Features Emerge as Discrete States: The First Application of SAEs to 3D Representations
Authors: Albert Miao
Chenliang Zhou
Jiawei Zhou
Cengiz Oztireli
Venue: ICLR 2026 Poster (2026)
Keywords: sparse autoencoders
mechanistic interpretability
computer vision
Primary area: interpretability and explainable AI
TL;DR: We present the first application of SAEs to the 3D domain, framing model internals as a state-based feature space governed by phase transitions.

Abstract: Sparse Autoencoders (SAEs) have found human-interpretable features in LLM activations, clarifying how LLMs transform input to output. However, they have rarely been applied outside of text, limiting explorations of feature dynamics. We present the first application of SAEs to the 3D domain, analyzing the features found in 53k 3D objects encoded by a state-of-the-art 3D reconstruction VAE. We observe that the model encodes discrete rather than continuous features, leading to our key finding: the model's feature activations approximate a discrete state space, driven by phase-like transitions. Through this state space framework, we address three otherwise unintuitive behaviors — the preference for positional encoding features, the sigmoidal relationship between feature ablation and reconstruction loss, and the bimodal distribution of phase transition points. This final observation suggests the model redistributes superposition interference to prioritize the high-importance features. Our work not only catalogs and explains unexpected feature dynamics, but also provides a framework to explain the model's learning dynamics. The code is available at https://feature3d.github.io/Dora-SAE/.

Published as a conference paper at ICLR 2026
FEATURES EMERGE AS DISCRETE STATES: THE FIRST APPLICATION OF SAES TO 3D REPRESENTATIONS
Albert Miao'
Chenliang Zhou'
Jiawei Zhou?
Cengiz Oztireli'
ABSTRACT
Sparse Autoencoders (SAEs) have found buman-interpretabe features in LLM activations, clarifying how LLMs transform input to output. However, they have rarely been applied outside of text, limiting explorations of feature dynamics. We present the first application of SAEs to the 3D domain, analyzing the features found in 53k 3D objects encoded by state-of-the the-tthe-rt 3D reconstruc- tion VAE. We observe that the model encodes discrete rather than continuous features, leading to our key finding: the model's feature activations approx- imate discrete state space, driven by phase-like transitions. Through this state space framework, we address three otherwise unintuitive behaviors the preference for positional encoding features, the sigmoidal relationship between feature ablation and reconstruction loss, and the bimodal distribution of phase transition points. This final observation suggests the model redistributes super- position interference not only catalogs and explains unexpected feature dynamics, but also provides a framework to explain the model's learning dynamics. The code available at attps://Eeatur github. io/ io/Doro-DSa SAE/.
prioritize the high-importance features. Our work
INTRODUCTION
Interpretability research has recently focused feature decomposition: the process of translating internal model activations into human-readable concepts. To this end, studies have used sparse autoencoders (SAEs) as dictionary-learinng tool (Bricken al. 2023). These publications find mantically interpretable pipelines in foundational models for a myriad of tasks, including arithmetic (Lindsey et al. 2025), protein characteristics (Garcia & Ansuini 2025), and image-text relationships (Yan et al. 2025). We cite further studies in Appendix A.
The success of these methods suggests that internal activations function as a concept space, where independent feature vectors point in different directions (Elhage et al. 2023). Furthermore, models can include a number of feature vectors far greater than the cardinality of the latent space through process called superposition, at the cost of interference (Hänni et al. 2024). However, research in feature decomposition is lacking in two key areas. First, the scope of data domains has been limited -imited-reeee recent feature decomposition techniques. particularly SAEs, have mostly focused text, raising questions on their generality. Second, existing research tends to empirically discuss what features contribute to model's performance, rather than explore why how these features were chosen by the model. In short, current SAE-based studies are text-focused and descriptive, rather than explanatory. We believe the field should aim for aggaiicagnostic framework explain how features are learned and organized.
To the best of our knowledge, we are the first work to apply an SAE to 3D data. We analyze 53k 3D objects sampled from Objaverse (Deitke et al. 2022) encoded with Dora-VAE (Chen et al. 2024). 3D data is well-suited for our purposes, because data points 3D space are drawn from an unordered, continuous domain. This is qualitatively distinct challenge to text. In addition, a) 3D data is visually obvious when detected features have semantic meaning, b) existing datasets have variety of recognizable objects with unique semantic combinations, c) many industries that use 3D data rely on Al tools, and d) we avoid datasets that are noisy, synthetie, homogeneous, sparse, and/or constructed for toy experiments.
University of Cambridge 2 Stony Brook University Corresponding Author

---

Published as a conference paper at ICLR 2026
Dora-VAF
3D Model
SAR
Feature Decomposttion (3.)
Enceder
Latent Vectors
H-2
Sumpled
Feature Vectors
Point Cloud Eniloldingg
Sanylee Embeediligs
Isolated Feature
e-ded
Vectors
Decoler
Point Quarko
Latesi Vastes
Onir Abrenso
Ocmppnyy Fiddi
Reconstruction
Festure Ablaionn and Addition (5.2)
Figure Our feature decomposition pipeline. Dora-VAB a 3D reconstruction model, encoding 3D objects M latent vectors each. We apply our SAE to these latent vectors, decomposing each vector to a linear combination of features. We can visualize the effects of an individual feature by plotting the its presence in each latent vector (Section 3.1) modifying the latent vector and observing the effects on the reconstruction (Section 3.2).
We extract numan-interpretatie features from latent vectors and report their semantic meaning. We also observe that Dora-VAK learns unorthodox feature representations. Namely, positional fea- tures are represented discretely rather than continuously, transition points of high-impact features follow a unimodal distribution, and transition points of low-impact features follow a bimodal distri- bution.
Where previous work prioritized cataloging features, we study the underlying learning dynamics. We deconstruct the optimization step and identify two terms that independently attend the pres- ence and identity of individual features. The dichotomy between these terms offers explanations to the learning behaviors of Dora- VAE highlighted earlier - particularly, we suggest the unimodality of high-impact transition states is explained through the presence term, and the bimodality of low- impact transition states is explained through the redistribution of superposition interference. This framework is potentially universally applicable, intended to provide context for future interpretabil- ity work to discuss how concepts form a discrete state space.
We substantiate framework through verifying experiments. First, substantiate learned features through targeted feature intervention. We show the clear semantic effect a feature has on reconstructed output. Second, observe and discuss several counter-intuitive behaviors that explained by our framework. We perform 848k independent feature interventions across set 53k 3D models observe patterns of changes in loss.
Ultimately, we provide the first application of SAES to 3D data, and explain unusual properties of the feature space through a novel framework. In future work, hope evaluate the universality and consistency of our framework on different models and modalities, as well as further investigate the feature dynamics models in the 3D domain.
FEATURE DECOMPOSITION PRELIMINARIES
We describe model the composition of functions f x and g : y, where is latent vector. Given a dataset ((x, y.) N the training objective is to optimize parameters 0, and 0 by minimizing loss function L over the dataset:
min Σ 1(()))
Several works have suggested theoretically (Bengio et al. 2013) and empirically (Elhage et al. 2022) that latent representations in both humans and models be viewed sets of semantic ideas
2

