The Independence Prior of SAEs Fragments Visual Concepts

Tommaso Mencattini; Giorgos Nikolaou; Donato Crisostomi; Thomas Fel; Francesco Montagna; Emanuele Rodolà; Francesco Locatello

cs.AIOct 2, 2026

The Independence Prior of SAEs Fragments Visual Concepts

Authors: Tommaso Mencattini, Giorgos Nikolaou, Donato Crisostomi, Thomas Fel, Francesco Montagna, Emanuele Rodolà, Francesco Locatello

Organizations: ISTA · EPFL · Goodfire · Sapienza University of Rome ·

Abstract

Sparse Autoencoders (SAEs) decompose model activations into sparse combinations of interpretable dictionary atoms. Although SAEs are grounded in the Linear Representation Hypothesis (LRH), their objective smuggles in an additional prior: concepts across patches are treated as independent, an assumption clearly violated by natural images and by the activations they induce. We therefore specialize LRH to vision through the Markov-Field Linear Representation Hypothesis (MFLRH), which adds the missing spatial dependencies to the LRH assumptions. We thus propose Spatial-SAE as an amortized MAP estimator under the MFLRH. Spatial-SAE consistently outperforms standard SAEs in concept recovery and interpretability. Across four variants, it achieves a 96% average win rate on synthetic concept recovery and improves interpretability on DINOv2 activations, at a reconstruction cost concentrated in high spatial frequencies.

Figures & tables

Panel of Figure 1 — Figure 1: Spatial organization at three levels. Top: Baseline and Spatial-sae feature maps across four sparsifiers, illustrating feature support for grass, ceiling, staircase, and painting (reading left to right, then top to bottom). Bottom: PCA-RGB views of original and reconstructed activations, alongside ground-truth segments and strongest-atom assignments. Together, these views assess feature support, spatial atom organization, and reconstruction structure.

Figure 2: Patch activations retain spatial structure. (a) PCA-RGB and 3D PCA views of DINOv2 activations show spatial organization. (b) Variance explained by Manhattan and Chebyshev neighborhoods. Solid bars are within-image shuffled controls; hatching is the additional contribution of the true layout. (c) Measured offset-similarity kernels across depth. (d) Measured offset-similarity kernels from an lrh model and the mf-lrh model used in our synthetic experiments.

Figure 3: From independent codes to attractive–repulsive coupling. Left: lrh + sae treats patch codes

\mathbf{z}_{i}

independently. Middle: attractive coupling of the full codes promotes spatial persistence but can oversmooth local departures. Right: mf-lrh decomposes each code as

\mathbf{z}_{i}=\mathbf{s}_{i}+\mathbf{r}_{i}

, coupling neighboring smooth codes attractively and neighboring innovation codes repulsively.

Figure 4: Generating synthetic data. From left to right: initialization (Gaussian noise / sampled center), spatial geometry (graph diffusion and rectification / distance from center), support shaping (quantile thresholding / compact radial envelope), concept construction (peak normalization / bipartite masking), grouped aggregation, and the final joint field $\mathbf{S}+\mathbf{R}$ . In each parenthetical pair, the smooth-stream operation is listed first and the innovation-stream operation second.

Figure 5: Isolating the prior’s effect on synthetic data. We compare spectral regularization at (a)

\mu=3\times 10^{-4}

, (b)

\mu=10^{-3}

, and (c)

\mu=3\times 10^{-3}

with the same two-stream model at

\mu=0

. Color intensity shows F1 gain; the prior helps across all smoothness and innovation settings.

Sparsifier	MCC	Poly.	F1	$R^{2}$
TopK	100%	100%	100%	0%
BatchTopK	100%	100%	100%	0%
ReLU	85.7%	87.7%	85.0%	0%
JumpReLU	98.7%	96.0%	100%	100%

Table 1: Win rate of Spatial-sae against baseline sae s on matched pairs (same generator configuration and seed); Spatial-sae wins the three identification metrics, conceding reconstruction.

