Identifiable Object Representations under Spatial Ambiguities
Authors: Avinash Kori
Francesca Toni
Ben Glocker
Venue: ICML 2025 poster (2025)
Keywords: Object-centric learning
identifiability
spatial ambiguities
Primary area: deep_learning->generative_models_and_autoencoders
TL;DR: We introduce a probabilistic model that resolves spatial ambiguities and provides theoretical guarantees for identifiability without additional viewpoint annotations.

Abstract: Modular object-centric representations are essential for *human-like reasoning* but are challenging to obtain under spatial ambiguities,  *e.g. due to occlusions and view ambiguities*. However, addressing challenges presents both theoretical and practical difficulties. We introduce a novel multi-view probabilistic approach that aggregates view-specific slots to capture *invariant content* information while simultaneously learning disentangled global *viewpoint-level* information. Unlike prior single-view methods, our approach resolves spatial ambiguities, provides theoretical guarantees for identifiability, and requires *no viewpoint annotations*. Extensive experiments on standard benchmarks and novel complex datasets validate our method's robustness and scalability.

Identifiable Object Representations under Spatial Ambiguities
Avinash Kori' Francesca Toni"
Ben Glocker
Abstract
Modular object-centric representations are essen- tial for human-like reasoning but are challeng- ing to obtain under spatial ambiguities, e.g. due to occlusions and view ambiguities. However, addressing challenges presents both theoretical and practical difficulties. We introduce novel multi-view probabilistic approach that aggregates view-specific slots to capture invariant content in- formation while simultaneously learning disentan- gled global viewpoint-lewe level information. Unlike prior single-view methods, our approach resolves spatial ambig subbiuuttes, provides theoretical guaran- tees for identifiability, and requires viewpoint annotations. Extensive experiments on standard benchmarks and novel complex datasets validate our method's robustness and scalability.
(b)
Figure 1. (a) Occlusion Ambiguity: the orange object, which occluded by the blue object, could any six plausible objects shown on the right. (b) View Ambiguity: the blue object is observed from two different viewpoints (represented with red arrow and dot), leading change in its overall shape. In general, identifiable representations resolve ambiguitics by determining the most plausible object under occlusion and correct object properties in case of view transformation by leveraging information from multiple viewpoints.
Introduction
The ability to capture the notion of objectness in learned rep- resentations considered to be critical aspect for develop- ing situation-aware AI systems with human-like reasoning capabilities (Schölkopf & von Kügelgen, 2022; Lake et al., 2017). Objectness can be characterised as understanding the enviromment from the perspective of its building blocks. These can further be divided into object-part composition (Hinton, 1979; 2022), which might be a potential reason why humans generalise across environments with few amples to learn from (Tenenbaum al., 201 Recent ad- vances in object-centric representation learning (OCL) have shown great potential in segregating objects in observed scenes (Locatello et al., 2020b; Kori et al., 2023; Löwe et al., 2024). Indeed, the goal of OCL is to enable agents to learn representations of objects an observed scene in the context of their environment, as opposed learning global representations as in the case of traditional generative mod- els such as variational auto-encoders (Kingma & Welling,
2013). OCL approaches enable agents to learn spatially disentangled representations, which is an important step in compositional scene generation (Bengio et al., 2013; Lake et al., 2017; Battaglia al., 2018; Greff et al., 2020) and understanding of causal (and physical) interactions between the objects (Marcus, 2003; Gerstenberg et al., 2021; Gopnik et al., 2004).
Recent progress in OCL has been limited to learning scene representations from single-viewpoints (Locatello et al., 2020b; Engelcke al., 2021: Singh al., 2021; Kori et al., 2023; Chang et al., 2022; Seitzer et al., 2022; Löwe et al., 2024). Although these approaches can learn meaningful object-specific representations, they encounter significant challenges stemming from spatial ambiguities such as oc- clusion and view ambiguities (see Fig. for examples). Additionally, while it has been hypothesised that these mod- els learn un-occluded object representations even in the case of occlusions. Learning from a single viewpoint fails cap- ture effective object representations, due to the presence of multiple plausibilities of partially or fully occluded objects and the effects of view transformations, as demonstrated in Fig. and highlighted by the results in Fig. 2 (we will revisit these results later in section 6). Another example of spatial ambiguities can be observed Fig. 3, where object
Department of Computing, Imperial College Lon- don, London, UK. Correspondence to: Avinash Kori <orior221 @imperial.ac.uk.
