methods across β∈{0,0.5,0.7,1.0}\beta\in\{0,0.5,0.7,1.0\} . We report sparsity ℓ0\ell_{0} , reconstruction ( R2R^{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.
Expansionℓ0\ell_{0} bandMethodSparsityReconstructionConsistencyDictionary Utilization
ℓ0\ell_{0}R2R^{2}DCSMCCUsable atom frac.# Dup.# Dead
4×4\timesℓ0≈32\ell_{0}\approx 32ReLU34 ±\pm 0.40.89 ±\pm 0.000.74 ±\pm 0.000.68 ±\pm 0.000.99 ±\pm 0.0036 ±\pm 4.10 ±\pm 0.00
TopK32 ±\pm 0.00.93 ±\pm 0.000.80 ±\pm 0.000.77 ±\pm 0.001.00 ±\pm 0.001 ±\pm 0.580 ±\pm 0.00
BatchTopK32 ±\pm 0.00.93 ±\pm 0.000.72 ±\pm 0.000.70 ±\pm 0.001.00 ±\pm 0.001 ±\pm 0.290 ±\pm 0.00
Matryoshka32 ±\pm 0.00.92 ±\pm 0.000.51 ±\pm 0.000.47 ±\pm 0.011.00 ±\pm 0.006 ±\pm 0.870 ±\pm 0.00
MP-SAE32 ±\pm 0.00.93 ±\pm 0.000.79 ±\pm 0.010.63 ±\pm 0.000.91 ±\pm 0.00275 ±\pm 90 ±\pm 0.00
Appendix
Table 15: Full results on CLIP ViT-B/16 across dictionary widths and sparsity levels. We evaluate three expansion factors ( 4×4\times , 8×8\times , 16×16\times ) and three target sparsity levels ( ℓ0≈32\ell_{0}\approx 32 , 6464 , 128128 ). We report achieved ℓ0\ell_{0} , reconstruction ( R2R^{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.
Expansionℓ0\ell_{0} bandMethodSparsityReconstructionConsistencyDictionary Utilization
ℓ0\ell_{0}R2R^{2}DCSMCCUsable atom frac.# Dup.# Dead
4×4\timesℓ0≈32\ell_{0}\approx 32ReLU29 ±\pm 0.00.93 ±\pm 0.000.79 ±\pm 0.000.75 ±\pm 0.000.99 ±\pm 0.0026 ±\pm 2.60 ±\pm 0.00
TopK32 ±\pm 0.00.96 ±\pm 0.000.90 ±\pm 0.000.88 ±\pm 0.001.00 ±\pm 0.003 ±\pm 2.020 ±\pm 0.00
BatchTopK32 ±\pm 0.00.96 ±\pm 0.000.83 ±\pm 0.000.81 ±\pm 0.001.00 ±\pm 0.000 ±\pm 0.290 ±\pm 0.00
Matryoshka32 ±\pm 0.00.95 ±\pm 0.000.67 ±\pm 0.000.67 ±\pm 0.001.00 ±\pm 0.009 ±\pm 2.840 ±\pm 0.00
MP-SAE32 ±\pm 0.00.96 ±\pm 0.000.87 ±\pm 0.020.69 ±\pm 0.010.89 ±\pm 0.01352 ±\pm 280 ±\pm 0.00
Appendix
Table 16: Full results on DINOv2-B/14 across dictionary widths and sparsity levels. We evaluate three expansion factors ( 4×4\times , 8×8\times , 16×16\times ) and three target sparsity levels ( ℓ0≈32\ell_{0}\approx 32 , 6464 , 128128 ). We report achieved ℓ0\ell_{0} , reconstruction ( R2R^{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.
Expansionℓ0\ell_{0} bandMethodSparsityReconstructionConsistencyDictionary Utilization
ℓ0\ell_{0}R2R^{2}DCSMCCUsable atom frac.# Dup.# Dead
4×4\timesℓ0≈32\ell_{0}\approx 32ReLU31 ±\pm 0.20.87 ±\pm 0.020.91 ±\pm 0.000.86 ±\pm 0.000.95 ±\pm 0.00161 ±\pm 86 ±\pm 3.00
TopK32 ±\pm 0.00.89 ±\pm 0.000.83 ±\pm 0.000.79 ±\pm 0.000.99 ±\pm 0.0027 ±\pm 3.40 ±\pm 0.00
BatchTopK32 ±\pm 0.10.90 ±\pm 0.000.83 ±\pm 0.000.79 ±\pm 0.000.99 ±\pm 0.0017 ±\pm 3.60 ±\pm 0.00
Matryoshka32 ±\pm 0.10.88 ±\pm 0.000.71 ±\pm 0.000.69 ±\pm 0.001.00 ±\pm 0.000 ±\pm 0.000 ±\pm 0.00
MP-SAE32 ±\pm 0.00.98 ±\pm 0.000.87 ±\pm 0.000.64 ±\pm 0.010.83 ±\pm 0.00517 ±\pm 130 ±\pm 0.00
Appendix
Table 17: Full results on GPT-2 Small across dictionary widths and sparsity levels. We evaluate three expansion factors ( 4×4\times , 8×8\times , 16×16\times ) and three target sparsity levels ( ℓ0≈32\ell_{0}\approx 32 , 6464 , 128128 ). We report achieved ℓ0\ell_{0} , reconstruction ( R2R^{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.