Figure 6: Concept alignment and spectral organization. (a) Normalized F1– $R^{2}$ frontier over Spatial-sae configurations. (b) Code spectrum relative to concept masks: zero denotes agreement with the masks, dashed curves the baseline, and solid curves Spatial-sae . (c) The baseline’s $R^{2}$ advantage over Spatial-sae (solid) and its innovation stream alone (dotted), by frequency. (d) TopK decoder atoms in a joint umap , colored by the graph frequency of their activation maps. (e–h) Normalized F1, Region Coherence, and automated map-quality and monosemanticity ratings. Error bars show one stddev over 5 seeds in (e–f) and $95\%$ class-bootstrap intervals from one seed in (g–h).

Appendix figures & tables16 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 7: Controlling smooth- and innovation-stream structure. Left: rows increase the coupling $\tau$ , making the raw field progressively smoother, while the two thresholded columns show how a larger support quantile $q$ reduces its active area. Right: rows increase the packet radius $r$ , expanding each innovation packet, while columns increase the number $m_{R}$ of active innovation concepts, producing more packets and greater overlap. Random draws and packet centers are held fixed across comparable panels.

Figure 8: Geometry sweep relative to B. Differences between DT and B across the

60

geometry cells. Columns show

\mathrm{L}

\mathrm{M}

, and

\mathrm{H}

; the upper row shows the polysemanticity difference and the lower row the F1 difference in percentage points. Each horizontal slice fixes

r

, while the in-plane axes vary

\tau

and

q

. Teal denotes an improvement, while the pale reference color denotes no change. The predominantly teal surfaces show that DT improves substantially over B throughout the grid.

Figure 9: Geometry sweep relative to D. Differences between DT and D across the

60

geometry cells. Columns show

\mathrm{L}

\mathrm{M}

, and

\mathrm{H}

; the upper row shows the polysemanticity difference and the lower row the F1 difference in percentage points. Each horizontal slice fixes

r

, while the in-plane axes vary

\tau

and

q

. Teal denotes an improvement, while the pale reference color denotes no change. The variation across

\mathrm{L}

\mathrm{M}

, and

\mathrm{H}

shows that the benefit over D depends on the trace strength.

Figure 10: Capacity sweep relative to B. Differences between DT and B across the

100

capacity cells. Columns show

\mathrm{L}

\mathrm{M}

, and

\mathrm{H}

; the upper row shows the polysemanticity difference and the lower row the F1 difference in percentage points. Each horizontal slice fixes

\nu

, while the in-plane axes vary

L_{0}

and

\rho

. Teal denotes an improvement and orange a deterioration.

Figure 11: Capacity sweep relative to D. Differences between DT and D across the

100

capacity cells. Columns show

\mathrm{L}

\mathrm{M}

, and

\mathrm{H}

; the upper row shows the polysemanticity difference and the lower row the F1 difference in percentage points. Each horizontal slice fixes

\nu

, while the in-plane axes vary

L_{0}

and

\rho

. Teal denotes an improvement and orange a deterioration.

	$\textbf{D}>\textbf{B}$	$\textbf{DT}>\textbf{B}$			$\textbf{DT}>\textbf{D}$			$\textbf{DT}>\textbf{D}>\textbf{B}$
	$\textbf{D}>\textbf{B}$	$\mathrm{L}$	$\mathrm{M}$	$\mathrm{H}$	$\mathrm{L}$	$\mathrm{M}$	$\mathrm{H}$	$\mathrm{L}$	$\mathrm{M}$	$\mathrm{H}$
MCC	$99\%$	$100\%$	$99\%$	$89\%$	$96\%$	$83\%$	$64\%$	$96\%$	$82\%$	$64\%$
Poly.	$99\%$	$100\%$	$99\%$	$89\%$	$96\%$	$80\%$	$64\%$	$94\%$	$79\%$	$63\%$
F1	$98\%$	$100\%$	$96\%$	$87\%$	$91\%$	$85\%$	$79\%$	$89\%$	$83\%$	$77\%$
$\mathbf{R}^{2}$	$13\%$	$10\%$	$1\%$	$0\%$	$15\%$	$1\%$	$0\%$	$4\%$	$0\%$	$0\%$