Proceedings of the 4rnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Identifiable Object Representations under Spatial Ambiguities
View Encoder
Scene
+0
Ti-
Network
M(V)
Transformation
Skat
(s" T (xe")
Optimal Prior p(c)
342
O
Content Encoder
qfc {o .....'"
Figure 2. Identifiability across a number of views measured with Slot Mean Correlation Coefficient (SMCC).
Posterior
C2
O₂ in x and x² can be interpreted as a cube, but only after considering x3 we can conclude that being a pyramid.
Figure 3. The figure illustrates scene with four objects O {O,, O2, O3, O4}, observed from three different viewpoints. each described with a set of clearly visible objects: O' {Oa, O.}, O² {O1, Oz, O4), Oa (O1, O2, O2, Ot}. The corresponding images are passed through view and content en- coders, and sampled global view vector is used to estimate transformation function To given by parameters 0" predicted us- ing a localisation network. We apply view-specific inverse Tg on respective images projecting them to an implicit space, which is used to learn view conditioned slot posterior corresponding to GMMs represented by g(s" |Tg= (x")), which are further aggre- gated to marginalize viewpoint information, resulting in a content posterior, also a GMM q(c {s'....,s s which is further accu- mulated across all samples resulting optimal prior p(c).
A handful of approaches, including MULMON(Li et al., 2020), DYMON(Li al., 2021), OCLOC(Yuan et al., 2024), have considered multiple viewpoints for extracting object representations. Additionally, methods such (Liu et al., 2025; Chen et al., 2021; Luo et al., 2024) effectively use NERF (Mildenlll et al., 2021) for constructing 3D environment from multi-viewpoint images, where the occlu- sions are addressed by construction. Among these methods, MULMON, DYMON, and all NERF based approaches assume that the viewpoint annotations are known, which simplifies the problem of learning to disentangle object rep- resentations conditioned on viewpoint information.
The problem setting in this work aligns with OCLOC, in that, our aim is to learn invariant object representations while simultancously learning global view information with respect to an implicit global coordinate frame. This elim- inates the requirement for paired viewpoint-image data. While OCLOC introduces innovative approach for learn- ing global view information independently of the scene, its primary focus is on achieving bbject-consistens uncondi- tional to views rather than explicitly learning view-invariant object representations. Additionally, learning global uncon- ditional view representations does not guarantee learning identifiable view/object representations, which was stud- ied for OCLOC In this work, provide novel model, where object representations satisfy view-invariance and view representations satisfy approximate equivariance prop- erties, allowing us to exploit objects' inherent geometry and semantics to establish correspondences across views.
knowledge, this is the first work addressing explicit formal- isations of assumptions and theory required for achieving identifiable object representations under occlusions with multi-view observational data. To this end, we make use of the spatial Gaussian mixture models(GMM) in latent distribution across viewpoints to encourage identifiability without additional auxiliary data. Our main contributions in this work can be summarised as follows:
(i) We propose a probabilistic slot attention variant, View- Invariant Slot Attention (VISA) for learning identifi- able object-centric representations from multiple view- points, resolving spacial ambiguities such as occlusions and view ambiguities (Section 4).
iii)
We prove that our object-centric representations are identifiable in the case of partial or full occlusions with- out additional view information up to equivalence relation with a mixture model specification (Section 5).
In single-view OCL. Kori et al. (2024); Brady et al. (2023): Lachapelle et al. (2023) make an effort in rigorously formal- ising the underpinning, explicit and implicit assumptions and provide conditions under which models result in learn ing identifiable slot representations, leaving out ambiguous scenarios. Unlike them, our approach resolves spatial am- biguities, provides theoretical guarantees for identifiability, and requires no viewpoint annotations. To the best of our
(iii) We provide conclusive evidence of our identifiabil- ity results, including visual verification on synthetic datasets; we also demonstrate the scalability of the pro- posed method on two new, carefully designed complex datasets MOVI-VC and MVMOVIDD (Section 6).