Expansionℓ0\ell_{0} bandMethodSparsityReconstructionConsistencyDictionary Utilization
ℓ0\ell_{0}R2R^{2}DCSMCCUsable atom frac.# Dup.# Dead
4×4\timesℓ0≈32\ell_{0}\approx 32ReLU31 ±\pm 0.30.84 ±\pm 0.000.90 ±\pm 0.000.82 ±\pm 0.010.92 ±\pm 0.00247 ±\pm 51 ±\pm 1.15
TopK32 ±\pm 0.00.90 ±\pm 0.000.84 ±\pm 0.000.78 ±\pm 0.001.00 ±\pm 0.0015 ±\pm 1.80 ±\pm 0.00
BatchTopK32 ±\pm 0.10.91 ±\pm 0.020.83 ±\pm 0.000.78 ±\pm 0.001.00 ±\pm 0.0013 ±\pm 1.00 ±\pm 0.00
Matryoshka32 ±\pm 0.00.90 ±\pm 0.020.71 ±\pm 0.000.67 ±\pm 0.001.00 ±\pm 0.000 ±\pm 0.290 ±\pm 0.00
MP-SAE32 ±\pm 0.00.96 ±\pm 0.000.84 ±\pm 0.010.51 ±\pm 0.000.72 ±\pm 0.00874 ±\pm 70 ±\pm 0.00
Appendix
Table 18: Full results on Pythia-160M across dictionary widths and sparsity levels. We evaluate three expansion factors ( 4×4\times , 8×8\times , 16×16\times ) and three target sparsity levels ( ℓ0≈32\ell_{0}\approx 32 , 6464 , 128128 ). We report achieved ℓ0\ell_{0} , reconstruction ( R2R^{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.
BackboneExpansionReLUConSDL
ℓ1\ell_{1}λ\lambdaTFISTAT_{\mathrm{FISTA}}TrefitT_{\mathrm{refit}}γsim\gamma_{\mathrm{sim}}
3232646412812832326464128128323264641281283232646412812832326464128128
CLIP ViT-B/16×4\times 41.4e-41.0e-47.5e-50.80.80.60.60.350.35303030303030101055551e-31e-31e-3
×8\times 81.5e-41.2e-48.4e-50.80.80.80.80.50.530303030303060601010555e-31e-31e-3
×16\times 161.6e-41.3e-41.0e-4111.251.250.750.755050303030305050101010105e-35e-32e-3
DINOv2-B/14×4\times 41.5e-48.5e-55.6e-51.761.761.081.080.740.743030303030301010101010105e-35e-31e-3
Appendix
Table 19: Hyperparameters for sparsity matching and ConSDL training. For each backbone, expansion, and target sparsity level ( ℓ0≈32,64,128\ell_{0}\approx 32,64,128 ), we report the ReLU ℓ1\ell_{1} coefficient and ConSDL sparse-coding penalty λ\lambda used for sparsity matching. For ConSDL, we additionally report the number of FISTA iterations TFISTAT_{\mathrm{FISTA}} , coefficient-refitting steps TrefitT_{\mathrm{refit}} , and the co-active similarity weight γsim\gamma_{\mathrm{sim}} .
β\bm{\beta}3K6K9K12K
ReLU ℓ1\ell_{1}Ours λ\lambdaOurs TrefitT_{\mathrm{refit}}ReLU ℓ1\ell_{1}Ours λ\lambdaOurs TrefitT_{\mathrm{refit}}ReLU ℓ1\ell_{1}Ours λ\lambdaOurs TrefitT_{\mathrm{refit}}ReLU ℓ1\ell_{1}Ours λ\lambdaOurs TrefitT_{\mathrm{refit}}
0.01.5e-40.3301.4e-40.5101.8e-40.4101.8e-40.55
0.51e-40.3301.2e-40.4101.8e-40.3101.6e-40.410
0.71.5e-40.35202.1e-40.25301.8e-40.35302e-40.310
1.02.1e-40.15302.3e-40.15202.1e-40.25302.1e-40.320
Appendix
Table 20: Hyperparameters for the synthetic experiments. For each ground-truth dictionary size and tail exponent β\beta , we report the ReLU sparsity coefficient ℓ1\ell_{1} and the ConSDL sparse-coding penalty λ\lambda and number of refitting steps TrefitT_{\mathrm{refit}} . ConSDL uses TFISTA=50T_{\mathrm{FISTA}}=50 , and γsim=1e-3\gamma_{\mathrm{sim}}=\texttt{1e-3} in all settings.
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