OrthoRF: Exploring Orthogonality in Object-Centric Representations
Authors: Despoina Touska
Bastiaan Onne Fagginger Auer
Alexandru Onose
Tejaswi Kasarla
Luis Armando Pérez Rey
Maximilian Lipp
Lyubov Amitonova
Martin R. Oswald
Pascal Cerfontaine
Venue: ICLR 2026 Poster (2026)
Keywords: Object Discovery
Object-Centric Representations
Structured Representation Learning
Orthogonality
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: Orthogonal Rotating Features (OrthoRF) is a synchrony-based object-centric model that enforces orthogonality, eliminating the post-processing steps required by other synchrony-based models and enabling recovery of occluded object parts.

Abstract: Neural synchrony is hypothesized to help the brain organize visual scenes into structured multi-object representations. In machine learning, synchrony-based models analogously learn object-centric representations by storing binding in the phase of complex-valued features. Rotating Features (RF) instantiate this idea with vector-valued activations, encoding object presence in magnitudes and affiliation in orientations. We propose Orthogonal Rotating Features (OrthoRF), which enforces orthogonality in RF’s orientation space via an inner-product loss and architectural modifications. This yields sharper phase alignment and more reliable grouping. In evaluations of unsupervised object discovery, including settings with overlapping objects, noise, and out-of-distribution tests, OrthoRF matches or outperforms current models while producing more interpretable representations, and it eliminates the post-hoc clustering required by many synchrony-based approaches. Unlike current models, OrthoRF also recovers occluded object parts, indicating stronger grouping under occlusion. Overall, orthogonality emerges as a simple, effective inductive bias for synchrony-based object-centric learning.

Published as a conference paper at ICLR 2026
ORTHORF: EXPLORING ORTHOGONALITY IN OBJECT- CENTRIC REPRESENTATIONS
Despoina Touska 1-, Bastiaan Onne Fagginger A Onose,,
Tejaswi Kasarla' Luis Armando Pérez Mexx,,aaiiillla Lipp", Lyubov Amitonova*
Martin R. Oswald' Pascal Cerfontaine*
Faaulty of Science, University of Amsterdam, Amsterdam, the Netherlands
2 AAMML Netherlands B. V., Veldhoven, the Netherlands
3 Advanced Research Center for Nanolithography, Amsterdam, the Netherlands
*Faculty of Information, Media and Electrical Engineering. TH Köln, Cologne, Germany
Correspondence: w.p..tosskkeeeee...p touukkggaill com, pascal. cerfont -ieetttttteetttteeteeteeeee
ABSTRACT
Neural synchrony is hypothesized to help the brain organize visual scenes into structured multi-object representations. machine learning, synchrony-bar -based models analogously learn object-centric representations by storing binding in the phase of complex-valued features. Rotating Features (RF) instantiate this idea with vector- valued activations, encoding object presence in magnitudes and affili- ation in orientations. We propose Orthogonal Rotating Features (OrthoRF), which enforces orthogonality in RF's orientation space via an inner-product loss and architectural modifications. This yields sharper phase alignment and more reli- able grouping. In evaluations of unsupervised object discovery, including settings with overlapping objects, noise. and -uist--distriuutioo tests, OrthoRF matches outperforms current models while producing more interpretable representations, and it eliminates the post-hoc clustering required by many synchrony-based ap- proaches. Unlike current models, OrthoRF also recovers occluded object parts, indicating stronger grouping under occlusion. Overall, orthogonality emerges a simple, effective inductive bias for synchrony- based object-centric learning.
INTRODUCTION
Decomposing scenes into constituent parts is a long-standing strategy in computer vision. Classi- cal approaches factorized images into surfaces and objects with properties like reflectance, albedo, and geometry (Coakley| 2003). Deep learning has renewed this effort by learning structured representations (Zhang al.] 2013; Dittadi (023). In practice, this is often operationalized as object-Centrie Learning (OCL) (Greff al.J (020). where models discover modular, com- positional object representations that support generalization and relational reasoning on many downstream visual tasks (Ding et al. 2021; Bapst et al. 2019; Mandikal & Grauman, 2021)
At its core, much like human percep- tion (Spelke 1990), OCL addresses the binding problem (Roskies 1999): flex- ibly integrating features (such as color, shape, texture) into unified percep- tion. This view aligns with cogni- tive and neuroscientific accounts that posit neural synchrony (Singer) (2077) key mechanism, whereby tem- porally synchronized oscillations bind distributed information into coherent objects (see Fig I).
35
Acmettoon
3.0
32 C2aar
Time
Figure 1: boaddnngby-sycchronn. Local excitation, inhi- bition partitions neurons into baase- groups. Same- phase firing (e.g.. B1-2) encodes the same object; out-of- phase firing (e.g., Al vs. B1-2) encodes different objects.
Research done during internship at ASML Netherlands B.V, Veldhoven, the Netherlands. *Research partially done ASML Netherlands B.V, Veldhoven, the Netherlands.