2

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Identifiable Object Representations under Spatial Ambiguities
2
Related Works
3 Preliminaries
Identifiable Object-centric learning. Extending nonlin- ear Independent Component Analysis (ICA) from represen- tation learning to object-specific representational learning has been heavily explored before (Burgess et al., 2019; En- gelcke et al., 2019; Greff et al., 2019) by employing an iter- ative variational inference approach (Marino et al., 2018), whereas Van Steenkiste et al. (2020); Lin et al. (2020) adopt more of a generative perspective, studied the effect of object binding and scene composition empirically. Recently, the use of iterative attention mechanisms has gained a signifi- cant interest (Locatello et al., 2020b; Engeleke et al., 2021; Singh et al., 2021; Wang et al., 2023; Singh et al., 2022: Emami et al., 2022). Most of these works operate in single- view setting, which causes fundamental issues of viewpoint ambiguities in terms of occlusions and uncertainties in bind- ing. Recent methods, including (Eslami et al., 2018; Ar- salan Soltami al., 2017; Tobin et al., 2019; Wu al., 2016) consider a single object from multiple views tackle this particular problem. Additionally, (Kosiorek al., 2018; Hsieh et al., 2018, 8; Li et al., 2020) explore multi-object bind- ing in videos and multiple views, tackling object binding issues across frames. Despite their empirical effectiveness, most of these works lack formal identifiability guarantees. In line with recent efforts analysing theoretical guarantees in object-centric representations (Lachapelle et al., 2023; Brady et al., 2023; Kori et al., 2024), we formally investigate the modelling assumptions and their implications for achiev- ing identifiability guarantees in the context of multi-object, multiview object-centric representation learning settings.
Probabilistic Slot Attention (PSA) as introduced by Kori et al. (2024), presents a probabilistic interpretation of the slot attention algorithm (Locatello et al., 2020b). In PSA, set of feature embeddings z R per input x is taken as input, and an iterative Expectation Maximization (EM) algorithm is applied over these embeddings. This process results in a Gaussian Mixture Model (GMM) characterized by mean (μ E R xd), variance (a² E R xd), and mixing coefficients ( E [0, 1] x]xx1). In summary, PSA employs the initial mean sampled from the prior distribution and initial variance initialized with unit vector, then iteratively updates the mean based assignment probabilities (Ank) using Eqn. 1, and adjusts the mean and variance accordingly as described in Eqn. 4, for T iterations.
Ank = T(t) (zni ((t) (t)
(Zni
(1)
E (1) (Zni (Zn:
Ank Ank/
Auii π(( + 1)x
Ant/N;
(2)
1=1
n=1
11l+ + 1)k- =
Σ
Ans Znn
(3)
N
at + 1) = Σ Ant (zn pu(t + 1)x)²
(4)
Identifiable representations. A model is considered iden- tifiable when different training iterations yield consistent latent distributions, thereby resulting in identical model pa- rameters (Khemakhem et al., 2020a;c). In the context of parameter space Q and a family of mixing functions F, identifiability of the model on the dataset X' is established if, for any 61, 02 ~ θ and fe, p(/o7((x) (x)) = p( (x)) holds for all x E X, implying 0 = 0g. However, in practical scenarios, exact equality or "strong" identifiability is often unnecessary, as establishing relationships to transformations, which can be manually recovered, proves equally effective, resulting in a notion of weak identifiability, where relationships are recovered up an affine transformation (Khemakhem et al., 2020c; Kivva et al., 2022). Similar identifiability relations have been elucidated for OCL in (Brady et al., 2023; Lachapelle et al., 2023; Kori et al., 2024; Mansouri et al., 2023). The notion of ~u equivalence relation is claborated in Dfn. 3. Definition 3.1. (~, equivalence (Kori et al., 2024)) Let So S + X' denote mapping from slot representation space S to image space X' (satisfying Assumption F.4,, the equivalence relation ~rs w.r.t to parameters 0 E θ is defined as: 01 ~e 02 +
Multiview nonlinear ICA. It has been noted that address- ing the challenge of nonlinear ICA can involve incorporating a learnable clustering task within the latent representations, thereby imposing asymmetry in the latent distribution (Wil- letts& & Paige, 2021; Kivva et al., 2022). Morcover, Gresele et al. (2020) delve into multiview nonlinear ICA, particu- larly in scenarios involving corrupted observations, where they aim recover invariant representations while account- ing for certain ambiguities. Along similar lines, Daunhawer et al. (2023); Von Kügelgeen et al. (2021) explore the concept of style-content identification using contrastive learning, fo- cusing on addressing the multiview nonlinear ICA prob- lem. Here, we work along similar lines by emphasising the learning of invariant content and identifiable object-centric representations. We achieve this by formulating recon- struction objective where the enforced invariance and equiv- ariance stem from the underlying probabilistic graphical model rather than relying on contrastive learning objective. Similar to the noiseless setting in (Gresele et al., 2020), we demonstrate the recovery of invariant content representa- tions using different subsets of viewpoints.