---

Published as a conference paper at ICLR 2026
borrowing terminology proposed by Kim et al. (2018), we denote these ideas set of vectors E, and the space spanned by these vectors as E. These works suggest that z E E and that z can be decomposed to linear combination of vectors from E:
= f(0 ((e)))a(0f;:))) where E [e1, .. eal C E, a (ai,..., an)
Abstractly, we state that model's latent space interprets a given input x as set of scalars a modifying a set of learned features E. We refer to aj as the presence of feature j and ej as the identity of feature j.
Even if an input is out of the domain of the training dataset, the model still attempts, and likely fails, to frame the input in these features. For example, image adversarial attacks use noise that are completely nut-of-distribution but classification model must still estimate the presence of each feature ej (Gorton & Lewis 2025). If the resulting feature presences are similar to an in-distribution input, the image is misclassified.
Recent LLM studies (Bricken et al. 2023) use a sparse autoencoder (SAE) to approximate this decomposition with the assumption that a is sparse; that is, the number of feature vectors in E (i.e. the dictionary size) is large compared to the number of features needed to represent a vector in E. Given collection of input and their corresponding latent vectors
attempt to approximate and E through the following parametrization, known as BatchTopK SAE (Bussmann et al. 2024);
For fixed 0y:
xx) R Enc(z)
where Enc(z) TopK (W Fac z + - Rray
(3)
E(O,) Wee
where 2 De b Doc
where TopK selects the top k nonzero values across the batch. The linear weight matrix Dec approximates the set of features E, forming an overcomplete dictionary. Thus, Enc(z) is sparse representation of z using Wee as the set of feature vectors. We train with standard reconstruction loss, alongside an auxiliary loss based on the reconstruction from dead features 2dead is the reconstruction using only dead features, and / is scalar hyperparameter (Gao et al. 2025).
L(0SAE) Lrecon (z, 2) + BC (z, Zetead)
Crecon (z, = Σ || 1/
(4)
2.1 THE LEARNING DYNAMICS OF FEATURE DECOMPOSITION
Existing applications of SAEs LLMs (Templeton et al. 2024) have shown extracted feature vectors that have clear semantic meaning. However, this begs the question: Why were these features learned by the model?
We examine the learning dynamics of a by deconstructing the gradient step. Optimization of f 0g performed using gradient descent:
0, 0 0, Ve, C, 0g x 0, Ve,.C
(5)
We take the gradient of Ciorel with respect to fy. substituting with our definition from Equation 2.
2C
0y
Oz
Oz
Σ 1=1
"
Oo,, (0,x(0);)) e,(0,) + (0,g))))) a0
de,
where
(6)
08,
Oz
00,
00,
08,
We that parameters are updated with respect to features in two ways: for each feature j. Ve, a modifies the magnitude and frequency it fires, and Ve,e, modifies the information carried. This framework allows us explain the following behaviors, through the lens of feature-based learning dynamics.
The model learns features with discrete, state-like presences rather than those with continu- ous spectrum of presences. We see that the a, term controls the learning rate for Vo, ej, suggesting that models prefer to learn features ej that naturally have high o.j. (Section
3

---

Published as a conference paper at ICLR 2026
High-impact features have phase transition points that form a centered, unimodal distribution. A transition point is where DC is highest, greatly affecting Ve,aj. The model is incentivized to 22 represent the presence of each feature in states that are far from this point. (Section 5.1)
Low-impact features have phase transition points that form a symmetric bimodal distribution. When condensing high-dimensional information into low-dimensional space, feature presences affected by interference from superposition. While transition points would typically be centered as above, the model minimizes damaging superposition by perturbing presences of low-impact fea- tures. (Section 5.2)
3 APPLICATIONS TO 3D RECONSTRUCTION
To verify our analysis, we apply an SAE to Dora-VAE (Chen et al. 2024). Dora-VAE is a Variational Autoencoder (VAE) that encodes point clouds Pa sampled from 3D models to condensed latent representations. These representations are then queried for diffusion-based reconstruction of the initial geometry. Rather than global latent for each shape, Dora-VAE selects set Pc of M poin features from Pa using furthest point sampling (FPS), which is passed through several cross- attention layers alongside Pa. This forms a set C of processed point features.
Pc FPS(Pa) C CrossAttn(PosEnc(PPR) PosEnc(Paa)
We take the provided Dora-VAS VAE network, pretrained objects from Objaverse-XL. These encodings form a dataset of pre-KL embedding network states. The number of latents in C is determined by the number of points initially sampled; we record set for M = 4096, where each pre-embedding is size 128.
subset of Objaverse, and encode 53k
After encoding, for each pre-embedding, Dora-VAL isolates mean μ4, E RR44 and variance or 8 by chunking.
C= (1r Thus, Vi (1, MY.j E (1,2, ...y 64), the KL embedding is:
Zi.j = Mi.j + Di.j
(8)
where e ~ NN, This embedding is fed through the decoder before querying for occupancy. Here, the term latent with respect to Dora-VAE will refer to the network state post-KL embedding.
3.1 SAE ON DORA-VAE
The dataset for our SAE is constructed from these recorded pre-embeddingss. Each epoch, through KL, we sample each recorded pre-embedding for new latents. Our latent space is thus extremely well-defined, as each epoch of training has 217 million newly sampled latents. In addition, since these latents are point cloud features initially downsampled from Pa. each latent will correspond to a point of the initial point cloud sample. This relationship allows us to interpret a feature based off of the position or structure of points with high presence for that feature.
We train our BatchTopK SAE with M = 4096, codebook size = 512, threshold k = 8, and = 0.125. We use batch size of 327680 latents, randomly selected regardless of which 3D model produced each latent. We use the Adam optimizer with an initial learning rate of le-3 and train for ten epochs. The model was trained on a single A100 and took hours to train. We also present metrics for variations on codebook size and threshold in Appendix B.
We highlight the qualitative performance of our feature extraction in Figure 2. For each encoded 3D object, we obtain set of latents {z, J.M. and a; for each latent by passing it through the SAE. We plot the M latents as points from their initially sampled positions Pa. Finally, to examine a feature color each point of each latent based off the presence Oi.j.
Most features display positional information along a single axis. Notably, features appear state-like, and store information in binary manner. Features emerge at striped intervals across models in a manner akin to positional encoding, suggesting latents form a discrete representation. In other words, feature doesn't 't have range of possible values ("As feature j increases, the point travels
4

---

Published as a conference paper at ICLR 2026
Feature
Model
Feuture 71
Feature 166
Feature 452
Figure 2: Overview of Dora- VAE features. Each point's color represents the presence of feature that point. Visually, most features congregate in stripes along single axis. This suggests that continuous position is represented by set of discrete elements, which each activate separate regions. We show further examples in Appendix C.
further along the axis") ") instead, feature tends towards of two states ("Iffeature j is present, is within this region" "). We discuss the discretization features further Section 4.
These positional features highly visually interpretable due the 3D medium. We that such features are applicable across all models, and activate with significant sensitivity and specificity. Some features, although they highly present across the model, have meanings that difficult interpret through observation. We can instead intervene on these features to determine their purpose and importance.
3.2 FEATURE ABLATION AND ADDITION
possible these features are simply vestiges of correlations between the sampled points; points that share close coordinates may simply propagate similarly across the encoder. To disprove this, and demonstrate these features are meaningful internal representations, we examine the downstream effects of modifying latents along feature axes.
We intervene on features through ablation and addition based on SAE decoder weights. our pipeline, inputs encoded by Dora-VAE set of latent vectors {z, Nei We recall Eq. 2; to visualize the effect modifying feature j on the reconstruction, we want to approximate modified set of latents such that:
z E where a' as, o,, (1 t) as
where (1 f) is the proportion of the original presence, set externally. Rather than rely on the reconstruction provided by our SAE, we modify the latents with the decoder weight for feature j.
Ablation:
2/50 RJ 2; - Enn((:) dec
MM Vz, 1
(10)
Addition: 2 2; + 0 weo