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Izzom ||
Zout
ENC
DE2
RF
Lac
lizonll
ENC
DEC
Lacc
our OrthoRF
LORTRO
Figure 2: RF vs. our OrthoRF: In RF, object content is distributed across orientation space (seen as multicolored bars in encoder activations and mixed final outputs). OrthoRF enforces orthogonal orientation axes, routing each object to distinct latent, yielding cleaner separation and improved handling of overlap regions.
A dominant OCL design to binding uses a collection of discrete latent vectors, named "slots" Lo- catello et al., 2020), each dedicated to the features of single object. As slot-based methods evolve, radically new ideas are emerging in OCL, some of which are inspired by synchrony (Mozer et al.) 1991 Reichert & Serre 2013). In this less-explored paradigm, binding is expressed via the rela- tive phases of complex-valun valued neural activations, with phase-space distances serving as an implicit relational metric between object instances. Recently, synchrony-based models (Löwe et al. [022] Gopalakrishnan et al.) 2024 Miyato et al. (222)) have achieved unsupervised object discovery on synthetic and more naturalistic scenes, surpassing prior supervised approaches (Mozer al. 1991).
Current state-of-in oifie--ffthearrr the-art synchrony-based models, such Complex-valued Autoencoders (CAE) (Löwe al., (022). use real-valued weights to process complex-valued activations, by sharing weights across the real and imaginary parts. In these models, magnitudes encode feature presence, whereas phases encode object affiliation. During training, they leverage the natural con- structive/destructive interference through the addition of complex activations in every layer, pro- moting phase alignment for features of the same object and phase separation for different objects. A related work, named Rotating Features (RF) (Löwe et al. 2024a), replaces complex-valued with vector-valuee valued (i.e., n-dimensional) activations that rotate on a hypersphere, extending representa- tional capacity beyond 2D complex planes. To (de)synchronize phases for object representation, these models use an additional inductive bias via a gating mechanism (Reichert & Serre 2013) that strengthens interactions among similarly oriented features and weakens those among dissimilarly oriented ones. However, this mechanism can be hard to interpret. To improve interpretability. Löwe al. (2024b) introduced cosine binding, more transparent alternative that based on cosine sim- ilarity between activations. This, however, entails substantial memory overhead, since it requires computing and storing many similarities between inputs and intermediate outputs.
Synchrony-based models have important limitations. Unlike slot-based methods, which yield dis- crete, object-aligned slots (one slot one object), state-of of the-art synchrony- based models (Löwe) et al. 2022 2024a, Stanić et al.) 2023 Miyato et al.) 2024 Gopalakrishnan et al.] (024) produce distributed representations. While this can more flexible, it makes the output hard to use with- out extra machinery. In practice, they require post-hoc clustering in phase space to recover objects, grouping features by phase. This distributed coding also degrades performance features from occluding objects become uncertain and drift farther from cluster centers (Löwe al.) 2024a), complicating object assignment. As a result, evaluations of these models often exclude over- lapping regions, potentially underrepresenting the regimes where robust binding most needed.
overlap regions:
In this paper, we advance synchrony-baat based models, specifically the RF autoencoder (Löwe et al. 2024a), by improving both interpretability and representational capacity. Motivated by evidence
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that orthogonality enhances efficiency and encourages disentanglement (Ranasinghe et al. 2021 we propose Orthogonal Rotating Features (OrthoRF), an RF extension that enforces orthogonal object encoding in the phase space. Our approach has two components: (i) softmax-based com- petitive binding that drives n--dimensional activations to specialize on distinct input components (objects), and (ii) an menrr-rroduct-based orthogonality loss that enforces 90 separation between object representations in phase space. Together, these yield sharper phase alignment and more re- liable grouping: features of the same object concentrate along single vector dimension, produc- ing one-hot-like object encodings. In unsupervised object discovery evaluation, across scenarios with overlapping objects, noise, and out-of-distribution tests, OrthoRF matches or surpasses current methods, eliminates post-hoc clustering, and recovers occluded object parts in intermediate repre- sentations (a capability not shown by slot-based or prior synchrony- -based models). These results underscore orthogonality as simple yet powerful inductive bias for synchrony- object-centric learning.
2 BACKGROUND: ROTATING FEATURES
We build upon the RF autoencoder (Löwe et al. 2024a), which replaces scalar features with n- dimensional vectors whose magnitudes encode feature presence and whose orientations encode ob- ject affiliation. Specifically, standard feature vector per-feature magnitude = [Zrotating lla R (the fo-norm over the n-dimension) plays the role of a standard neural activation. This lifting applics both to input images (initialized with zeros along the orientation dimension) and to activations any layer. Given a neural layer fw with din inputs and dous outputs, an input Zin E Rexdd is transformed using a weight matrix w E Rda Xdout, shared across the n components, and bias b Roxdou follows:
Rd is lifted to Zrotating E Rn whose
s = (Zin) + Ruxdows
To ensure that similar oriented features are processed together, RF uses a gating mechanism ichert & Serre 2013). Specifically, it applies the shared weights w E Rdin to both the inputs Zin Ruxda and their per-feature magnitudes |zznll Rdi,, then combines the results:
U Re-
fw (l l12) E god..
(2)
Mbln = 0.5 [lf((()))l)))) + 0.5 X E Idow.
(3)
The gated magnitude mbind is passed through ReLU to enforce non-negativity, and then used to rescale 1b ensuring the output of the layer retains wts orientation, as follows:
Most ReLUU(BachhNoom((mhaaa))) dout (4) Zout = Mout O
E Raxdow. (5)
1|
The reconstructed image is obtained by computing the per-pixel magnitude of the final-layer activa- tions, || E Admaage with dimage = c x h x w (where c, h, are channels. height, and width), scaling with learnable scalar weight w/ and bias D' R, and then applying a sigmoid:
x Sigmoid (w'l Izailll + 8) E Rddingo.
(6)
During training, an MSE loss is used between the input and reconstructed images, LREC MSE(x, x). The vector-valued activations add across layers, producing constructive interference for features of the same object and destructive interference for different objects. Because regions of the same object exhibit high pointwise mutual information, destructive interference would hurt reconstruction, so training implicitly encourages within-object alignment and across-object anti- alignment in features. For object discovery. k-means is applied to the output Zinal E Randdannee, assigning each pixel an object-cluster. See Löwe al. (2024a) for further details on RF.
TThere is an inconsistency between the description in the paper and the implementation the code regarding the calculation of misnd. While the paper states that mand = 0.5 0.5.|| +0.5 x. the codee utilizes the formula described in Equation for Moind E B We used the latter implementation it performs better.
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3 METHOD
3.1 MOTIVATION
Synchrony- -hase architectures such as RF (Löwe al. 2024a) show that these n-dimensional fea- tures can support object discovery, yet their representations are distributed (see Fig. 2 across dimen- sions, which in turn demands post-hoc clustering (e.g.. k-means) to recover objects. This depen- dence makes the pipeline fragile and less practical; a single object may occupy multiple dimensions, creating redundancy and blurring boundaries, particularly in overlap regions where features drift away from cluster centers (as noted by Löwe et al. (2024a)) and assignments become uncertain. That uncertainty, however, carries informative cues: RF's behavior overlaps reveals occlusion signals that slot-based OCL methods (Anciukevicius et al. 2020) seldom exploit. We investigate whether an architectural bias can preserve RF's strengths while addressing these drawbacks. Guided by evi- dence that orthogonality sharpens discrimination and fosters disentanglement (Lezama et al. 2018
Ranasinghe et al. 2021] Sun et al.] 2017 Chen et al. 2020 Liu et al.) 2018 Wang et al.] 2018). impose orthogonality constraints in the RF orientation space that each object collapses onto single component in this n--dimensional orientation space, reducing redundancy, removing the need for clustering, and converting overlap-driven uncertainty into reliable cue for occlusion recovery.
3.2 ORTHIOGONAL ROTATING FEATURES
In this section, we present the architectural modifications that yield competitive binding, and an orthogonality loss enforcing 90 separation among latents in the orientation space. We also high- light key properties that emerge from these modifications. Fig. 2 illustraaees the overall OrthoRF architecture.
Competitive binding in orientation space We model object-componet assignment as discrete competition: each object should map to one component in the n-dimensional orientation space. Inspired by multi-class classification, where softmax layer maps logits to categorical distribu- tion, and by OCL methods such as Slot Attention (SA) (Locatello et al. 2020). we use the same mechanism to induce competition that drives object-oriented specialization across components. In OrthoRF autoencoder, we apply a per-layer softmax over orientation components, yielding winner- take-most assignments and object-wiss specialization. To improve stability and prevent component collapse (c.g., all features mapped to one component while others are never used), we apply center- ing before the softmax, but only to the encoder's output vectors. Empirically, this removes biases that would otherwise let a single component dominate (Caron et al., (02T). Specifically, after Eq. we use the intermediate output vb "Rx (rows i: orientation components; columns j: features). For each feature index j, apply softmax over components i after subtracting the per-feature mean logit to obtain assignment probabilities, as follows:
exp(1' v5,)
n
where J
Ukj-
k= exp Ukj w5y)
12
k=
The remaining steps follow Section 2
Orthogonality regularization We enforce orthogonality among latent orientation components at the encoder output, since this stage aggregates global features and offers lower-dimensional representation, reducing computational cost. implement this, we use the encoder's ss output z IR bebaxxxxxax (bs: batch size, n: orientation components, 2dim: feature dimension). For each sample i (I....,,br bs} and feature j E {l,..., Zdim), we center across orientation components:
Zskj
Zikj
Zig.
where
Zi
Zimj'
m=1
After centering, we stack the n latent vectors for a sample as the rows of Z, E Ra Zeimn and define the Gram matrix, follows:
G; = 2,2, E Raxn
Here, (G.)Re = (Z.... Z.e.) is the (unnormalized) inner product between the component vectors l and f. If different components encode distinct information, these inner products should be small
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Input
Dim.2 2
Dim. 3
Dim.
Dim.
Input
Dim.
Dim. 2
Dim.
Dim. 4
O
0
A
D
O
A
O
O
A
0
0
Figure 3: Qualitative OrthoRF results on 4Shapes, after thresholding V'mal- Objects occupy distinct
dimensions, and occluded parts recovered.
(ideally zero) off the diagonal. Then, penalize the square off-diagonal mass, averaged samples and unique component pairs:
Lortho
1
(fffding(G))) bs n(n
(G)) Right
(10)
bs n(n 1)
By squaring and averaging the (Gi)æ terms, drive cossscompnnent similarities toward zero, thereby decorrelating the embeddings and promoting orthogonality. The factor n(n 1) normalizes by the number of ordered pairs (or twice the number of unique pairs). Finally, the full objective includes the orthogonality term weighted by X:
Liotal
PECC + À Lortho:
A O.
(11)
Equivariance The OrthoRF autoencoder exhibits key property, analogous to SA (Locatello et al, 2020), namely permutation equivariance over orientation components. For representation Rasxaxd (e.g. output of encoder) with the orientation axis, any permutation II acting on this axis satisfies f[II x) = II fx) This property arises from weight sharing across orientation components at every layer, which guarantees identical processing for each component.
Magnitude gating and occlusion completion In the final binding step (Eq. 5) the output is Zout
Moatt T where the magnitude Mont serves as a visibility gate: visible regions pass, oC-
+
cluded regions are suppressed. The pre-gated content & shows ocelusion-complete shapes (Fig. 2 plausible explanation that vb is predicted from learned shape priors under the reconstruction ob- jective, yielding completion behind occluders, while mout encodes visibility. This selective behavior depends on the softmax over orientation channels (competitive binding), which enables clean gating at the final layer.
4 EXPERIMENTS
4.1 EXPERIMENTAL SETTINGS
Datasets We follow the evaluation protocol of Löwe al. (2024a] discovery on datasets with varying numbers of geometric shapes. From RF, we use the 4Shapes dataset (binary variant), containing four object types placed at random locations with possible over- lap. We choose the binary form due to RF's reported sensitivity to color variation. We also eval- uate on MNIST&Shape (Löwe et al., 2022) pairing one MNIST digit with a randomly positioned geometric shape, and on the Shapes dataset of Miyato et al. (2024), which contains 2-4 objects randomly sampled from four basic shape classes. To assess performance a more realistic domain, introduce synthetic SEM dataset of four stacked material layers, reflecting those used in semi- conductor manufacturing, each containing repeated patterns from a single shape class. Layers are laterally shifted and heavily occlude each other. We provide clean and noisy variants to mimic SEM imaging conditions. This dataset emphasizes challenges such as occlusion, layer disentanglement, and robustness noise. For additional dataset details, see the Appendix.
2022). benchmarking object
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Table 1: Object discovery on 4Shapes dataset (mean + standard deviation over 4 seeds). OrthoRF matches RF under identical output post-processing but surpasses it on the shape-conpletion task, measured by MBO; (OV: overlapping regions), when evaluated on Innal- "MSE, ARI-BG, and MBO, are taken from Löwe al. (2024a); RF's ARI-GG and MBO, were recomputed (see Table Appendix). Best results are highlighted as first, second, and third.
Model
n
A
MSE 4
ARI-BG
MBO; 1
MBO 1
AE
5,492c-03 =:2 9.393e-04
'CAE
3,435e-03 = 0.8
0.694 + 0.041
0.628 + 0.039
2
2.198c-03 + 1.110e-04
0.667 1 0.046
0.589 1 0.026
0.5076 0.032
3
1.112e-03 = 2.698c-04
0.937 + 0.018
0.861 + 0.020
0.6379 0.017
4
5.893e-04 + 4.350c-05
0.944 + 0.016
0.908 : 0.012
0.6981 - 0.025
5
5,439e-04 + 6.984e-05
0.975 + 0.003
0.934 : 0.006
0.8049 + 0.013
'Rikmeans
6
2.526c-04 + 1.416c-05
0.991 + 0.002
0.970 + 0.003
0.8111 0.009
7
1.642e-04 + L.810e-05
0.992 + 0.002
0.974 + 0.003
0,8172 0.001
8
1.360e-04 = 7.644e-06
0.987 = 0.003
0.968 1 0.008
0.8196 0.001
9
1.119e-03 5.715e-04
0.9949 + 0.002
0.989 + 0.002
0.8170 0.001
20
1.000e-04 0.000e-04
0.7129 + 0.283
0.7340 0.242
0.5822
0.179
2
0.5
3 3.050e-03 + 1.202e-03
0.4987 + 0.058
0.4499 + 0.041
0,3719 + 0.037
3
0.5
1.00e-003 = 0.000e-04
0.6530 + 0.009
0.6576 + 0.029
0.5127 + 0.016
4
0.5
4.500c-004 + 0.707c-04
0.9994 + 0.0007
0.8915 + 0.029
0.7205 + 0.024
5
0.8
2.330e-104 + 0.942e-03
0.9995 + 0.001
0.9887 + 0.003
0.8204 + 0.002
OrthoRkkmae
6
0.5
1.963e-04 = 0.855e-03
0.9941 + 0.009
0.9856 + 0.006
0.8151 + 0.006
Zoul
7
0.1
1.660e-(04 + 0.942c-03
0.9947 + 0.006
0.9856 0.008
0.8161 0.004
8
0.09
3.714e-04 = 0.768e-03
0.9924 + 0.009
0.9527
0.018
0.8022
0.003
9
0.08
2.133c-04 0.991c-03
0.9955 + 0.002
0.9849
0.005
0.8183
0.001
20
0.1
0.003c-05
0.141c-03
0.9974 + 0.001
0.9764
0.016
0.005 0.7426
4
0.5
4.500e-004
0.7077-04
0,9870 + 0.004
0,9760
0.005
0.9661
0.021
5
0.8
2.330c-044
0..942-003
0,9934 + 0.001
0.9843
0.009
0.9832
0.006
6
0.5
1,963e-04 = 0.855e-03
0.9869 1 0.004
0.9845 1 0.004
0,9853 + 0.003
OrthoRFilresh
7
0.1
1.660-044 = 0.942e-03
0.9763 1 0.001
0.9730 + 0.002
0.9794 0.005
Pizza
8
0.09
3.714e-04 + 0.768c-03
0.9682 + 0.005
0.9604 + 0.002
0.9680 + 0.002
9
0.08
2.133e-04 + 0.991e-03
0.9631 + 0.003
0.9678 + 0.002
0.9875
0.009
20
0.1
0.003e-05 = 0.141c-03
0.9822 + 0.008
0.9670 + 0.011
0.9696 0.001
Evaluation metrics Following standard practices in object discovery, we evaluate performance using the Adjusted Rand Index (ARI) (Hubert & Arabie 1985 Rand [971] Greff et al. 2019) and Mean Best Overlap (MBO) (Pont-Tuset et | 2016 Seitzer et al. 2022). ARI measures clustering similarity, ranging from (chance) to I (perfect match). We compute it using decoder predicted ob- ject masks against ground truth, either excluding background (ARI-BG) or including it (ARI+BG). MBO matches predicted and ground-truth masks by overlap and averages their Intersection-over- Union (IoU) scores.
decoder-predicted
Implementation details The OrthoRF implementation follows RF (Löwe et al., 2024a) using convolutional autoencoder. Architectural details appear in Table 9 and Table [0 (Appendix). The models were trained with Adam (Kingma & Ba 2015) using a batch size of 16 for 100-200k steps across different datasets, with a CosineAmnealingLR scheduler applied for learning-rai -aat decay. Ex- periments were in PyTorch (Paszke al. 2019) single NVIDIA Tesla T4 (16 GB). Addi- tional training settings are listed in Table (Appendix).
4.2 RESULTS
Evaluation on the 4Shapes dataset Table T compares OrthoRF with RF (Löwe et al. 2024a). AE (Löwe et al.] 2024a),, and CAE (Löwe et al.) 2022) on 4Shapes under two protocols. First, visible-only object discovery evaluates model outputs (Ziinal for RF/OrthoRF) against ground-truth masks that exclude overlapping regions (as defined by Löwe al. (2024a)). Second, evalu- ate shape completion using MBO,V which measures full-object recovery including overlapping (OV) regions- by comparing predictions to instance-level ground -ruth masks (labeling scheme in Appendix B.3 In Table [ first report RF's Zinal using its standard processsnss (output normalization, magnitude masking, and m-manss) We then apply the same post- protepsocsssing to Or- thoRF's Zrinal for fair comparison. Finally, we evaluate OrthoRF's intermediate map e'inal the