F, the condition
(x:v) + +a), 3P. H. fo. (x:)) P(Fo,
ax VX E V
X,
3

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Identifiable Object Representations under Spatial Ambiguities
puts x [2, v), its inverse is denoted by x = / (x; v), which signifies the reversal of f conditioned on a variable In the rest of this section, we introduce all the components involved in our model. We also introduce assumptions, ex- amples, and intuition wherever necessary. Considering the generative model Eqn. 5, which is overviewed in graphical model Fig. 4, any scene x is generated using view v and content c. Here, both с and v are latent variables learned with variational (ifenence(KKngmaa & Welling, 2013).
u
T
c = To (c)
x
e
K
N
V
Figure 4. Graphical model for multi-view probabilistic slot at- tention: For every image in dataset a view v E Rd= ~ p(v), this view is used to compute transformation To.. Similarly, desired number (<. K) of content representations e E B are sampled content distribution p(c). Finally, the image x is generated using the transformed content To- (c) and view v.
p(x) |f p(x" To.(c), v") p(c) p(v") dv" de (5)
where P E P S (0, K is permutation matrix, H E Bdxd is an affine matrix, and E Rd.
View model. Given that the view property remains consis- tent across all objects, we treat the view as a global, image- level property as opposed to Yuan et al. (2024), where view is treated as an object-level property. Assuming access to a discrete set of viewpoints denoted by A, we consider prior over a view distribution to be a GMM represented by p(v) = -E T (y; ters of this GMM, we consider the posterior of the form (v x") V E A In both prior and posterior, we con- sider the covariance to be diagonal, implicitly making an ICA assumption (Khemakhem al., 2020a). The sampled variable
4 VISA Formalism
Let x = [xxx....'"' e X" x... x XV, V views of the same scene observed from different viewpoints with an observational space X C IR xVxHxW/ xc, We consider [V] as a shorthand notation for {L...., V). Let O = O' u .UO correspond to an abstract notion of object sets of an environment, while O", Ve E [V] is a set of objects present in considered viewpoint v. Importantly, we consider that the number of objects per viewpoint can vary, i.e., |0 U -. .. 2 |0 A v E [V]. allowing for partial or full occlusion in some viewpoints. Let v.V E =V=V X .... X VV C V xd. be inferred viewpoint-sppecfiic information' E = St x... SV C V xxx, correspond to a viewpoint-speeifie slot representation. Let C1.K E C C kxxd capture the notion of an aggregate, effectively accumulating the object knowledge across viewpoints. For any subset A of [V]. we represent scene observations as xa ===fxx= [xx : Vi E A} E XiE The inferred viewpoints and the view specific slots are denoted as yd = (v' Vi E A} E XseAV', and $i.x = {si:x : Vi E A} E XI, respectively. We define (c) as the distribution of e over A. A more comprehensive summary of notations and terminologies is provided in App. A.
To learn the parame-
(o | x²) is used to estimate transforma- tion parameters 0 E RR32 as in Jaderberg et al. (2015) which makes an affine transformation map To which is later applied on content c and on view-specific slots s. It is important to note that we use the same set of parameters & across all viewpoints in A for inferring view information v.
while V
Viewpoint specific slots. As illustrated in Fig. the in- ference of depends on the view-specific slots For considered image x", v A, we first apply an inverse view transformation Tg- and model the slot distribution as a spatial mixture model represented by ((1 Tx- The inverse transformation makes sure that the estimated object representations across all view in A are in com- mon implicit representation space. As this is an inter- mediate variable which does not show up in our genera- tive model in Eqn. 5, we update the corresponding pa- rameters with closed-form equations via expectation max- imisation algorithm as in (Kori et al., 2024). The result- ing slot posterior is a conditional GMM as described in Eqn. 6, where x" = TT='( is a transformed inputs, (x²(), of (x*), (x")) are mean, diagonal covariance, and mixing coefficients for the considered a view and object.