---

Published as a conference paper at ICLR 2026
100%
67%
33%
0%
Feature 313
Festure 363
Figure 3: Top: Example ablation on feature 313. @ This feature primarily attends to points on the positive end the z-axis. As the feature removed, shapes disappear and appear spontaneously, rather than moving along the object (.. This suggests the feature represents discrete region in space, rather than continuous range of positions. Bottom: Example of feature 363 ablation and feature 426 addition. Both features attend to regions along the y-axis. ! Shapes that had their region fixed by feature 363 are moved to regions defined by feature 426. In addition, they preserve their local structure. See Appendix B.2.
During ablation, the portion of feature removed is scaled by value t; here, f = 1 implies the feature completely removed. During addition, feature j is added by amount manually set a'.- Each modified set of latents is passed through the the Dora- VAE decoder to be compared to the original model. record the mean squared error (MSE) of the decoded reconstruction.
ablation of feature 313. As shown by the red points in @ this feature primarily attends to points on the positive end of the z-axis, with small region on negative end. When the feature is ablated, shapes whose position relied on it are rendered elsewhere this indicative of a causal relationship between this feature and the shape's 's position. In addition, shown in D rendered points appear spontaneously, rather than moving across the model. This again suggests that features represent discrete states, and presences do not have a continuous range of information.
The top of Figure 3 demonstrates
At the bottom of Figure 3, we demonstrate an ablation of feature 363 alongside an addition feature 426. Rather than applying constant presence of feature 426 all latents, instead replace every

---

Published as a conference paper at ICLR 2026
Example ARCs
Quantiles ARCs
Loss per Arg. Presence
Loss per Density
1.5
1.5
RICL
Scalad Dallornee in Lose.
Scalad Dallorane in Lose
2.0
1.0
Deaelly
0.6
Dauely
0.6
0.5
0.5
0.2
0.2
0.0
0.0
0.0
0.0
7 /xolue
7 +xalic
Lous Diffceenoe
Lous Diffeeenco
Intermed. Loss Loss Diff
Slopes By Loss Didl
Min Slope Loss Diff
Max Slopes By Loss Difl
4.0
2.0
0.8
1.5
Daneyy
Danely
Denacy
Danndy
0.0
1.0
2.0
0.5
0.2
1.0
0.2
0.0
0.0
0.0
0.0
0.0
0.5
Intermediote Losses
Stogs
Point of Minimum Siope
Standwrtoodd Mix Slops
Figure 4: Top left: Examples and quantiles of ARCs. ARCs display almost sigmoidal behavior, with a varied transition point. Top right: Correlations of loss with average value and feature density. Bottom: Various experiments to demonstrate discretization. ARCs will typically have stagnant MSE near the beginning and end, and change most rapidly in small interval.
presence of feature 363 with an equal presence of feature 426.
Vz: (z:)
2 R z: Enc (. 3363 w dec
363 + Enc )an3 (2))33) W 426
dec
Note that, because both features 363 and 426 attend to positions on the y-axis, points displaced by the removal of feature 363 are anchored by the addition of feature 426. Rendered shapes preserve their form even after moving. We discuss further observations in Appendix B.2.
4 DO FEATURES SHOW STATE-BASED BEHAVIOR?
significant portion of features in Dora-VAS are dedicated to representing the position of the la- tent. Intuitively, one would assume that these features should be continuous, as points are relatively uniformly distributed across 3D space, Each latent's position could be represented by only three features, with others for additional fidelity. Despite this, the model chooses to represent features discretely; if the feature has a high presence, the position is within a defined region. This method of representation is akin to binary positional encoding. Some features make the similarity more explicit by representing a set of multiple regions across an axis, rather than a single one.
To verify whether these features are truly discrete, lations over our dataset and measure the change in loss. As above, we pass the M latents of 53k 3D objects through our BatchTopK SAE with h = 8 and codebook size 512. Each 3D object thus has a set of presence vectors {ax; }/d where each 04 E RR512 shows the presences of 512 fea- tures and has 8 nonzero values on average. For each 3D object, we randomly select 16 features to intervene on, preferring features that are present in more latents. We perform each ablation with t E {0.00, 0.05, 0.10. ..., 1.0], recording the MSE of the decoded reconstruction for each t. We evaluated 848k ablations in total.
perform series of systematic feature ab-
The model's response to feature ablation displays interesting recurring behaviors. Given our set {a we define the feature density of feature j as 1 Σi" 1aa,,, * 0). and the average pres- ence of feature as 1 M
M - aij- We also define the impact AL of an ablation as the difference in MSE between t 0 and t = 1. Figure 4 shows kernel density estimations (KDEs) for these properties of each feature ablation. We note that
of ablations shows wide variety of im-
7

---

Published as a conference paper at ICLR 2026
pact, feature density, and average presence. In addition, impact is positively correlated with both the feature density and average presence.
We also plot several ablation- reeppionn- curves (ARCs). Each curve represents single ablation, and shows the change in MSE as t increases. We normalize MSE such that the plotted error at t = is O, and at f = is 1. We also record the transition point of ablation as the value of t when the normalized MSE is 0.5. Note that the ARCs do not show a linear relationship between change in latents and MSE. Rather, they exhibit variable curvature, with two inflection points initial changes in loss are below our projected linear growth, then accelerate at the transition point, before again slowing down.
We find ARCs with greater impact exhibit more discrete behavior. To demonstrate this, we group ARCs together based on AL and perform four experiments. First, for each group of similar AL, we plot a KDE of all normalized MSE for 0.05 S S 0.95. Notably, as impact increases, intermediate MSE values tend to cluster towards the initial and final MSE. Second, we analyze the maximum slope of each ARC to determine if it is an outlier in the distribution of slopes. For each ARC, we estimate slope as the difference of normalized MSE between every two consecutive / values. We then z-score the greatest slope of each ARC relative to the distribution formed by all slopes in the group, and plot the average for each group. As the AL of an ARC increases, the greatest slope of the ARC trends further from the group distribution, suggesting the transition point is more well defined as impact increases. Third, we find the point of flattest slope for each ablation, record the value of and plot KDE of these values. We that points the beginning and end have typically the flattest slope. Finally, we note that the distribution of slopes across ARCs leans further left as AL increases, showing that ARCs are typically flatter, with sharper jumps, when impact increases.
The discretization of features can be explained through the learning dynamics defined in Section 2.1. The signal the identity of feature j. Ve,e, ey, is scaled by the presence aj. Thus, the identity of feature j is most influenced when aj is high, while, when ay is low, the signal to identity is diluted by other signals for features with higher presence. We further discuss this intuition Appendix B.3.
5 THE BIMODALITY OF TRANSITION POINTS
If we interpret feature activations as discrete state space with distinct phase transitions, we can follow up by investigating when these phase transitions oceur. We plot KDEs of two properties transition points and the points of greatest slope for all ARCs in Figure 5. Surprisingly, both distributions are bimodal.
We further investigate this behavior by again grouping ARCs by AL and plotting the KDEs of transition points and points of greatest slope for each group. From this figure, we see that the transition points of high-impact ARCs form unimodal distribution around the center (t 0.5), while the transition points of low-impact ARCs form a bimodal distribution roughly symmetric about this center.
We then further group ablations based on which feature j is removed and repeat the same investiga- tions as above, Again, for high-impact ablations, each feature has a peak near the center. However, low-impact ablations are no longer bimodal, and instead there is instead a single peak that strays from the center. Some fall closer to the beginning, while others are nearer the end. Thus, it is only when all ARCs are aggregated together, regardless of feature, that we observe bimodality of transition points in low-impact features.
5.1 UNIMODAL TRANSITION POINTS OF
ABLATIONS
We explain the distribution of high-impact transition points using the learning dynamics defined in Equation 6. Because our feature activations, especially high-impact ones, approximate discrete behavior, we can consider a feature to be on (high presence) or off (low presence). During the gradient step, the effect on Ve, aj is scaled by 3% This gradient, by definition, reaches peak near the transition point. Because the rapid change in loss associated with the transition point, Vo, a; is incentivized to adjust ay such that both on and off states of the feature
Oc
distance from the transition point. The magnitude of 8C at the transition point is not necessarily equivalent for when 8g
8