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Table 2: Object discovery on the SEM dataset. OrthoRF outperforms RF under severe occlusions and noise, and shows stronger out- out-of-distributin generalization (noisy testing after clean training, and vice versa). Best results are highlighted as first, second, and third.
Train
Noise-free
Noisy
Test
Model
MSE
ARI-EGT
MBO;
MSE
ARE-G 1 MBO;
K-maans
0.8427
0.6546
Histogram
0.8011
0.6582
Noise-free
R Saud
5
0.0001
0.9551
0.6834
0.0057
0.6942
0.4146
OrthoRFFlnes
5
0.0002
0.9908
0.7171
0.0624
0.7610
0.5644
K X-means
0.3381
0.3442
Noisy
Histogram
0.3333
0.3612
RIJamens
5
0.0042
0.8816
0.6044
0.0007
0.7043
0.4154
FilheR
5
0.0051
0.9836
0.6705
0.0007
0.8356
0.6268
Slaad
Input
Dtm.
Dtm.2 2
Dim. 3
Dim.
Dlm.
RF -oor
OrthoRF: dp
Figure 4: Qualitative results on noise-free SEM test images for 5. RF's output (top row) is spread across the orientation components, whereas the thresholded OrthoRF's output (bottom row) separates the SEM layers and reveals occluded structures.
gating mechanism's intermediate representation in the output layer-by thresholding it at 0.1 (see Appendix BB44 for details), without any additional post-pre post-processing, to obtain binarized masks for metric computation. Note that thresholding is used solely for binarization; the objects in 3lmmll are already disentangled, unlike in RF where k-means is required.
Table [ shows that OrthoRF matches RF on visible-only discovery for ARI-BG and MBO, metrics, but outperforms on shape completion (last column) when evaluated on Vimml- With a global thresh- old, OrthoRI's Viinal reaches approximately 0.98 MBO
5, while the same value for Zou of both RF and OrthoRF is approximately 0.80. Post-processing largely explains these outcomes: be- cause RF doesn't cleanly separate objects in Zout. it uses k-means to recover memberships. However, k-means enforces one label per pixel. overlaps get credit for only a single object. Thresholding eltinal instead permits multi-label pixels in overlapping regions, improving MBO.V, Overall, per- formance for both models remains stable across n (minor fluctuations). Finally, OrthoRF and RF substantially outperform AE and CAE. Fig. 3 shows qualitative OrthoRF results after thresholding Viimat objects separate into distinct dimensions, and occluded parts are recovered.
In Table D also compare performance across different orientation dimensionalities n. When smaller than the number of objects, RF's distributed representations outperform OrthoRF. In con- trast, when substantially larger (c.g.. = 20), OrthoRF maintains high performance, whereas RF shows a clear degradation. For the thresholded vinnal variant, we omit results when n is smaller than the number of objects, since a one-to-one correspondence between predicted and ground-truth masks is not achievable under these settings. Additional analysis of varying the orthogonality loss weight À is provided in Appendix C.2
Evaluation on the SEM datasets Table reports visible-only object discovery results for Or- thoRF, RF, and two non-neural baselines, k-means and histogram-based method, on clean and noisy SEM datasets. For k-means, we use scikit-leam''s 's implementation to cluster pixel intensities. The histogram baseline mode/peak assignment (nearest-peak clustering); in the intensity his-