Appendix

Table 2: Ablating Spatial-sae components. Each entry is the strict win rate of the indicated comparison across the combined Geometry and Capacity Sweeps, rounded to the nearest integer. The no-trace

\textbf{D}>\textbf{B}

column isolates decomposition. The remaining groups introduce spectral regularization:

\textbf{DT}>\textbf{D}

isolates its incremental effect,

\textbf{DT}>\textbf{B}

measures the combined effect, and the final group requires both improvements.

Figure 12: Spatial annotations for concept evaluation. Images from ADE20K (left) and COCO-Stuff (right), paired with their semantic segmentation masks. The annotations identify the spatial extent of objects and scene regions, allowing us to evaluate whether learned features activate on the concepts they represent.

$\bm{\mathbb{E}}\mathbf{[L_{0}]}$	Model	TopK		Vanilla		JumpReLU		BatchTopK
$\bm{\mathbb{E}}\mathbf{[L_{0}]}$	Model	$R^{2}$	Norm. F1	$R^{2}$	Norm. F1	$R^{2}$	Norm. F1	$R^{2}$	Norm. F1
	Baseline	$61.1{\scriptstyle\,\pm\,0.2}$	$58.0{\scriptstyle\,\pm\,1.4}$	$40.8{\scriptstyle\,\pm\,0.8}$	$51.5{\scriptstyle\,\pm\,1.0}$	$38.4{\scriptstyle\,\pm\,0.7}$	$52.9{\scriptstyle\,\pm\,0.7}$	$57.9{\scriptstyle\,\pm\,0.9}$	$54.2{\scriptstyle\,\pm\,0.5}$
$8$	Spatial-sae	$45.1{\scriptstyle\,\pm\,0.1}$	$67.0{\scriptstyle\,\pm\,1.2}$	$27.7{\scriptstyle\,\pm\,0.5}$	$55.7{\scriptstyle\,\pm\,2.0}$	$30.4{\scriptstyle\,\pm\,0.3}$	$57.2{\scriptstyle\,\pm\,1.2}$	$43.9{\scriptstyle\,\pm\,0.3}$	$60.2{\scriptstyle\,\pm\,1.6}$
	Baseline	$70.4{\scriptstyle\,\pm\,0.2}$	$57.5{\scriptstyle\,\pm\,1.1}$	$53.7{\scriptstyle\,\pm\,0.7}$	$50.5{\scriptstyle\,\pm\,1.0}$	$52.2{\scriptstyle\,\pm\,0.8}$	$52.1{\scriptstyle\,\pm\,0.5}$	$69.2{\scriptstyle\,\pm\,0.9}$	$55.9{\scriptstyle\,\pm\,0.7}$
$16$	Spatial-sae	$52.4{\scriptstyle\,\pm\,0.3}$	$65.4{\scriptstyle\,\pm\,1.9}$	$39.1{\scriptstyle\,\pm\,0.7}$	$55.5{\scriptstyle\,\pm\,2.0}$	$41.4{\scriptstyle\,\pm\,0.3}$	$56.7{\scriptstyle\,\pm\,1.6}$	$52.4{\scriptstyle\,\pm\,0.4}$	$60.3{\scriptstyle\,\pm\,2.0}$
	Baseline	$78.1{\scriptstyle\,\pm\,0.2}$	$56.1{\scriptstyle\,\pm\,0.7}$	$66.1{\scriptstyle\,\pm\,0.6}$	$49.1{\scriptstyle\,\pm\,0.7}$	$67.5{\scriptstyle\,\pm\,0.5}$	$49.8{\scriptstyle\,\pm\,0.3}$	$78.0{\scriptstyle\,\pm\,0.2}$	$54.9{\scriptstyle\,\pm\,0.7}$