In modelling, w. w.l.o.g g, we consider access to certain subset A C [V], ensuring the model's 's applicability across different scenarios. Furthermore, to simplify notation, we sometimes do not include the superscript denoting the full set of views, thereby using x = xA, $1.K six, and 4 interchangeably. Likewise, if we do not specify the subscripts for and s, it implies they represent the entire collection of objects, specifically as = $1.x and CLKK- Lastly, for any function ff that operates on two distinct in-
K
((s") x") =
Σ n (x (x") (sssH)()))) (((²))
(6)
We abuse the terminology by considering viewpoint, lighting, object dimension, be encoded in a representation v. Note that the vin is inferred by the model.
We consider the parametric form of q to Gaussian.
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Identifiable Object Representations under Spatial Ambiguities
Representation matching. Given the permutation equiv- ariance property of slot representations, we use matching function with a permutation matrix P m, S + 8 such that (s(.k.)) = . mapping representation axis w. w.r.t. t. P The permutation matrix p is estimated by considering the slots of the first viewpoint s as a base repre- sentation, and other representations s" Vu E A are matched to align with it. We utilise Hungarian matching, as illus- trated in (Locatello et al., 2020b; Wang al., 2023), to estimate this permutation matrix P to control the noise in the matching algorithm, we introduce view-warmup strategy, which we detail in App. G.5.
both additive and non-additive (ref. definition E.1) mixing functions fa C x V - x For additive decoders, we use ppatialbroadccsttnng (Greff' et al., 2019) and MLP decoders, and for non-additive mixing function, we use auto-regressive transformers (Vaswani et al., 2017). We use the shared decoder fa for all views and objects, modelling the conditional distribution p(x" Ta- (c), v*). To train our model in an end-to-end fashion, we maximise the log- likelihood of the joint p(x*), which results in the evidence lower bound (ELBO), Eqn. 9, check Lemma F.I. Here, we consider the distribution form of p(x" c, v") to be Gaussian with learnable mean with isotropic covariance.
Content aggregator. We consider g : S + C as a con- tent aggregator function, which marginalises the effect of view conditioning. To achieve this, we consider convex combination of all the aligned slot representations (aligned to base representation), considering mixing coefficients TT& (xx" (we use T. for simplicity) in Eqn. 6 as a combina- tion weight. The convex combination accounts for potential object occlusions, which may cause objects to be absent in particular views ensuring only active representations are combined (refer to an intuition below), resulting in a con- tent posterior (g(c s)), which is GMM with mixing
E log (x| To (c), v) - (gV x) ll P(V)) (9)
achieves O(VTN(Kd) with the added complexity of (0(VV)) for inverse and forward view point transformation given by Te. while it retains the complexity per view to be ((TAKdd) which is the same slot attention and probabilistic slot attention. Additionally, the representation matching function contributes O(VIC d): this term does not alter the dominant term, in the general case when K. << Similar to PSA, when VISA is combined with an additive decoder, the complexity of the decoder can be lowered due to the property of automatic relevance determination (ARD), eliminating the need to decode inactive slots.
Computational
complexity: VISA
coefficients 7e (E m&) /IA| and the parameters described in Eqn. 8 (w.1.o. we consider s, π to rep- resent aligned representations), refer to Lemma E3, with w, 1/| Vi A. Additionally, algorithm I details the
entire forward process.
5 Theoretical Analysis
Intuition: Content aggregation
In this section, we leverage the properties of the proposed model to theoretically demonstrate the learning of identifi- able representations under challenging spatial ambiguities. In this work, we consider our data-generating process to satisfy a viewpoint sufficiency assumption (refer to 5.1).
Based on illustrated example in Fig. 3, for images x², x²,x²,x the resulting matched slots and mixing coef- ficients correspond to s²
{s), bb. sb.. sb....) s = =(S6 .og. S...