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Input
Dim. 1
Dim.
Table 3: Object discovery results on MNIST&&Sapes (mean
+ standard deviation over 8 seeds). OrthoRF matches RF and CAE on most metrics and achieves the highest ARI-BG, while DBM and SA fail to reliably discover objects. Best results are highlighted as first, second, and third.
8
Model
MSE 4
ARI+BG f
ARI-BG f
DBM
1.56c-02 + 8.07c-05
0.718 + 0.002
0.175 + 0.006
CAE
3.19c-03 + 1.51c-04
0.783 + 0.007
0,971 # 0.011
RF
6.00e-03 + 0.00e-04
0.699 + 0.008
0.972 1 0.009
OrthoRF
6.00e-03 + 0.00e-04
0.667 + 0.002
0.996 + 0.003
Figure 5: Qualitative results of Or-
SA
5.44e-03 = 1.61e-03
0.047 + 0.013
0.089 1 0.028
thoRF on MNIST&Shapes dataset.
togram, detect prominent peaks, and assign each pixel to the closest peak. These non-neural baselines were chosen because they are unsupervised, fast, and interpretable. They also show how much segmentation is possible from intensity alone uniform-color SEM shapes, without heavy chitectures, or additional training. Both OrthoRF and RF use n = 5 (4 object layers + background) dimensionality in the orientation space.
Table 2hooss that OrthoRF outperforms RF on both the noise-free (ARI-BG: 0.9908 vs 0.9551) and the noisy test set (ARI-BG: 0.8356 vs 0.7043), indicating greater robustness under heavy occlusion and noise. Compared with k-means and the histogram baseline, the neural models excel especially in noisy settings, where edge blurring potentially hinders intensity-only clustering. We also evaluated out-of-distriie out-of-distribution generalization for OrthoRF. Training on clean data and testing on noisy yields only a minor drop (ARI-BG from 0.9908 to 0,9836), indicating strong noise tolerance. In contrast, training on noisy data and testing clean degrades more (from 0.8356 to 0.7610), likely because noise-trained models learn smoothed boundaries that underfit sharp, clean edges. Fig. 4 shows qualitative results on noise-free SEM test images. RF (top) distributes content across orientation components, whereas OrthoRF (bottom) separates SEM layers and recovers occluded structures.
Evaluation the MNIST&Shape dataset Since OrthoRF converts distributed representa- tions into discrete ones, compare it slot- based methods (SA (Locatello et al. 2020)), synchrony-based approaches (RF, CAE), and the Deep Boltzmann Machine (DBM; (Salakhutdi-
Table 4: Object discovery on Shapes (mean + standard deviation over 3 seeds). OrthoRF out- performs other models when object count and shape vary randomly. Best results are high- lighted as first, second, and third.
nov & Hinton 2009). following the setup of Re-
ichert & Serre 2013): Löwe et al. 2022). on the MNIST&Shape dataset. As shown Table Or- thoRF matches RF's performance while achiev- ing clear object separation (Fig.
Model
n
ARI-BG 1
MBO,
IttSA
0.570 0.055
0.348
0.046
AKOrNatt
0.713
0.042
0.469
0.014
RF OrthoRP
8
0.744
0.079
0.780
0.065
5). DBM and SA both fail &NhST&Shape. SA, in particular, struggles because (1) the MNIST digits exceed its receptive field and (2) handles grayscale inputs poorly, reported by Löwe al. (0022)
8
0.833 + 0.019
0.865
0.013
Effect of Shape and Count Variability Ta- presentts results on the Shapes dataset, where both the number of objects (ranging from 2 to 4) and the combination of shape types vary per image. The comparison includes Iterative Self- Attention (ItrAA) and AKOrN troduced by Miyato et al.) (2024), with the lat- ter being a synchrony-based approach inspired by Kuramoto oscillatory neurons-as as well as RF. Among these methods, OrthoRF achieves the best performance. Its discrete representation likely contributes to improved object separation in this challenging, variable setting.
Table 5: Architectural ablations on 4Shapes. Variants differ by the use of softmax with cen- tering (SC) and the orthogonality loss (A), Best results are highlighted as first, second, and third.
models in-
Model
SC
MSE.,
ARI
MBO, f
RF
No
0.0005
0.9750
0.9340
No
0.1
0,0002
0.8530
0,8680
OrthoRF
Yes
0
0.0034
0.6280
0.6880
Yes
0.1
0,0002
0.9995
0,9887
8

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Table 6: Quantifying similarity in phase space via mean pairwise cosine angles (degrees). On 4Shapes, OrthoRF is near-orthoonnal with lower variance; on SEM. RF has a slightly higher mean but larger variance. Best results are highlighted as first, and second.
Data
Model
A
Dim.1
Dim.2
Dim.3
Dim.4
Dim.5
Avg.
RF
46.7
0.96
80.3 + 2.1
64.7 = 1.5
67.7 1 2.4
87.0 1.7
69.28
13.91
4Shapes
orthoRF
0.5
88.5
1.29
88.8 + 1.3
89.6 = 1.0
89.3 1: 1.0
78.1
1.7
86.86 4.39
RF
82.6 +
88.6
2.6
69.9 2.1
82,4
4.7
87.1
3.3
82.12
6.57
SEM
orthoRF
80.0 + 79.0
85.8
75.5 3.1
80.7
2.2
80.2 3.32
Ablation Architectural Modifications T shows ablation 4Shapes, assessing the effect of softmax with centering (SC) and the orthogonality loss, controlled by À (where À = disables the loss). Neither component alone yields strong performance, but their combination leads to substantial boost, achieving near perfeeteeet ARI and the highest MBO, for 5.
Quantitative evaluation of similarity We quantify how the orthogonality constraint shapes the encoder output =ou by averaging pairwise cosine angles across all orientation components (Table 6). On 4Shapes, OrthoRF yields angles near 90º with far lower variabil- ity than RF (std 4.39 vs. 13.91), indicating cleaner phase separation. On SEM, OrthoRF averages 80" (softer constraint due to lower A), while RF attains a higher mean (82.129) but with greater variance. Across datasets, the background dimension (Dim. 5 in 4Shapes; Dim. 4 in SEM) shows the smallest angles, flecting weaker distinctiveness but is included in all statistics.
Figure 6: Separability in phase space on 4Shapes using inter- and intra-cluster metries. OrthoRF forms tighter clusters, while RF shows higher mean inter-angles but with large variability. Best results are highlighted as first, and second.
Model
Inter-cluster 1
Intra-cluster +
Min
Mean
82.89
106.47
23.58
17.29 1
OrthoRF
89.85
89.92
0.067
1.09
4.38
140
020
200
Quantitative evaluation of separability To assess the orthogonality constraint in the out- put Zfiual we compute inter- and intra-cluster angular metrics following /Stanié et al. (2023). The inter-cluster metric measures separation between objects by computing unit-normalized centroids per object and evaluating pairwise centroid angles; angles near 90" indicate strong, orthogonal separation. The intra-cluster metric measures compactness by computing the angu- lar deviation of each pixel feature from its ob- ject's centroid. Metrics
-1.00
-031
4633
-221
375
3.5
computed per im- age and averaged over the 4Shapes dataset. Ta- blows that OrthoRF achieves consistently lower intra-cluster dispersion than RF, indicat- ing tighter object representations. RF attains higher mean inter-cluster angle, but with large variability (std. 23.57%). suggesting unsta- ble separation. In contrast, OrthoRF produces netr-orthogonal inter-cluster angles with sub- stantially lower variance.
2
400
-65
Figure Principal component visualization of 4Shapes. OrthoRF (bottom) forms tighter clusters, while RF's (top) are more dispersed. Background is always label 0.
To aid interpretation, we visualize per-pixel embeddings from the models' outputs (#fina) by projecting them to 2D with PCA and normalizing them on the unit circle, then overlaying class cen- troids as arrows from the origin (see Fig. (7.. OrthoRF forms tighter, better- setter-separated clusters, while RF appears more dispersed consistent with the quantitative metrics. The background cluster (label 0) is naturally more diffuse.