---

Published as a conference paper at ICLR 2026
Transition Points By Loss Diff
Max Slope Pt By Loss Diff
10 ba Ml Le
10r is Ml0 di
10 AlR) Lit
100-ft i MDO Lit-tt
Danity
Donsity
1.5
1.0
0.5
0.5
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Transition Point
Point of Maximum Slope
36 Pts Loss Diff
Ft T Loss Diff
PC76 T Pts By Loss Diff
329 T By Loss Diff'
Densty
Donsity
Dandly
0.0
0.5
0.0
0.5
1.0 0.0 0.5 1.0
0.0
0.5
1.0
Transiiion
Tiaasition
Figure Top: Distributions of transition points and points of maximum slope over all ARCs, grouped by AL. ARCs with greater A form unimodal distribution of transition points centered about 0.5, while those with less AT. have a symmetrie, bimodal distribution. Bottom:
Distributions of transition points of individual features, grouped by AL. ARCs with greater ÁL still form somewhat centered, unimodal distribution, but those with less AL also unimodal, further from the center. This polarization across individual features causes the bimodal distribution when considered in aggregate.
feature j is on and off - however, over many different transition points, we can assume it is roughly symmetric. Thus, the overall distribution of high-impact transition points located the center.
5.2 BIMODAL TRANSITION POINTS OF LOW-IMPACT ABLATIONS
The bimodality of low-impact transition points is a more complex property. When examining dividual features, the distribution of low-impact transition points forms unimodal peak that drifts away from the center. This behavior is not caused by a weaker ac 8u the transition point, which would've only increased the variance of the distribution. Instead, the peak itself is off'set to the left or right - as a polarizing effect drives transition points away from the center as AL decreases.
We speculate that this effect is caused by variable offset applied to the feature presence, and provide a visual aid in Figure 6. We suggest that as impact decreases, the larger this offset becomes. this way, the transition point is moved earlier or later in the ablation. This effect does not make the feature presence estimation less accurate; again, in that case, we would see the low-impact distribution have greater variance, but retain the same center. Rather, the peak itself moves. We hypothesize that this offset caused by the model learning to redistribute interference from superposition.
Superposition occurs when high-ddieesioonll features constrained causing interference between features in previously distinct dimensions. (Elhage et al. 2022). We show another visual aid in Figure 7. Suppose the model has determined to represent a latent through high-impact feature 1, low-impact feature 2, and a set of several other features *. We refer to the presences as 011, Q2 and a.. As shown at the top of Figure due superposition, the features add interference in the direction of (1 and a2 We suggest that the model selects features such that feature a low-impact feature, is affected by this superposition more than feature high-impact feature. This causes the offset shown at the bottom of Figure 7 - while the transition point of feature I still centered, the transition point of feature 2 moved.
low dimensional space,

---

Published as a conference paper at ICLR 2026
Ablation I, High Impact
Ablation Low Impact
Ablation 3, Low Impact
Zero Prousee
Ioauston Pore
Uransttion Phrne
Tranction Poins
Offlct
Offict
Origial Poneass
Drfbct Presence
Figure 6: Left: A visual aid of high-impact ARC. As feature presence is ablated, the loss (MSE) increases. It is fastest when passing through the transition point, then stagnates. Center and Right: A visual aid of two low-impact ARCs. We suggest that low-impact ARCs are affected by a polarizing offset. the presence is decreased by the offset, the relative transition point is moved left. If increased, the point is moved right.
Even though AL has correlations with other variables, they are not strong enough to cause a third variable contamination that affects SAE estimations of presences (see Appendix B.4). Thus, we can reasonably conclude the offset is present even be- fore the latent is passed to the SAE. In other words, we specu- late the model itself is actively preserving high-impace features by passing interference features that are lower impact.
Tranciion
Odter
Figure 7: ference affects feature presences. We suggest this offsets low-impact features, shifting the relative tran- sition point.
Superposition inter-
Previous works have shown that model learns set of features that minimizes interference from superposition during train- ing. However, this work suggests that a model can redistribute interference from superposition inference time; that not only does the model select the most important features with respect to the dataset, but it dynamically es- timates the importance of features input imput.
FUTURE WORK AND CONCLUSION
These results are motivating for feature learning dynamics research, but further work is necessary generalize and substantiate our findings. First, by validating our observations across other domains (text, image) and models (PointNet++ (Qi al. 2017). LION (Zeng al. 2022)), we can confirm our results and broaden the scope of our work. Second, examining Dora-VAE with circuit detection techniques (Ameisen et 2025) and extracting an attribution graph may show how features flow through the architecture and form discrete patterns. Third, probing gradients of toy models, similar to Elhage et al. (2022), will evaluate the feature learning framework we present here. Finally, can identify what influences redistribution of interference among features, we can potentially perform feature decomposition at training time to develop meta- learning module.
Our work is the first apply SAE to 3D data, highlighting specific discovered features and showing the causally related downstream effects. We then take advantage of the continuous and unstructured nature of the domain to investigate the model's feature decomposition, confirming that the latent space can be interpreted as a discrete, state-based feature space driven by phase transi- tions. We then provide a potentially general framework of feature learning dynamics that explains the unexpected discretization we observe. Finally, we explain the counter-intuitive property of the bimodality of transition points by proposing mechanism by which the model redistributes super- position to only affect tow-impact features.
10