(s² s² b.,. sio,. to..s,. . s").. where S',,S,, Sy, and Se correspond to slot representa- tion for object O., random slot representation and back- ground information. respectively, with mixing coefficients x¹ {0. 1,,, = (1,0, 1, and न 1,,1, Proposed aggregation merges the slots ig- noring the random slots s", resulting in co, (86, + $ 12, co. = s0, and on.
3
3 so s.).
Assumption 5.1. (View-point sufficiency) For any set A C [V]. we consider set A to be view-poing wew-point sufficient iff (A|
10% This basically means that all the objects are visible across all the considered views A, even when an individual view may not contain all the objects.
Example: Viewpoint-sufficien
Example 1. Based on illustrated example in Figure
|
3, the scene is composition of four objects O {O,,0s O2, O3 Oa). view point subset A == = = (,) is considered to be view point sufficient since UPE O" 10s, O.) U (O,, Os, Oa} U (Oi, Oz, O+} = O".
L. n I. g(11
Σ
iikk
SiKi
A A
AA1
(xxx =
A p4 (x"):
Given that we learn the parameters of our view-specific spatial GMM with closed-form updates, we do not use ( an explicit prior minimising KL divergence. Instead, we rely on the fact that marginalising the effect of data points from posterior (aggregate posterior) is an optimal prior (Hoffman & Johnson, 2016; Kori et al., 2024), resulting in
[A|
2
8 ox(x1) = Σ
π
o
(x");
A T
v=1
Mixing function and training objective. We consider
5

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Identifiable Object Representations under Spatial Ambiguities
p(c) - fs q(c]s4, dx Given that GMMs are uni- versal density approximates given enough components (even GMMs with diagonal covariances), the resulting aggregate posterior g(c) = p(c) is highly flexible and modal. It often suffices to approximate it using a sufficiently large subset of the dataset if marginalising out the entire dataset becomes computationally restrictive.
Theorem 5.4. (Invariance of aggregate content) For any
subset A, B C [V], such that |A] > 0, [B] 0 and both A, B satisfy assumption 5.1, consider aggregate con- tent to be invariant if fA ~ fs for data XA x X²
Intuition: Invariant slots
Considering an example 1, with A = (1, 11 (2.3), 3}, such that sets A, B are viewpoint sufficient. Let fA and fr. be trained models on X and X respectively, Resulting in C = =J (x v²) = {co,, CO2, CO3, Co., co,) and = {co,, Co2, CO2, Co2 , Thm. 5.4 states that the representations Tg (Co, ) can be mapped Taa (co,, by permuting object indices and an affine transformation.
Lemma 5.2 (Optimal Prior). For A E [V], given the a local content distribution q(C1.K s1:K:x4) x²) (per-scene XA E {x { 1),, which can be expressed as GMM with K components, the aggregate posterior q(c) is obtained by marginalizing out x, s is a non-degenerate global Gaussian mixture with MR K components:
(xA;
C. = -E (x=:x"vvv
M K
p(c) = 9())=
1 M
Σ
Σ
N (c: (ikk 0kx)) (10)
Proof Sketch. To prove this, we extend the proof of Thm. 5.3, and establish that there exist two inevitable affine func- tions A, hB for mixing functions fA. fB : C x V + X to map representations e with a given view set v to observa- tions x Later, we show that, in the case of invariance, an affine mapping exists from A to hg.
Proof Sketch. The result is obtained by integrating the prod- uct of involved posterior densities q(c () x)p(x)) Fur- ther, we verify if the mixing coefficients sum to one in the new mixture, proving the aggregate to be well-defined. D
Theorem 5.5. (Approximate representational equivariance) For a given aggregate content c. for any two views V, V Pa(v), resulting in respective scenes x PA(x v.c) and & ~ PA(X v, for any homeomorphic transfor- mation ht E Hz such that lx(x)) = 8, their exists an- other homeomorphic transformation hy E Hy such that He S Hr C RRdim(x)) and v = he (fz ()x((x))):
With this, we show three main results: firstly, we show that aggregate content representations (c) are identifiable with- out supervision (up to ~2). Secondly, we show that these representations are invariant to the choice of viewpoints un- der assumption 5.1. Finally, we show that the model exhibits in an approximate view equivariance.