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5 RELATED WORK
Object-centric learning Over the years, a wide range of OCL models (Eslami al., 2016 Greff
ee al.
2016 2017] Jiang et al. 2019 Lin et al. 2020; Prabhudesai et al. 20022] Stelzner et al. 2019) have been proposed. MONet (Burgess et al. 2019), LODINE (Greff et al. 2019). and GENESIS (En- gelcke et al. 2019 2021) propose unsupervised approaches to disentangle scenes into objects, with MONet and IODINE modeling objects independently and GENESIS capturing their interactions. SA Locatello et al. 2020) extends this line of work by introducing an iterative attention mech- anism where slots compete bind to distinct objects. This design has inspired numerous exten- sions, including SLATE (Singh et al. 2021) and DINOSAUR (Seitzer et al. 2022), which integrate Transformer- based encoders and decoders Vaswani et al. 2017) to better handle real-world images.
OCL has been extended to both 3D (Chen et al. 2021 Sajjadi et al. [202]] Stelzner et al. [02 1) and video data (Lai al., 2021 Elsayed et al.j 2022; Singh [l.]222 ROOTS (Chen et al.) 2021) disentangles objects via 3D-to-2D multi-view projections, while SAVi (Lai et al. 2021), SAVi++ (EI- sayed al. 2022), and STEVE (Singh et al. (022)) extend SA to videos, leveraging temporal dynam- ics to separate objects from each other and the background. Similarly, PathTracker (Linsley et al. 2021) disentangles objects via trajectory- -bssed grouping, and its extension FeatureTrackee (Muzel- lec et al.
2024) further separates in feature and color space.
Synchrony-based learning Recent OCL works (Löwe et 2022 Reichert & Serre 2013 Löwe
ict al. 2024a; Stanić et al., (024) move beyond slot-based models, inspired by neuroscience. While early studies utilized supervised (Mozer et al., 199T) or weakly supervised (Ravishankar Rao Cec- chil (2100) settings, modern methods emphasize unsupervised discovery. CAE (Löwe et al. 2022)
uses complex-valued activations for phase clustering, a framework recently extended via contrastive losses in CICAE (Stanić et al. 2024) or nigher-dimensions rotations in RF (Löwe et al. 2024a). Un- like these models where synchrony emerges dynamically, other works (Miyato et al. [2024] Muzelled et al. 2025) enforce phase synchrony through explicit synchronizers (e.g.. Kuramoto oscillators).
Orthogonality Orthogonality has been widely exploited in deep learning. It has been used in net- work initialization (Saxe et al. 2013) and during training (Achour et al. 2022 Li et al.) 2019) to
improve stability and generalization. Orthogonality also has been used create discriminative fea- ture representations (Lezama et al., 2018 Ranasinghe et al., )02T) and disentangle features (Wang) al. 2018). In open-world object detection (Sun et al.] 2024). a study utilized multiple levels of orthogonality throughout the training process to mitigate catastrophic interference and facilitate incremental learning of previously unseen objects.
6 CONCLUSION
Summary In this paper, we introduce the OrthoRF autoencoder address central RF (Löwe et al. 2024a) limitation: distributed object-centric representations break down in overlaps, where features from different objects become uncertain and undermine phase-space clustering. OrthoRF couples competitive binding with an inner-product orthogonality loss to align each object to a dis- tinct phase axis, yielding sharper alignment and removing the clustering step. Across unsupervised object discovery OrthoRF matches surpasses relevant models and recovers occluded parts. More broadly, orthogonality provides an effective inductive bias that regularizes distributed representa- tions into discrete, directly resolving overlap ambiguity and improving the downstream usability.
Limitations and future work OrthoRF converges more slowly than RF (=220k vs. 100k steps), due to its stricter object discovery and separation objective. Rarely, training can get stuck in sub- optimal states (incomplete separation or multi-object collapse onto one phase axis), a phenomenon also noted in slot-based setups (Locatello et al., 2020). Increasing À or lowering the learning rate stabilizes training. Future research will explore integrating synchrony-based binding into attention mechanisms, mitigating RF's performance degradation RGB datasets, and extending evaluations to video and realistic datasets. To offset the computational cost of vector-valued "lifting," we aim to explore more parameter-efiice efffccceeee encodings. Furthermore, inspired by Extreme Image Transforms (EIT) (Crowder & Malik 2022 Malik et al. (202)), exploring EIT- augmentations and severe corruptions (e.g., salt-and- salt-and-pepper pepper noise) remains promising direction for enhancing robustness.
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Acknowledgements The presentation of this paper the conference financially supported by the Amsterdam ELLIS Unit.
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APPENDIX
A REPRODUCIBILITY STATEMENT
To ensure the reproducibility of our experiments, we provide detailed overview of the model architectures, hyperparameters, evaluation procedures, datasets, and baselines in Appendix[B] While code not yet publicly available, we note that our method builds upon RF, allowing others to reproduce our approach by starting from their implementation. The required modifications include adding the orthogonality loss (see Algorithm|] I and implementing changes related to the competitive binding mechanism. All necessary details for the latter described in the corresponding section of the main paper.
Algorithm Orthogonality Loss Pseudocode
Require: Batch of rotation encoder outputs R R BxNxD, where B is batch size, N is number of slots/rotations. D is feature dimension.
Ensure: Scalar loss Lurtho- Center the representations:
1: μ f- N E R.
Compute mean across the N dimension Broadcast subtraction
2: R R--R-A
Compute pairwise inner product:
3: C < 1-mmm R, R
b. Batch matrix multiplication: [B, N, N]
Mask and Compute Loss:
4: M + mask( -I)
Select only off-diagonal elements Mean squared value of off-diagonals
5: Lonto [M ZzEM 22
6: return Lartho
B IMPLEMENTATION DETAILS
B.1 ARCHITECTURE AND HYPERPARAMETERS
We implement OrthoRF a convolutional autoencoder with some fully connected layers, closely following the RF architecture Lôwe al. (2024a). Below, we specify the exact model configuration and training setup. Tables and 10 provide the architectures used for all datasets and for Shape, respectively (using LeakyReLU in place of ReLU). Training hyperparameters are listed Table
B.2 DATASET DETAILS
We follow the evaluation setup of Löwe et al. (2024a 2022], which benchmark object discovery across datasets with varying shape compositions.
Figure 8: Official label format for the 4Shapes dataset. Occluded regions are labeled with -1 and excluded from eval- uation.
4Shapes From the RF paper, we adopt the 4Shapes dataset (see Fig. 3. We use its binary variant rather than RGB, motivated by RF's reported sensitivity to color vari- ation: accuracy drops substantially-eee colors-wher when depth masks or pretrained features (c.g., DINO et al. (2021) are omitted. Each image contains four objects (square, up-pointing triangle, down-pointing triangle, and circle) placed at random locations, with possible overlap and shared color.
with only five
&hIST&Shape Following Löwe et al. (022)) this dataset pairs a single MNIST digit with randomly positioned geometrie shape, increasing both visual diversity and difficulty.

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Figure 9:
Our labeling scheme used to evaluate occluded regions in the 4Shapes dataset. The leftmost image the input image and all the rest are binary labels for each of the four shapes.
Shapes We include the dataset of MMyato al. 2024), which generates images with 2-4 objects randomly sampled (with repetition) from triangle, square, circle, and diamond classes.
SEM Dataset For a more realistic application domain, we construct a synthetic Scanning Electron Microscope (SEM) dataset of semiconductor materials (see Fig. 4 Each image shows a four-layer vertical stack viewed from above. Each layer contains single shape class (horizontal lines, circles, vertical lines, ellipses) repeatedly tiled across the layer. Layers are horizontally displaced relative to one another, and the displacement varies across images. We provide two variants: (i) noise-free and (ii) noisy (Gaussian blur + additive Gaussian noise) mimic SEM acquisition artifacts.
This dataset designed to stress-test object-centric models under (i) severe inter-layer occlusion from higher z-layers, (ii) relevance semiconductor metrology, where layer separation is crucial, and (iii) robustness to realistic acquisition noise. While not publicly released, the dataset can be reproduced from the provided description.
B.3 EVALUATION DETAILS
We use settings in the our evaluation protocol. (i) Visible-only discovery: follow Löwe et al. (2024)) and evaluate using their labels, which mark occluded regions with -1. This scheme is applied to both binary and RGB data, though we find this setting less fair, especially for boundaries and overlapping regions evaluation. (ii) Shape completion: to assess recovery under occlusion, we create per-shape labels for the 4Shapes dataset, enabling evaluation of both RF and OrthoRF with
the ( ov metric. Fig. 8 illustrates the labeling schemes introduced by Löwe et al. (2024a) and Fig. 9 shows our labeling format for the shape completion evaluation.
B.4 THRESHOLDING
Thresholding is used as a simple post-processing step to binarize the model's output for metric computation. The gi-thresholding procedure is largely insensitive to the choice of threshold. As illustrated Figs. 10 we first discard background channels (i.e., feature maps with very low vari- ance) and then apply fixed threshold to the remaining maps. Because competitive binding creates a strong separation between active and inactive regions, wide range of thresholds produces nearly identical binarizations; 0.1 serves convenient default, though other values yield similarly strong results. An adaptive, statistically motivated thresholding scheme is a natural extension for future work.
C ADDITIONAL EXPERIMENTAL RESULTS
C.1 EFFECT ORTHOGONALITY MECHANISM
Fig. presents three supplementary results from the 4Shapes dataset demonstrating the effect of our modifications in the network. The first row showcases the RF model's output, where object content is distributed across orientations. The second visualizes the 1b output of the last layer of the OrthoRF model, demonstrating clear object separation and reconstruction of occluded object
Public source descrihing the 3D-layer structure used generate synthetic SEM images.
2

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Published as a conference paper at ICLR 2026
0nd-003004
500 0.403
sadw61822 1822
stom Splb
04
0 1671
ond - 31.33
de 2277
7004
aod-2.3033
steno 2570
addm0.3220
(a) Vinmall before thresholding.
sidwo 0020
SAdeO 6413
sndwo 0000
TA 1.770
Pot e
RA
Input
Fot
0020
for 10
004
sodiet 3764
ood-0 0000
as-008200
sodvi 9020
(b) Vinna after thresholding.
Figure 10: (a) Network output for 20: object-related dimensions exhibit noticeably higher variance, making them easy to identify. The strong separation between object and background pixels allows wide range of effective thresholds. (b) Result after applying fixed threshold to obtain binarized masks.