---

www.ciis...pu2222ooooooooobbbbb. Published as a conference paper at ICLR 2026
AUTHOR CONTRIBUTIONS
Albert Miao designed the project, performed the experiments and analysis, and wrote the publica- tion. All authors contributed through discussions and comments on the paper.
REFERENCES
Emmanuel Ameisen, Jack Lindsey, Adam Pearce, Wes Gurnee, Nicholas Turner, Brian Chen, Craig Citro, David Abrahams, Shan Carter, Basil Hosmer, Jonathan Marcus, Michael Sklar, Adly Templeton, Trenton Bricken, Callum McDougall, Hoagy Cunningham, Thomas Henighan, Adam Jermyn, Andy Jones, Andrew Persic, Zhenyi Qi, T. Ben Thompson, Sam Zimmerman, Kelley Rivoire, Thomas Conerly, Chris Olah, and Joshua Batson. Circuit tracing: Revealing computational graphs in language models. Transformer Circuits Thread, 2025, URL https: //transformer circciist...... puub
pub 22025/atttit htt///attiibiiion graphs hoossss. html.
Yoshua Bengio, Aaron Courville, and Pascal Vincent. Representation learning: A review and new perspectives. JEEE Transactions Pattern Analysis and Machine Intelligence, 35(8): 1790-182888 2013. doi: 10. 20.1009/MMMM..013...0. URL https:t///// /oi. org 10. 1011109TPPPTT 2013. 50.
Trenton Bricken, Adly Templeton, Joshua Batson, Brian Chen, Adam Jermyn, Tom Con- erly, Nick Turner, Cem Anil. Carson Denison, Amanda Askell, Robert Lasenby. Yifan Wu, Shauna Kravec, Nicholas Schiefer, Tim Maxwell, Nicholas Joseph, Zac Hatfield-Dodds, Alex Tamkin, Karina Nguyen, Brayden McLean, Josiah Burke, Tristan Hume, Shan Carter, Tom Henighan, and Christopher Olah. Towards monosemanticity: Decomposing language models with dictionary learning. Transformer Circuits Thread, 2023. https://trassfommer- htwpa///////mooomttttaaaaarreere
Bart Bussmann, Patrick Leask, and Neel Nanda. Batchtopk sparse autoencoders, 2024. URL
https://arxiv: org/ www.aas./////ooooooo/oooooooooar 2212.06411.
Rui Chen, Jianfeng Zhang, Yixun Liang, Guan Luo, Weiyu Li, Jiarui Liu, Xiu Li, Xiaoxiao Long, Jiashi Feng, and Ping Tan. Dora: Sampling and benchmarking for 3d shape variational auto- encoders. arXiv preprint arXiv:2412.17808, 2024. doi: 24..55/
URL
https://arxiv. http:////aaxii.../aaaioo///s//// wwt/oorg///oooooooooooooo.oooooo 24.es/242...0 17808. Accepted to CVPR 2025.
Hoagy Cunningham, Aidan Ewart, Logan Riggs, Robert Huben, and Lee Sharkey. Sparse autoen- coders find highly interpretable features in language models. In Proceedings the International Conference on Learning Representations (ICLR 2024), 2024. doi: 10.48550/arXiv.23399oo 08660..
URL https:tp/////. //ap:::://rxxa aaapaaxivvvo.////..//aaa org/ abs/2309 08600.
Matt Deitke, Dustin Schwenk, Jordi Salvador, Luca Weihs, Oscar Michel, Eli VanderBilt, Ludwig Schmidt, Kiana Ehsani, Aniruddha Kembhavi, and Ali Farhadi. Objaverse: A universe of anno- tated 3d objects. arXiv preprint 22.iv:22222000555 2022. doi: 104485500axXx.221220006 URL https: / //tpx/aaavvv////////a org/ alos/ 2212 08051.
Nelson Elhage, Tristan Hume, Catherine Olsson, Nicholas Schiefer, Tom Henighan, Shauna Kravec, Zac Hatfeld-Dodds. Robert Lasenby, Dawn Drain, Carol Chen, Roger Grosse, Sam McCandlish, Jared Kaplan, Dario Amodei, Martin Wattenberg, and Christopher Olah. Toy models of super- position. Transformer Circuits Thread, 2022. URL pubb2022 htte:/tyooodde/eeeee html..
circitts.
Nelson Elhage, Tristan Hume, Catherine Olsson, Nicholas Schiefer, Tom Henighan, .... and Christopher Olah. Superposition, memorization, and double descent. Transformer Circuits Thread, 2023.
html.
Thomas Fel, Victor Boutin, Mazda Moayeri, Rémi Cadène, Louis Bethune, Léo Andéol, Math- ieu Chalvidal, and Thomas Serre. A holistic approach to unifying automatic concept traction and concept importance estimation. arXiv preprint arXiv:2306.072 2023a. URL https://ee /taes///rxxiie http//iww.....o/b/aa/////////aas wtgg 02300

---

Published as a conference paper at ICLR 2026
Thomas Fel, Agustin Picard, Louis Bethune, Thibaut Boissin, David Vigouroux, Julien Colin, Rémi Cadène, and Thomas Serre. Craft: Concept recursive activation factorization for explainability. In Proceedings of the IEEE/CVF Conference Computer Vision and Pattern Recognition (CVPR), pp. 18526-18536, 2023b.
Thomas Fel, Ekdeep Singh Lubana, Jacob S. Prince, Matthew Kowal, Victor Boutin, Isabel Pa- padimitriou, Binxu Wang, Martin Wattenberg, Demba Ba, and Talia Konkle. Archetypal sae: Adaptive and stable dictionary learning for concept extraction in large vision models, 2025. URL https://arxee org/ 25t../252.... 12892.
Leo Gao, Tom Dupré la Tour, Henk Tillman, Gabriel Goh, Rajan Troll, Alec Radford, Ilya Sutskever, Jan Leike, and Jeffrey Wu. Scaling and evaluating sparse autoencoders. Ope- nAI Technical Report, 2025. Preprint available at https: /dn.......// wptp://a..ooooiiioooo/eoeoaaaa.a www. papprs/ sparse autoencderss. pdf..
Edith Natalia Villegas Garcia and Alessio Ansuini. Interpreting and steering protein language mod- els through sparse autoencoders, 2025. URL https: http://aaavvvvvieeeeeee org/ aalaab///55 2502. 09135.
Amirata Ghorbani, James Wexler, James Y Zou, and Been Kim. Towards automatic concept-basee based explanations. In Advances in Neural Information Processing Systems, volume 32, pp. 9273- 9282, 2019, URL https: /ttp:::://pppsaaaeeeee . nips. wttpppssccpppppppppeepepee wttps/ffeesssppeeeeeeeeee (1122affc33166633333f3fffffe
9/hash/
-ete-rbbstttttt htmll.
Liv Gorton and Owen Lewis. Adversarial examples are not bugs, they are superposition, 2025. URL attps://arxiv.orr/33t
17456.
Kaarel Hänni, Jake Mendel, Dmitry Vaintrob, and Lawrence Chan. Mathematical models com- putation in superposition, 2024. URL httphttss///arii..ooo org/ 22082408. 05451.
Been Kim, Martin Wattenberg, Justin Gilmer, Carrie Cai, James Wexler, Fernanda Viégas, and Rory Sayres. Interpretability beyond feature attribution: Quantitative testing with concept acti- vation vectors (tcav). In Jennifer Dy and Andreas Krause (eds.), Proceedings of the 35th Inter- national Conference on Machine Learning, volume 80 Proceedings of Machine Learning Re- search, pp. 2668-2677. PMLR, 2018. URL httpt:httpocceeeeeeeeeee http:///occeeeeinneeeee mlr wtt.sssaww.........emmmmmmmmmeee press/v8C/ kiml 8d. html.
Jack Lindsey, Wes Gurnee, Emmanuel Ameisen, Brian Chen, Adam Pearce, Nicholas L. Turner, Craig Citro, David Abrahams, Shan Carter, Basil Hosmer, Jonathan Marcus, Michael Sklar, Adly Templeton, Trenton Bricken, Callum McDougall, Hoagy Cunningham, Thomas Henighan, Adam Jermyn, Andy Jones, Andrew Persic, Zhenyi Qi, T. Ben Thompson, Sam Zimmerman, Kelley Rivoire, Thomas Conerly, Chris Olah, and Joshua Batson. On the biology of large language model. Transformer Circuits Thread, 2025. URL pub/ 2025//attrbi itttriiutiioo wwwettrriitiiingggrggeeee graphs htnphhs/bblll ciolooy.. html.
ci reuits.
Francesco Locatello, Stefan Bauer, Mario Lucie, Gunnar Rätsch, Sylvain Gelly, Bernhard Schölkopf, and Olivier Bachem. Challenging common assumptions the unsupervised learn- ing of disentangled representations. In Proceedings of the 36th International Conference Ma- chine Learning, pp. 4114-4124, 2019. URL https: / htt.siinssmmm..pes////.s. presss/997
mlr
locatellol9a. html.
Samuel Marks, Tegan Chevalier, Jacob Steinhardt, Chris Olah, Rishi Bommasani, et al. Sparse feature circuits: Discovering and editing interpretable causal graphs language models. arXiv preprint arXiv:2403. 2024.
/ttetts://arxiie www.ii.oor//bb..... /20///403..c 1000..19647
Tomas Mikolov, Wen-tau Yih, and Geoffrey Zweig. Linguistic regularities in continuous space word representations. In Proceedings of the 2013 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pp. 746-751,
2013. URL https::////.. htttss://aclatthllte
org /e///N13333 N1901090.
Core Francisco Park, Maya Okawa, Andrew Lee, Ekdeep Singh Lubana, and Hidenori Tanaka. Emergence of hidden capabilities: Exploring learning dynamics in concept space. In Ad- vances in Neural Information Processing Systems 37 (NeurIPS 2024), June 2024. doi: 10. 48550/arXii.2406.1933 URL https:/ arrxiv.... org/ alss//as//aaa 2406.0..00 19370. arXiv preprint
arXiv:2406.1999 19370.
12