Appendix

Table 3: Sparsity-budget ablation on ADE20K . We measure reconstruction with

R^{2}

and concept recovery with Normalized F1, relative to the corresponding linear probe. Entries are mean

\pm

standard deviation over five seeds and are reported as percentages. We use an expected sparsity budget of

16

: compared with a budget of

8

, it improves baseline reconstruction for every sparsifier while retaining at least

98\%

of each baseline’s best Normalized F1, and is itself optimal for BatchTopK. Spatial-sae has higher Normalized F1 throughout the sweep.

Controls for smooth-field estimation. The static control (left) removes neighbor-based estimation, while the convolutional control (right) aggregates neighboring patches without the content-dependent weighting used by context attention. Both retain the same two-stream innovation and decoding structure, allowing the ablation to isolate how the smooth field is estimated.

Panel of Figure 17 — Rubric 1: Monosemanticity scoring rubric. Scores measure whether one coherent concept explains the highlighted regions across the panel.

Panel of Figure 18 — Rubric 2: Map-quality scoring rubric. Scores measure the coverage, tightness, and consistency of the highlighted regions relative to the concept inferred by the rater.

	Two Streams $-$ Single Stream			Norm. F1
$\mu$	$\Delta R^{2}$	$\Delta$ Norm. F1	$\Delta$ Region coh.	Win rate $\uparrow$
$10^{-3}$	$-0.80\pm 0.63$	$\mathbf{+6.12}\pm 1.86$	$-2.90\pm 4.07$	$\mathbf{100\%}$
$10^{-2}$	$-7.25\pm 3.19$	$+0.23\pm 3.33$	$-4.49\pm 3.79$	$57\%$
$10^{-1}$	$-5.12\pm 2.59$	$-5.60\pm 4.30$	$+5.51\pm 12.21$	$14\%$

Appendix

Table 4: Effect of code decomposition at matched regularization strength. Static TopK on ADE20K ,

\mathbb{E}[L_{0}]=16

. The single-stream control applies the attractive Laplacian penalty to the complete code; the two-stream model additionally represents an innovation field with the repulsive penalty. Differences are two-stream minus single-stream at the same

\mu

, reported as mean

\pm

sample standard deviation across seven budget allocations and two initializations each.

R^{2}

and Normalized F1 changes are in percentage points; coherence changes are multiplied by

100

. The last column reports the percentage with higher Normalized F1, rounded to the nearest integer. Shading marks the selected regularization strength.

	Transition	Reconstruction		Normalized F1		Coherence
Regularization Strength $\mu$	$0\to 10^{-3}$	$-5.52$ $[-5.77,-5.09]$	$0\%$	$+6.53$ $[+4.77,+6.94]$	$\mathbf{100\%}$	$+2.51$ $[-2.69,+4.10]$	$86\%$
	$10^{-3}\to 10^{-2}$	$-25.91$ $[-29.85,-23.46]$	$0\%$	$+6.75$ $[+2.60,+8.61]$	$86\%$	$+0.13$ $[-1.23,+2.10]$	$64\%$
	$10^{-2}\to 10^{-1}$	$-27.94$ $[-28.20,-27.23]$	$0\%$	$-43.14$ $[-47.46,-35.75]$	$0\%$	$+16.04$ $[+7.95,+24.17]$	$\mathbf{100\%}$
Smooth Field Estimator	$\mathrm{S}\to\mathrm{C}$	$-3.52$ $[-6.90,-1.78]$	$0\%$	$-0.34$ $[-1.79,+1.01]$	$43\%$	$+2.23$ $[+1.21,+3.00]$	$86\%$
	$\mathrm{S}\to\mathrm{CA}$	$-2.55$ $[-5.16,-1.48]$	$0\%$	$+1.95$ $[+0.09,+5.11]$	$71\%$	$+2.69$ $[+2.14,+3.88]$	$93\%$
	$\mathrm{C}\to\mathrm{CA}$	$+0.85$ $[+0.42,+1.55]$	$79\%$	$+2.67$ $[+1.51,+5.23]$	$86\%$	$+0.79$ $[+0.16,+0.94]$	$79\%$