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Published as a conference paper at ICLR 2026
Table 7: Object discovery on 4Shapes (mean = standard deviation over 4 seeds). We reran RF for E {2, 3, 4, 5, 6, 7 and additionally evaluated n 9. For fair comparison, Table | reports the values from Löwe al. (2024a); our reruns closely match the official results.
Model
n
ARI-BG 1
MBO; 1
2
0.7198 + 0.038
0.6526 + 0.065
3
0,9326 + 0.028
0.8615+ 0.014
4
0.9767 = 0.028
0.9382 = 0.010
R Fkmcans
5
0,9931 + 0.007
0,9835 = 0.007
Zhaal
6 0.9950 + 0.005
0.9885 + 0.006
7 0.9980 + 0.001
0.9927 + 0.001
8 0.9979 + 0.001
0.9928 + 0.001
parts. Finally, the last row depicts the Zout output from the OrthoRF's final layer, which reveals object separation but with some holes due occlusions.
Table 8: Effeet of varying orthogonality loss coefficient À and dimensionality of the orientation space on reconstruction loss, ARI, and MBO.
(a) Reconstruction Loss
A
6
7
9
20
0.05
0.0002
0.0001
0.0001
0.0001
0.0001
0.0002
0.1
0.0002
0.0002
0.0001
0.0001
0.0001
0.0002
0.5
0.0002
0.0001
0.0001
0.0001
0.0002
0.0002
0.8
0.0002
0.0002
0.0001
0.0001
0.0001
0.0002
0.0034 0
0.0054
0.0043
0.0043
0.0046
0.0001
(b) ARI f
A
6
7
9
20
0.05
0.8478
0.9983
0.9996
0.9944
0.9964
0.9971
0.1
0.9997
0.9992
0.9995
0.9994
0.9996
0.9988
0.5
0.9996
0.9912
0.9989
0.9938
0.9806
0.9985
0.8
0.9995
0.8179
0.9992
0.9975
0.9993
0.9433
0.6279 0
0.6001
0.9676
0.7896
0.9645
0.9975
(c) MBO,
A
5
6
7
9
20
0.05
0.8872
0.8938
0.9896
0.9832
0.9879
0.9875
0.1
0.9854
0.8758
0.9891
0.9876
0.9887
0.9877
0.5
0.9791
0.9762
0.9853
0.9818
0.9669
0.9859
0.8
0.9843
0.8613
0.9868
0.9880
0,9933
0.8930
0
0.6881
0.6363
0.8152
0.7071
0.8117
0.9918
C.2 VARYING A
We analyze how performance varies with the orthogonality loss weight across different numbers of slots The results in Table 8 and Fig. [] reveal that for larger n, performance remains stable across a wide range of values. This suggests that higher-dimensiooal spaces naturally support object separation. However, for smaller (c.g., ARI and MBO, indicating that orthogonality constraints become essential when representational capacity is limited.
6), low À values lead to significantly worse
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Published as a conference paper at ICLR 2026
ARI vs. Number of Slots (n)
1.0
(orthogonality loss)
A=0.05
0.9
A=
A=
A=
ARI
0.8
A=0.0
0.7
0.6
MBO; vs. Number of Slots (n)
1.00
0.90
0.85
MBO,
0.80
0.70
0.65
8
10
12
14
16
18
20
Number of slots (n)
Figure 11: Effect varying À and number of slots n on ARI and MBO. With larger n, performance remains high across À values, while for smaller sufficient orthogonality regularization (A >>>>> 0) is crucial for object separation.
C.3 QUANTITATIVE RESULTS FOR THE 4SHAPES DATASET
In Table I. we report RF results on 4Shapes as published by Löwe al. (2024a). For a fair compar- ison in our main results, we also replicated their experiments; the outcomes are shown in Table 7 Our rerun closely matches the official numbers.
C.4 QUALITATIVE RESULTS FOR THE 4SHAPES DATASET
In Table 1 we present results of OrthoRF model in comparison with other models for the 4Shapes dataset. Specifically, we vary the orientation-space dimensionality and find that performance re- mains largely stable. In this section, we show qualitative results for orientation-spaa dimensionali- ties n E (6, 7, 8, 9} in Figs. 13 The visualizations demonstrate that only necessary number of dimensions are used while the rest stay empty.
C.5 QUALITATIVE RESULTS FOR THE SEM DATASET
This section presents qualitative results for the noise-free and noisy SEM datasets. Fig. 17 com- pares the Zout output of the OrthoRF and RF models trained and tested on noise-free SEM images. The leftmost image in each row represents the input image. As shown, the RF model distributes layer content across dimensions, while the OrthoRF model effectively separates SEM layers into distinct dimensions. Fig. 18 displays the OrthoRF output alongside its thresholded version, used for binarization in metric calculations. Fig. shows qualitative results for OrthoRF and RF trained and tested on the noisy SEM dataset. As in the noise-free -fre case, RF distributes content across orientations, whereas OrthoRF cleanly separates layers into distinct dimensions.

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Published as a conference paper at ICLR 2026
C.6 QUALITATIVE RESULTS FOR THE SHAPES DATASET
In this section, we present qualitative results for orientation-space dimensionality 8 on the Shapes dataset (see Fig. 21). The visualizations show that OrthoRF activates only the necessary dimensions, while the remaining ones remain unused, reflecting sparse object representations.
C.7 EFFECT OF MAGNITUDE
Fig. 20 presents v and Zout outputs of the OrthoRF model. Notably, the Rout output exhibits gaps in the shapes of deeper layers, such as the ellipsoid layer. However, the 1 output (top row) provides information about these missing parts, enabling more complete layer reconstruction.
D USE OF LARGE LANGUAGE MODELS
We used large language models solely to polish the writing. All scientific claims and analyses were conceived and validated by the authors.
Table 9: OrthoRF architecture with input dimensions h, w, and feature dimension d. All fractions are rounded up. We followed the same architecture as RF Löwe et al. (2024a).
Layer
Feature Dimension
Kernel
Stride
Padding
Input Output
Input
h x w X
Conv
h/2 x w/2 x d
3
2
1/0
Conv
h/2 w/2 d
3
1
1/0
Conv
h/4 x w/4 2d
3
2
1/0
Encoder
Conv
h/4 x w/4x 2d
3
1
1/0
Conv
h/8 x w/8 2d
3
2
1/0
Reshape
1 (h/8 * w// 2d)
Linear
1 x 2d
Linear
1 1 X (h/8 * w/8 2d)
Reshape
h/8 x w/8 2d
TransConv
h/4 x w/4 2d
3
2
1/1
Decoder
Conv
h/4 x w/4 2d
3
1/0
TransConv
h/2 x w/2 x
3
2
1/1
Conv
h/2 w/2
3
1
1/0
TransConv
h W
3
2
1/1

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Published as a conference paper at ICLR 2026
(a) Example
(b) Example 2
(c) Example 3
(d) Example 4
Figure 12: Comparison of OrthoRF and RF performance for Row 1: RF (Löwe et al., 2024a) displays distributed object content. Row 2: OrthoRF's & displays distinct objects, including occluded regions. Row 3: OrthoRF's zout shows distinct objects; holes indicate occluded regions.