---

Published as a conference paper at ICLR 2026
Charles R. Qi, Li Yi, Hao Su, and Leonidas J. Guibas. learning on point sets in a metric space, 2017. URL 02413.
Pointnet++: Deep hierarchical feature
Senthooran Rajamanoharan, Tom Lieberum, Nicolas Sonnerat, Arthur Conmy, Vikrant Varma, János Kramár, and Neel Nanda. Jumping ahead: Improving reconstruction fidelity with jumprelu sparse autoencoders, 2024. URL https: //aaiiiiii. wttp/or///////////////////////// 14435.
Sukrut Rao, Sweta Mahajan, Moritz Böhle, and Bernt Schiele. Discover-then-name: agnostgi concept bottlenecks via automated concept discovery, 2024. URL https:/ https::/aaxxi..rrrr abbs/2407. 14499.
Adly Templeton, Tom Conerly, Jonathan Marcus, Jack Lindsey, Trenton Bricken, Brian Chen, Adam Pearce, Craig Citro, Emmanuel Ameisen, Andy Jones, Hoagy Cunningham, Nicholas L Turner, Callum McDougall, Monte MacDiarmid, C. Daniel Freeman, Theodore R. Sumers, Edward Rees, Joshua Batson, Adam Jermyn, Shan Carter, Chris Olah, and Tom Henighan. Scaling monosemanticity: Extracting interpretable features from claude sonnet. Trans- former Circuits Thread, 2024. URL sttps://transfot scal inninnooonn sog-mooossematt ci ty/ index. html.
ceaerr-cccuiits... pub/2244/
Harrish Thasarathan, Julian Forsyth, Thomas Fel, Matthew Kowal, and Konstantinos Derpanis. Uni- versal sparse autoencoders: Interpretable cross-model concept alignment, 2025. URL https:
//a//aaa //t.xiiicrgg/aaaaaaaao/.a/.aaa.a /25/02/ 03714.
Eric Todd, Millicent L. Li, Arnab Sen Sharma, Aaron Mueller, Byron C. Wallace, and David Bau. Function vectors in large language models. In Proceedings of the 2024 International Confer- ence on Learning Representations, 2024. URL https::/openreeee AwyxtyMwaas 11X1v:2310.15213.
net forum?id=
Hanqi Yan, Xiangxiang Cui, Lu Yin, Paul Pu Liang, Yulan He, and Yifei Wang. The multi-faceted monosemanticity in multimodal representations. arXiv preprint arXiv:2502. 14888, 2025. URL
https://arxiv.orre
2550...888.
Xiaohui Zeng. Arash Vahdat, Francis Williams, Zan Gojeic, Or Litany, Sanja Fidler, and Karsten Kreis. Lion: Latent point diffusion models for 3d shape generation, 2022. URL https://
arxiv. wwwoorg/bbbooo abs 02202221.2.00 06978.
Ruihan Zhang, Prashan Madumal, Tim Miller, Krista A. Ehinger, and Benjamin I. Rubinstein. Invertible concept-based based explanations for emn models with non-negative concept activation vec- tors. In Proceedings of the Thirty-Fifth AAAI Conference on Artificial Intelligence (AAAI 2021), pp. 11682-11690, 2021. doi: 10. 13099//3...35113....... URL httpt:tt///eee http///axvvvvv// wtwp:iis.rgg/s//////// 2006. 15417.
13

---

Published as a conference paper at ICLR 2026
A FURTHER RELATED WORKS
The theory behind neural networks performing feature decomposition has seen several variations in previous years (Bengio et al. 2013), (Locatello et al. 2019). Even prior to studies of latent space decomposition, studies found interpretable axes in concept embeddings such as language vectors (Mikolov et al. 2013). However, in recent years, we can generally divide discussions of feature decomposition into two camps.
Empirical papers discuss the decomposition of features for interpretability of specifie models or domains. Most notable of these are applications of SAEs and similar techniques on LLMS (Bricken et al. 2023), (Ameisen et al. 2025), (Gao et al. 2025), which followed the initial results of SAEs (Cunningham et al. 2024). These include analyses extract internal representations true and false statements (Marks et al. 2024), or discovering function vectors by analysing the cumulative impact of attention heads (Todd al. 2024). Similarly, image classification and reconstruction studies have proposed new CNN decomposition methods, highlighting segmentations of the input image that led to appropriate classification (Ghorbani et al. 2019), (Zhang et al. 2021), (Fel et al. 2023b), (Fel et al. 2025), (Rao et al. 2024). Particularly, we point out Thasarathan et al. (2025), which suggests that text and image features can operate in the same feature space. This suggests that features can bridge modalities, and although it doesn't 't address it independently, the paper invites further investigation into the dynamics of cross-modal models. Still other papers propose variations or improvements on feature decomposition techniques (Rajamanoharan et al. 2024), (Bussmann al. 2024) draw comparisons between them (Fel et al. 2023a). Empirical papers typically focus on either the method of feature extraction or the application of specifically extracted features towards robustness, safety, or interpretability; however, they do not discuss how this feature space was learned by the model or generally functions, and, as said before, have left unstructured data domains relatively unexplored.
Theoretical papers discuss the structure and formation of the feature space. These papers are fewer and further between. We primarily draw on these for our framework of superposition, as these papers provided abstracted experiments on feature learning dynamics to build intuition (Elhage et al. 2022), (Elhage et al. 2023). Rarely, other works have investigated concept learning dynamics through accuracy evaluation of individual concepts each stage of the model (Park et al. 2024). However, this body of work has been significantly abstracted away from current state-of-the-art models, relying on controlled toy experiments.
In short, there has been a gap in establishing general dynamics of weal-world feature spaces. Al- though feature decomposition itself has significantly improved with the advent of SAEs, the field lacks an equivalent explanation of how these feature spaces are formed. Our work addresses this absence though a thorough investigation of a real-world feature space, analyzing the overall trends in presence and identity of learned features in addition to their specific function. We also believe the gap in generalized feature space research is the result of heavy investment in LLM interpretability research. While such studies are clearly highly salient and fruitful, relationships between tokens are not explicit, and can be difficult to intuit. In contrast, unstructured and unordered data, while difficult to work worth, have clear spatial relationships that allow for intuitive interpretation. We hope to further discussion regarding properties of the feature space general, as any insights are likely to inform improvements in transparency, robustness, and meta-training
B ADDITIONAL DISCUSSION
B.1
ADDITIONAL SAE VARIATIONS
n
Relative &g (L)
Universality (t)
Dora-VAL VAE Loss (L)
256
0.518 / 0.366 / 0.4
0.420 0.421 0.420
0.538 0.300 / 0.228
512
0.507 / 0.355 / 0.187
0.293 / 0.297 / 0.295
0.427 / 0.409 / 0.216
1024
0.501 / 0.356 / 0.182
0.206 0.208 0.209
0.407 0.414 / 0.239
Table 1: SAE Variations, where threshold k: =4/ 16.
14