Appendix

Table 5: Effects of regularization and smooth-field estimation. TopK on ADE20K ,

\mathbb{E}[L_{0}]=16

. Differences are destination minus source, matched by budget allocation and initialization (

14

pairs). For each metric, the left column reports the median change (

\Delta

), with its interquartile range beneath; the right reports the win rate (WR), the percentage of positive differences rounded to the nearest integer.

R^{2}

and Normalized F1 changes are in percentage points; coherence changes are multiplied by

100

. Regularization transitions use the static encoder; estimator transitions fix

\mu=10^{-3}

. S: static; C: convolutional; CA: context attention.

Figure 13: Class-best feature maps on ADE20K . For each sparsifier (labels at right), two representative classes (labels at left) compare the baseline (left block) with Spatial-sae (right block). Each model contributes its highest-concept-F1 feature for the labeled class. The four displayed images are selected for high per-image F1 of the baseline feature and then held fixed across models, so the selection deliberately favors the baseline. Warmer colors indicate stronger activation.

Figure 14: Class-best feature maps on COCO-Stuff . For each sparsifier (labels at right), two representative classes (labels at left) compare the baseline (left block) with Spatial-sae (right block). Each model contributes its highest-concept-F1 feature for the labeled class. Images are selected for high per-image F1 of the baseline feature and then held fixed across models, so the selection deliberately favors the baseline. Warmer colors indicate stronger activation.

Figure 15: Quantitative results on COCO-Stuff . (a) Normalized F1 against reconstruction $R^{2}$ ; points show training seeds and arrows connect model means. (b) Base- $2$ log ratio between code and concept-mask spectral density; zero denotes agreement, with dashed baseline and solid Spatial-sae curves. (c) The baseline’s $R^{2}$ advantage over the combined Spatial-sae code (solid) and its innovation stream alone (dotted), resolved by graph frequency. (d) TopK decoder atoms in a joint umap , colored by activation-map frequency. (e–h) Normalized F1, region coherence, automated map quality, and monosemanticity. Error bars show one standard deviation across five training seeds in (e–f) and $95\%$ class-bootstrap intervals in (g–h).

Sparse Autoencoders

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

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Vision Sparse Autoencoders (SAEs) have become a popular tool in Mechanistic Interpretability due to their presumed ability to disentangle complex features learned by a model into monosemantic concepts. Despite their growing popularity, evaluating their interpretability remains an active topic of research. The bedrock motivating the adoption of SAEs is the Linear Representation Hypothesis (LRH), which claims that polysemantic features can be projected onto a (near) orthogonal basis of sparse, human-understandable representations. Yet, most current frameworks evaluate proxies such as the sparsity of SAE features or the coherence of the inferred dictionary, implicitly assuming that these reflect alignment with human perception. In this paper, we provide empirical evidence that measuring the interpretability of SAE concepts is more difficult than these proxies suggest. To this end, we adapt the Autointerpretability Score (AIS) - previously shown to align with human judgments in Natural Language Processing - to vision tasks and validate our approach in a dedicated user study. We evaluate SAE concept quality using both standard metrics and our adapted AIS. We find that established interpretability metrics for SAEs correlate neither with one another nor with AIS, indicating that no single reference-free metric, whether grounded in the LRH or not, is sufficient for verifying the interpretability of vision SAEs. We argue these findings support recent calls for more verifiable, ground-truth-anchored design and evaluation of explanation methods.

Abstract

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