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Published as a conference paper at ICLR 2026
Table OrthoRF architecture with input dimensions (h, w, c) and feature dimension d. presented for the &NIST&Shape dataset. All fractional values are rounded upward.
Layer
Feature Dimension
Kernel
Stride Input Output
Padding
Input
h x w C
Conv
h/2 w/2 d
5
2
1/0
Conv
h/2 x w/2 2 x
5
1/0
Conv
1/4 x w/4 x 2d
5
2
1/0
Encoder
Conv
h/4 x w/4x 2d
5
1
1/0
Conv
h/4 X w/4 2d
5
1
1/0
Reshape
1 X (h/4 w//4
Linear
x 1x1x x 2d
Linear
1 1 X (h/4 w/4 x 2d)
Reshape
h/4 x w/4 2d
TransConv
h/4 x w/4x x 2d
5
1/0
Decoder
Conv
h/4 x w/4 2d
5
1/0
TransConv
h/2 w/2
5
2
1/1
Conv
h/2 x w/2
5
1/0
TransConv
h X W d
5
2
1/1
Table 11: Hyperparameters of the OrthoRF for the five different datasets.
Dataset
SEM
noisy SEM
4Shapes
MNIST&Shape
Shapes
Training Steps
200k
200k
200k
100k
100k
Batch Size
16
16
16
16
16
LR Warmup Steps
1000
1000
1000
Peak LR
0.001
0.001
0.001
0.001
0.001
Gradient Norm Clipping
0.1
0.1
0.1
0.1
0.1
Feature dim d
64
64
64
128
128
Rotating dim n
5-9
Image Size
256
256
32
32
40
Crop Size
200
200
32
32
40
Cropping Strategy
CenterCrop
CenterCrop
Rescale Size
64
64
32
40
Augmentations
Objects/Image
2
2-4
No. Train Images
5k
10k
50k
50k
40k
No. Test Images
6400
6400
10000
10000
1000
8

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0
A
0
A
0
A
O
O
O
4
A
O
0
0
Figure 13: Qualitative results of OrthoRF's fimal for n
0
O
D
O
A
0
0
0
0
O
0
0
Figure 14: Qualitative results of OrthoRF's s vo for n 7.

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Published as a conference paper at ICLR 2026
4
0
O
O
0
Figure 15: Qualitative results of OrthoRF's final for 8.
A
0
0
0
0
O
O
0
4
A
O
O
A
Figure 16: Qualitative results of OrthoRF's ti for n
10

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Published as a conference paper at ICLR 2026
(a) Example
(b) Example 2
(c) Example 3
dd) Example
(c) Example 5
Figure 17: Qualitative comparison between RF (top row) and OrthoRF (bottom row) for n = when trained and tested in the noise-free SEM dataset. We used zfinal for a fair comparison. The leftmost image in every row is the input image.
11

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Published as a conference paper at ICLR 2026
(a) Example
(b) Example 2
(c) Example 3
(d) Example 4
(e) Example 5
Figure 18: Qualitative results of OrthoRF for n = 5. The top row displays the model's output (z (z)imat), while the bottom row shows the thresholded rotations that are used for the metrics calcula- tion. The leftmost image in every row is the input image.
12

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Published as a conference paper at ICLR 2026
(a) Example
(b) Example 2
(c) Example 3
(d) Example
(e) Example 5
Figure 19: Qualitative comparison between of RF (top row) and OrthoRF (bottom row) for n when trained and tested in the noisy dataset. We used zfina for a fair comparison. The leftmost image in every row is the input image.
13

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Published as a conference paper at ICLR 2026
(a) Example
(b) Example 2
(c) Example 3
(d) Example 4
(c) Example 5
Figure 20: Qualitative results of OrthoRF for 5. The top row displays the than and the bottom
row fiinat The finat in this case serves to fill the gaps for a layer such as ellipsoids layer. The leftmost image in every row is the input image.
14

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Published as a conference paper at ICLR 2026
Irput
Bct
Bat
Rot
cp
c
Irput
Rar.
Rot
Nist
Rot
Itd
Irput
For
2ct
Fot
Rot
Ret 5
6ot
Rot 7
Rot
Irput
Rct
Rot
Rot
Ret
hot 6
Rot
Irptt
Rar
Bet
Rot
Ret
Rot 5
Bct 8
Rod
Irput
Irput
2
Rot
Rot
Rot 5
Bot 6
Rot
Rot
Irput
Figure 21: Visualizations of OrthoRF's Zinul (top of each pair) and a/ninal (bottom of each pair) for samples from the Shapes dataset.
15