---

Published as a conference paper at ICLR 2026
In addition to the SAE trained in the main paper, also train several variations codebook size and threshold k to establish general intuition regarding the effects of modifying these hyperpa- rameters. We report the reconstruction loss of the latent, the loss produced by the reconstructed latent passed through the Dora- VAE decoder, and universality. Universality is measurement of the similarity between features of separately trained SAEs, inspired by Fel et al. (2023a) and Bricken et al. (2023). To measure universality, for each set of hyperparameters, we train 10 identical SAEs on 10-fold subsets of the data. We then use pairwise Procrustes alignment to align feature vectors between two trained SAEs. Finally, we report the average cosine similarity of vectors among paired models as universality.
We present our results in Table 1. Performance is as expected - as l increases, the reconstruction is allowed higher fidelity, improving the reconstruction loss both the SAE and Dora- VAE. Similarly, codebook size improves our metrics, as the dictionary of vectors becomes larger.
The same cannot be said for universality, however; as codebook size increases, features become less universal. We suspect this is due to the many possible tilings of positional encoding as features are allowed to become more specific, there a greater variation of feature collections that cover similar space. In future work, we explore how codebook size affects the patterns of features learned.
B.2 FURTHER FEATURE ABLATION AND ADDITION INSIGHTS
100%
67%
33%
0%
Feature 313
Feature 363
Pesture 126
Figure Reprint of Figure 3. Top: Example ablation on feature 313. Bottom: Example of feature 363 ablation and feature 426 addition.
15

---

Published as a conference paper at ICLR 2026
We discuss some interesting properties in Figure 3 here. By gradually removing single feature, affect the reconstruction by Dora- VAE. In this case, we remove feature 313, which is responsible for points along the positive end of the z-axis, as well small section of points towards the negative end. When we remove feature 313, these points move towards the center of the model, forming distinct shapes. We frame the discussion around several key properties.
As seen in b the appearance of these shapes is not reflected as a continuous shift in position. In- stead, points are spontaneously instantiated in their final position, with more of the shape becoming visible over time. This supports our claim that features represent discrete states.
In C we that points, even after shifts, can merge with their neighbors if they share compatible latents. The upper right corner of the bench, originally higher on the z-axis, merges successfully with the lower corner of the bench after transposition. This as opposed to ( and @ where the edges of the tree and bush prevent shapes that would render at that location after shift. We suggest this because latents in the bench have a collaborative relationship with other nearby latents, ensuring that nearby points with the same properties will attempt to merge. On the other hand, points on the edge of the tree and bush have features for non-occupancy, ensuring that other latents that would conflict with the model do not render.
@ shows an example of latent redundancy within shapes. We note that, even though a section of the bush was removed, the remaining latents were able to reconstruct the local scene with strong accuracy. In addition, the new shape rendered by the shifted latents also recreates the bush with similar success. We suggest this is due to a of redundancies within the model decoder help preserve shapes, such that only major changes to latents can affect the model.
In the latter figure, refer to the replacement of feature 363 with feature 426. In short, substitute every presence of feature 363 in the model latents with a presence of equal magnitude in feature 426. As both features attend positions shapes previously located at the region marked in feature 363, and shift them to feature 426. We highlight I here note that the cobblestone path now elevated a region marked by feature 426, while maintaining its shape. We use this substitution to demonstrate the independence of these features, able affect position while still preserving the majority of the form.
the y-axis, we see that through this substitution, we displace
B.3 LEARNING DYNAMICS' EFFECT ON DISCRETIZATION
One might assume the discretization of features arrives solely from the interference caused by the superposition. When a model attempts to represent a large number of features in a small number of dimensions, features that appear with low presence more likely to be construed as interference. Thus, the model learns to prefer features that only fire with a significant presence.
However, this doesn't address the effect across all features, parameters that brings about discretization. Notably, models prefer to superimpose features that are a) sparse and b) less important. Given that positional information is highly relevant across the entire span of inputs, it is unlikely the model would prioritize the information presented by other features the expense of position information.
does it describe the movement of
Our framework in Eq. 6 offers a possible explanation to this counter-intuitive behavior, through the second term ag- ez,e,. We see that the signal to the identity of feature j. e scaled by the presence a.j. Suppose we had continuous positional feature j, where az (x) was higher and lower based on whether x was closer or further along a designated axis. Across x at variety of positions, the identity of feature j would receive the strongest signal positions that have high presence. Conversely, during optimization, positions with lower presence of feature j have to contend with other features that have a higher presence, diluting the signal to identity. Over time, we suspect that this dynamic drives the identity of j to solely consider x at a specific position with the highest presence, localizing the feature. This effect contributes to the discretization of all features, not just positional ones.
B.4 CHOSS-CONTAMINATION OF LOSS DIFFERENCE
One might object to the supposition that the reconstruction model itself is shifting interference from superposition, and instead suggest these shifts reflect bias in the SAE's estimations. This objection
16

---

Published as a conference paper at ICLR 2026
doesn't 't argue that the SAE is misestimating based on the importance of the feature (as the SAE does not optimize based off the effect of the feature on the final Dora- VAE reconstruction loss); rather, proposes there exists correlating property with feature importance that consistently increases the emror of the SAE, causing an offset in its estimation.
Given the SAE is simple two-layer encoder structure and that encoder and decoder weights share a high cosine similarity, if there exists a correlating property, such a feature would affect the input with relatively linear correlation similar to the effect on the estimation observed from the output - as this feature increases/decreases, the error of the SAE increases. The two candidates with the highest potential for cross- contamiaonniinn with loss difference in this manner are average presence and feature density. We investigate if these two properties drive the relationship between loss difference and transition point.
To do so, grouping ARCs by feature, we compile loss difference, average presence on active features, and feature density as dependent variables, to test their relationship with transition point location. For each feature, we perform linear regression through OLS on these variables, as well as with their log forms, to determine which variables scale nonlinearly. We find only loss difference per- forms better in log. Then, again for each feature, we perform partial R2 analysis to quantify the improvement each variable has on predicting transition point. Finally, we take the average R value for each of the investigated variables over all features in defined subset. We repeat this investigation four times, considering subsets of features with a number of ablations Dasl > (500, 1 3k, 6k) ablations, and report our results in Table 2.
Tabll
# of Features
Log Loss Diff H²2
Avg Val H²
Density R²
500
150
0.188+00100 1004
0.025+:0.027 0.042++00056
lk
128
0.18110.105
0.046:10.059
0.02310.025
3k
91
0.183:10.116
0.053-10.064
0.020=0.018
6k
59
0.188=0.119
0.058=00.068
0.017=0.017
Table 2: Partial R analysis of variables contributing to transition point.
Log loss difference consistently contributes to transition point estimation more than other variables. This suggests that, rather than either average presence feature density driving the location of the transition point, both variables' effect are likely the result of the correlation with loss difference. Thus, the offset transition points is not caused by an artifact of the SAE.
This distinction is important. As opposed to average presence or feature density, neither the dif- ference in loss caused by moving the latent in a chosen direction nor the location of the transition point is apparent from examining the latent individually. This suggests the model has a complex, non-linear relationship with both feature importance and relative transition point, and the correlation between these two properties is highly relevant to model behavior.
C FURTHER FEATURE EXAMINATIONS
We demonstrate further examinations of individual features here. Each feature, randomly selected, is highlighted across three objects, and ablated for single object. We also give our qualitative impression of each feature's purpose. Note that some ablations have little effect, due to the redundancy discussed in Appendix B.2, section @.
17

---

Published as a conference paper at ICLR 2026
Feature 424
Feature 425
Feature 119
Feature 130
Feature 13
Feature 300
Figure Feature 424: Smoothness right-facing surface. Feature 425: x-axis encoding. right side. Feature 119: Unknown. Feature 130: Left-facing regional smoothness. Feature 13: Edge encoding. Feature 300: x-axis encoding, right side.
18

---

Published as a conference paper at ICLR 2026
Fenture
Feature 248
Feature 64
Feature 415
Fenture 453
Figure 10: Feature 313: z-axis encoding. Feature 248: Smoothness of faright-facing
surface. Feature 60: Unknown. Feature 64: Unknown. Feature 415: z-axis encoding. Feature 453: Upper edge encoding.
19