Reviews:
[{"review_id": "6O5pD4gp0w", "reviewer": "Reviewer_wHKK", "summary": "The paper introduces OrthoRF, a simple tweak to Rotating Features that makes each “orientation channel” specialize in a single object by adding a softmax competition and an orthogonality penalty. This reduces the usual phase-mixing and removes the need for post-hoc k-means, especially helping in overlapping/occluded regions where they can directly threshold an intermediate map to complete hidden parts. On synthetic 4Shapes and a custom SEM-style dataset, OrthoRF matches RF on standard object discovery and improves occlusion completion, with cleaner, near-orthogonal channels.", "questions": "- Can you report both models with the same decoding strategy so we can separate model effects from decoding choices? A small table would clarify this quickly.\n- Will you release code, configs, and a generator for the SEM dataset so others can reproduce the results?", "limitations": "", "rating": 4, "confidence": 2, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- The idea is intuitive: make channels compete and push them orthogonal so each object lands on its own axis. This yields cleaner, more interpretable features and avoids fragile clustering steps.\n- The method is lightweight to implement (softmax + inner-product loss) and shows consistent angle statistics and tighter clusters. --- Qualitative plots match the quantitative gains, which is good to see. Table 2 shows substantial gains over RF in noisy conditions", "weaknesses": "- Most evidence is on binary synthetic images. It’s hard to judge real-world usefulness without RGB/textures or common OCL benchmarks.\n- The headline occlusion gains rely on changing the decoding step. This makes it unclear how much of the win comes from the model vs. the decoding choice.\n- The method still needs n set to at least the number of objects plus background. It’s not clear how it behaves when object count varies or is unknown at test time. The paper doesn't show what happens when n is too small (fewer components than objects), n is too large (many unused components) or object count changes at test time\n- There’s no clear orthogonality ablation—it would help to see results when varying the orthogonality term, or comparing to alternative orthogonality formulations, to verify that this term is the main driver of the improvements.", "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": "xqQ4RnNlqR", "reviewer": "Reviewer_wSwc", "summary": "This paper introduces Orthogonal Rotating Features (OrthoRF), an extension of the Rotating Features (RF) autoencoder framework for unsupervised object-centric learning (OCL). The original RF model, inspired by neural synchrony, uses vector-valued activations where magnitude encodes feature presence and orientation/phase encodes object affiliation, but often results in distributed representations requiring post-hoc clustering.\n\nOrthoRF addresses this by imposing orthogonality as an inductive bias in the orientation space. It achieves this through two main architectural modifications:\n\n\n1. Competitive Binding: A per-layer, centered softmax over orientation components to encourage \"winner-take-most\" specialization, mapping each object to a single, distinct component. \n\n2. Orthogonality Regularization: An inner-product based $\\mathcal{L}_{ortho}$ loss that penalizes the squared off-diagonal mass of the Gram matrix formed by the latent vectors, explicitly driving cross-component similarities toward zero, effectively enforcing a $90^{\\circ}$ separation.  \n\nThese modifications produce one-hot-like object encodings, which eliminates the need for post-hoc clustering and results in more interpretable representations. In experiments on the 4Shapes and synthetic SEM datasets, OrthoRF matches or outperforms current synchrony-based models on object discovery metrics (ARI-BG, MBO). Crucially, OrthoRF demonstrates superior performance on the shape completion task ($MBO_{i}^{OV}$), especially in overlapping/occluded regions, and shows a unique capability to recover occluded object parts in its intermediate representations—a strength not observed in prior slot-based or synchrony-based models.", "questions": "1. The OrthoRF$^{\\text{thresh}}$ results, particularly the striking $MBO_{i}^{OV}$ performance, rely on using the intermediate $\\psi_{final}$ with a manually set global threshold of 0.1. Have the authors explored making this threshold adaptive, for instance, by linking it to the statistical properties of the activations (e.g., a multiple of the standard deviation) to maintain the method's unsupervised nature?  \n\n2. The paper uses a synthetic SEM dataset to show robustness to noise and occlusions. Could the authors elaborate on how they would apply the OrthoRF framework to a real, complex dataset, like CLEVR or MOVi-C, and what modifications might be necessary to the architecture or training protocol?  \n\n3. Given the effectiveness of orthogonality, have the authors considered applying the orthogonality loss constraint directly to slot-based models (like Slot Attention) in conjunction with their reconstruction or binding losses? This might provide further insight into the general utility of this inductive bias in OCL.\n\n4. In relevant works, the authors have missed a few important papers to cite/compare their method with. When talking about ROOTS and SAVi, the work on PathTracker and InT [Linsley et al., 2021, NeurIPS] also disentangles objects in space. Similarly the extension of this work, FeatureTracker [Muzellec et al., 2025, ICLR] disentangles the changes in feature and colorspace. \n\n5. I would highly encourage the authors to release the code publicly for further experimentation by the community.\n\n6. The authors should also mention about the training details and hardware setup. \n\n7. The citation for paper Rotating features for object discovery is repeated twice.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* Originality and Significance: The core idea of enforcing orthogonality in the orientation space of RF is highly original and delivers a substantial improvement in the interpretability and performance of synchrony-based models. The result is a robust model that eliminates post-hoc clustering and yields one-hot-like encodings.    \n\n* Handling of Occlusion: OrthoRF exhibits a unique and powerful capability: the recovery of occluded object parts in its internal representations. This is demonstrated on the challenging, realistic SEM dataset and provides a crucial step toward more robust scene decomposition.    \n\n* Clarity and Efficacy: The method is conceptually clean (competitive binding + orthogonality loss) and empirically highly effective, with strong results across various conditions, noise, and out-of-distribution tests.", "weaknesses": "* Hyperparameter Sensitivity: The orthogonality loss is controlled by a weighting factor, $\\lambda$. Table 1 shows that optimal performance varies across different settings of $n$ (orientation dimensionality), requiring a hyperparameter search for the specific dataset. For instance, $n=7$ prefers $\\lambda=0.1$ while $n=5$ prefers $\\lambda=0.8$. A more thorough sensitivity analysis or a proposed dynamic weighting strategy could improve the method's generalizability.   \n\n* Dependency on Final-Layer Output: The best-performing OrthoRF method (OrthoRF$^{\\text{thresh}}$ on $\\psi_{final}$) requires binarizing the intermediate map with a threshold (0.1) and no further post-processing. The dependence on a hand-tuned global threshold (0.1) for peak performance might limit its applicability compared to the fully unsupervised k-means approach used by RF and OrthoRF$^{\\text{kmeans}}$. The reliance on an *intermediate* layer ($\\psi_{final}$) rather than the final $z_{final}$ output for the $MBO_{i}^{OV}$ metric should be discussed as a limitation or characteristic. \n\n* The authors claim training on noisy and testing on clean data degrades performance. I encourage authors to look at Extreme Image Transforms (EIT) [Crowder and Malik, 2022; Malik, Crowder and Mingolla, 2023, Biol Cybernetics]. They show that training on EITs helps with robustness on object detection tasks. \n\n* The authors do not compare how their methods compare to other non-RF methods. I would encourage the authors to also compare their findings with methods like Stable and expressive recurrent vision models [Linsley et al., 2020, NeurIPS].", "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": "0hWDtG9HIm", "reviewer": "Reviewer_WVWM", "summary": "For the problem of forming multi-object representations, a previously proposed framework, \"Rotating Features,\" can be improved by requiring the learned features to be orthogonal. This immediately improves the interpretability of the features, but surprisingly also improves performance in various tasks of synthetic scene understanding benchmarks.", "questions": "* Can you suggest in retrospect for which problems is the original RF method still superior (as seen in part in Tables 1, 2, 3, 4)?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "* A straightforward method to add orthogonality constraints to a previously suggested loss.\n* Improved performance (e.g., better handle occlusion), using a method which is much more scalable than other algorithms in the field (e.g., slot-based methods, synchrony-based methods).\n* Improved interpretability, as the features are distinct rather than mixed (Figure 2 is impressive).", "weaknesses": "* The relative contribution of the two technical improvements ('Competitive binding in orientation space' and 'Orthogonality regularization') is not systematically analyzed, and it is unclear if both are needed. It seems the manuscript emphasizes the orthogonalization, but some of the heavy lifting is done by the improved 'gating' from the softmax.\n* Inconsistent improvement over the previous method (as seen in Tables 1, 2, 3, 4).", "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": "zK5ALe3ZnH", "reviewer": "Reviewer_WoQj", "summary": "The authors introduce OrthoRF, an extension of RotatingFeatures improving the state-of-the-art in unsupervised object discovery using synchrony-based representations. This improvement consists in enforcing orthogonality between the different latent spaces of the model, yielding internal object separation akin to Slot-based models. The authors evaluate the performance of their model on 2 synthetic datasets: 4Shapes and SEM and show that OrthoRF tends to outperform RF.", "questions": "**Noise Robustness.** Gaussian blur and additive Gaussian noise may preserve the structure of the image. Have the authors tested more disruptive corruptions (e.g. salt-and-pepper)? Also, do the results still hold under various SNR?\n\n**Figure 2 (n=5).** The figure indicates n=5, seemingly tying the number of channels to the number of slots. Why is this augmentation necessary? \nAlso, in Figure 7, the panel shows one full scene plus four single-object images. Are these single-object images used only for evaluation (e.g., to compute metrics) or also during training as part of the n channels? If used in training, how are they generated without label leakage, and what supervision signal (if any) do they introduce?", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "**Timely and important problem.** Object-centric learning is a central goal in vision; advancing synchrony-based models is particularly valuable because they promise binding without supervision or heavy architectural machinery.\n\n**Clarity and presentation.** The paper is overall clear and reads well; the motivation, method, and experimental setup are easy to follow.\n\n**Simple, intuitive, and promising idea.** Orthogonality serves as a practical geometric surrogate that encourages non-interfering latent spaces. It’s easy to implement,  and serves as a strong regularizer that constrains networks toward clean, object-separated representations.\n\n**Good quantitative evaluation of learned representation.** The similarity and separability analyses (Section 4.2) give a concrete, interpretable view of internal representation geometry. These measurements constitute useful standardized evaluation practices for synchrony in future work, beyond task metrics alone.", "weaknesses": "*Major*\n\n**Loss of flexible dynamic binding.** A core weakness of slot-based methods is their fixed capacity: a preset slot budget limits generalization when scenes contain fewer or more objects than during training. Synchrony-based models explicitly aim to avoid this by distributing object clusters in a continuous phase space, enabling dynamic, per-image allocation. By enforcing orthogonality across latent components, OrthoRF effectively funnels objects back into component-specific subspaces, re-introducing slot-like rigidity and sacrificing one of synchrony’s key advantages. \n\nIs it possible to add empirical comparisons with slot-based models showing that OrthoRF still performs better? For example: the authors could train OrthoRF on 4Shapes and evaluate with 1–8 (or more?) objects per image to demonstrate that OrthoRF can still outperform slot-based solutions when object count varies. Additionally, give the striking similarity between OrthoRF and slot-models, could the authors include the latter as baseline in addition to RF?\n\n**Decoder comparability on 4Shapes.** OrthoRF does not clearly outperform RF on 4Shapes (especially considering RF’s performance in Table 5). The gains appear mainly when thresholding psi_final for OrthoRF while using k-means on z for RF, and performance is better only when including overlapping regions in the evaluation. This mixes architectural effects with decoder choices. \n\nCan RF be evaluated in the same way—thresholding psi—so that both models use comparable decoding strategies, on both datasets? Or could the authors propose ways to cleanly separate improvements due to OrthoRF’s architecture from those due to decoding?\n\n**Benchmark scope and claim accuracy.** The Stanić et al., 2023 evaluation benchmark cited does not include 4Shapes (contrary to the statement at L268); their datasets are high-resolution, colored images (Tetrominoes, dSprites, and CLEVR). Later synchrony work (e.g., Gopalakrishnan, 2024) also reports strong phase separation on those more complex datasets. RF, even though includes some tests on 4Shapes, is also evaluated on colored naturalistic scenes (Pascal, FoodSeg). While simplified grayscale setups can be valuable diagnostically, most work on unsupervised object discovery with synchrony-based models since 2023 evaluates on RGB images, so returning to grayscale toy shapes can feel not fully aligned with current practice; framing this choice explicitly and adding at least one RGB benchmark would help align with field norms.\n\nCould the authors (i) clarify/correct the claims, (ii) either evaluate OrthoRF on the standard color/RGB benchmarks or demonstrate that prior synchrony models specifically fail under the proposed overlap/noisy regimes, (iii) discuss scalability to known failure modes, especially same-color object collisions in RGB (a classic challenge for synchrony). Does orthogonality alleviate phase ambiguity when two objects share hue/texture, or does it merely enforce cleaner axes without resolving color-based binding failures?\n\n*Minor*\n\n**Notation consistency.** The authors define z_out in Eq. 5, then switch to z_final in the subsequent sentence and Eq. 6, and the two are used interchangeably elsewhere. Please clarify whether they are identical; if yes, standardize on one term throughout (update figures/captions accordingly). If not, define z_final precisely and maintain the distinction.\n\n**Typo.** Table 2: “thesh” → “thresh”.\n\n**Capacity fairness.** In Table 1, AE/CAE may not be directly comparable to RF/OrthoRF if they have fewer parameters. Please report parameter counts and, ideally, include capacity-matched baselines or parameter-controlled ablations (and note any compute/step differences) to ensure that performance differences are not confounded by model size.", "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)