---

Published as a conference paper at ICLR 2026
Feature
Feature 351
Feature
Fealure 379
Feature 381
Feature 303
Figure 11: Feature 22: z-axis encoding. Feature 351: z-axis encoding. Feature 97: Right- facing smoothness. Feature 379; y-axis encoding. Feature 38I: z-axis encoding. Feature 303: 7-axis encoding.
20


Reviews:
[{"review_id": "w1qf44pfIF", "reviewer": "Reviewer_y1WU", "summary": "This paper investigates the internal features of a 3D reconstruction model by being the first to apply Sparse Autoencoders (SAEs) to the 3D domain. The authors use an SAE to analyze the latent vectors of a Dora-VAE (a 3D reconstruction VAE) trained on the Objaverse dataset.  The central finding is that the Dora-VAE's internal features are **discrete, not continuous**.\n\nThis paper proposes a novel theory framework oo explain \"unintuitive\" behaviours of VAEs. They analyze the gradient and decompose it into two components: the feature's **\"presence** ($\\alpha_j$) and its **\"identity\"** ($e_j$). The framework suggests that the model prefers to learn discrete features because the learning signal for a feature's *identity* is scaled by its *presence*. Specifically, the **unimodal** (single-peaked) distribution of transition points for **high-impact features** is explained by the feature's \"presence\" term. In contrast, the **bimodal** (two-peaked) distribution observed for **low-impact features** is explained by the model's active manipulation of superposition. The authors further apply SAEs into VAEs for 3D data and explain the unusual properties.", "questions": "How important it is for the application of SAE to 3D data? What is performance of the proposed method in Dora-bench?", "limitations": "", "rating": 4, "confidence": 1, "soundness": 3, "presentation": 1, "contribution": 3, "strengths": "1. New Theoretical Framework: The decomposition of the learning gradient into \"presence\" ($\\alpha_j$) and \"identity\" ($e_j$) provides a new and powerful lens for why features emerge in a certain way, moving beyond just observing what features are learned.\n2. Well-motivated. The application of SAEs to the 3D domain is motivated by the theory analysis to extend interpretability.\n3. Strong Explanatory Power: The proposed framework successfully connects several counter-intuitive empirical observations (discrete features, sigmoidal loss curves, and bimodal transition points) into a single, cohesive theory.", "weaknesses": "1. The writing can be improved for clarity, such as the definition of ARC and loss difference. It is not clear how to calculate them. What the colored points representing are also not depicted. \n2. The findings are based on the Dora-VAE architecture, which processes 3D models by sampling points and using point features and positional encodings. The introduction of SAE is an incremental contribution.\n3. The paper uses the terms \"discrete state space\" and \"phase transition\" heavily. While this is a helpful analogy for the observed sigmoidal loss curves, it may overstate the case.", "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": "WQ3VkmAdKd", "reviewer": "Reviewer_nEVZ", "summary": "This work provides the first application of Sparse Autoencoders (SAEs) to 3D data, decomposing the internal representations of a 3D Variational Autoencoder (VAE). The paper's central finding is that the model learns features as discrete states rather than continuous values, which are activated via phase transitions. This claim is supported by a large-scale analysis of 848k feature ablations, which reveal that: (1) High-impact features have a unimodal (single-peak) distribution of transition points. (2) Low-impact features have a bimodal (two-peak) distribution of transition points. The authors hypothesize this bimodality is evidence that the model actively manipulates superposition interference at inference time, pushing interference onto low-impact features to preserve the saliency of high-impact ones.", "questions": "- Do these findings hold across different models and data types? Or is this only for Dora-VAE and 3D pointcloud data? Showing these findings hold on another model/domain would increase the contribution strength for the learning dynamic theory section a lot. Or, this could be added as a weakness/limitation to the paper since it is only applied to 3D data (even though applying SAEs to 3D data is a contribution itself, it does not mean all general findings on 3D data will hold to other domains).\n- What is Figure 6 showing? It is not clear if it is a real experiment or just a visual aid. It is just mentioned in the first line of Line 454 and 459, but the text and caption are insufficient to parse the figure. \n- Moreover, could the authors kindly explain in more clear terms how the bimodal transition arises? I think this section could even be dropped, or written as a 'hypothesis' rather than making stronger claims about superposition/interference.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 4, "strengths": "- Interpretability for 3D data is underexplored and interesting!\n- Lots of experiment runs (e.g., 848k feature interventions) which makes the results robust.\n- The theoretical explanation of how models see features as a decomposition of presence and identity is also interesting. \n- The bimodal experiments (Fig 5) and visualization (Fig 3) are insightful. \n- Validating the threshold t with max slope experiments is really nice!", "weaknesses": "- The learning dynamics and 3D contributions seem completely disjoint (although both interesting). Moreover, the paper makes broad claims about \"a generally applicable, state-based feature framework.\" However, all the evidence is derived from a single model architecture (Dora-VAE) on a single data modality (3D point clouds). It's impossible to know if these findings (especially the bimodal transitions) are a fundamental property of feature learning, or a specific quirk of the Dora-VAE architecture e.g., and its cross-attention mechanisms.\n- I am having difficulty understanding Figure 6 and more generally, the explanation for the bimodality. I generally get the intuition that the model would put more interference with the low-impact concepts, but have difficulty following the explanation in Lines 459-475. \n\nSmall stuff\n\n- Figure captions should be more detailed and explain (i) why this plot is being shown and (ii) what takeaways the reader should make. Not just a short description of what is shown.", "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": "sxpeQFmVbn", "reviewer": "Reviewer_JL4Z", "summary": "This is a poorly written paper that seems to claim to be the first study on the application of SAEs on 3D datasets and presents corresponding findings and interpretations of the learned features.", "questions": "In generanl, the authors are encouraged to re-setup the goal and scope of this research and rewrite the whole paper for the future submissions.\n\nPlease also respond and provide clear explanations for the confusions I described above, which may change my opinions.", "limitations": "", "rating": 2, "confidence": 3, "soundness": 2, "presentation": 1, "contribution": 1, "strengths": "Unfortunately, given the current organization and writing of the main content in this paper, it is extremely difficult to identify any valuable insights for readers to learn.", "weaknesses": "(W1) The contributions of this paper are vague and poorly discussed. For example, in line 41, the author states that \"the scope of data domains has been limited — recent feature interpretability studies have focused on discrete and structured data, like image and text, rather than continuous or unordered data, ...\" However, I don't see why the solution to this would be to study SAE applications on 3D data, as the authors mentioned in lines 48–49 and highlighted in the abstract. Why don't the authors try continuous and unordered data in 1D and 2D spaces first? Besides, a contradictory and confusing point is that the 3D point cloud data used in this paper's experiments is also discrete.\n\n(W2) The writing of the methods section is very confusing and difficult to follow. It seems that the author proposes a summary formula for SAE works in Eqn (2). However, first, it is not clear how the authors derive and denote the well-known representation learning formula of Eqn (1) into Eqn (2). Second, it is unclear how the authors convert the related SAE works into their formula. For example, in lines 113–114, the authors state that \"Recent LLM studies (Cunningham et al., 2024) use a sparse autoencoder (SAE) to approximate this decomposition with the assumption that $\\bf{α}$ is sparse.\" However, it seems that the SAE formula in Cunningham et al. (2024) has nothing to do with the left part of Eqn (3) of this paper, and their α is a hyperparameter in the final loss function to control the sparsity of the reconstruction. And it has a completely different meaning from the proposed set of scalars $\\bf{α}$ in this paper. Similar confusing descriptions and links to related works are everywhere in the methods section. Lastly, it is unclear how the authors \"apply\" an SAE to Dora-VAE in their major 3D data application. What is FPS in Eqn (7)? Where are the proposed E and $\\bf{α}$ notations in Section 3? How exactly do the authors modify a VAE framework using an SAE structure?", "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)