Theorem 5.3. (Affine Equivalence) For any subset A C [V], such that [A] > O, given a set of images x E x and a corresponding aggregate content c E C and a non- degenerate content posterior q(c s²), considering two mixing function fa: fa satisfying assumption F.4, with a shared image, then e are identifiable up ~u equivalence.
Remark 5.6. Note that the theorem only says that the trans- formation function transforming the view representations v as an effect of the homeomorphic transformation of x lies in the same subspace of input transformations.
Intuition: Approximate equivariance
Intuition: Affine equivalence
In the scenario when the cameras are positioned such that they have overlapping fields of view, and their relative pose (rotation and translation) must avoid degeneracies like align- ing on the same plane or mapping points to infinity. This results in the transfornation between views being smooth, invertible, and consistent. the scene is planar or depth variations are minimal, the homography can capture the transformation accurately without the need for inverse ren- dering. Notably, the cameras should have non-zero rotation and translation to avoid collapsing the scene, and their in- trinsic parameters must be known or identical to prevent distortions. When the scenario satisfies all the above prop- ertiex, the 2D homography transformation H between two camera views can be learned as homeomorphic trunsfor- mation (Hartley & Zisserman, 2003).
Considering an example I, with two perfectly trained mod- els fa and fa. Resulting aggregate contents are described Co2, 1- (x (xA:v") v²) coo,, Co2y Co2, Co,} and ē= 1=x
{coz, čo,, Co,; Co., Co,} for A = [V] 31, ~o equivalence states that there exists a permutation matrix P which aligns the object order in C. to match with e and there exists and invertible affine mapping A such that Co2- Aco, vk E (142,3,4)
Proof Sketch. Sketch. tiple steps as described below: (i). We demonstrate the distri- bution p(c) obtained as a result of lemma 5.2 is non-degen- erate and a valid distribution, (ii). With the above results, we demonstrate invertibility restrictions on mixing functions, (iii). Finally, we constrain the subspace to affine, demon- strating ~n of aggregate content c.
To prove the following result, we follow mul-
Proof Sketch. We prove the following result by following the steps in Thm. 5.4, over a view distribution p(v) but for fixed content vector c.
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Identifiable Object Representations under Spatial Ambiguities
6 Empirical Evaluation
shared CNN to produce an output of size W x H 4, where W and H are width and height of the image, respectively, The output channels encode RGB color channels and an (unnormalized) alpha mask. Further, we normalize the alpha masks with Softmax and per- form convex combinations to obtain reconstruction. Shared CNN architecture: 3 [Conv(kernel = x 5, stride = 2), Leaky + Conv(kernel 3 x 3, stride 1), (0.0))
Given the work's theoretical focus, experimentally, we aim to provide strong empirical evidence of our identifiability, invariance, and equivariance claims in a multiview setting. We also extend our experiments to standard imaging bench- marks, including CLEVR-MV, CLEVR- et al., 2020): we additionally demonstrate the framework's scalability to highly diverse setting with GSO (Downs et al., 2022) and proposed datasets MV-MoVIC MV-MOVID which are multiview versions of MoViC dataset with fixed and varying scene-specific cameras (Greff al., 2022).
GQN (Li
2. MLP decoders- Input/Output: similar to the spatial broadcasting decoder, here, each slot representation is broadcasted onto N tokens (resulting in N x d) and augmented with position embeddings. Then the individual slot representation transformed with a shared MLP decoder to generate representation corresponding to feature dimension along with addi- tional alpha mask, which is further normalised with Softmax and used in creating convex combinations to obtain reconstruction. Shared MLP architecture: [Linear d, bias = False), LayerNorm (d) + 3 x [Linear(de (d, dnadden), LeakyeeL (dhidedd d + 1)
Experimental setup. To verify our claims on (i) identi- fiability claim, we train our model on a given view subset A S [V] and compare view averaged slot mean correlation coefficient (SMCC) measure as defined (Kori et al., 2024) (SMCC(s, B) = R. Ei= -i p(s.y. (0)()) for some permutation map T and affine transformation A), (ii) invari- ance claim, we train multiple models on different subsets of viewpoints A, B S [V] and compare the aggregate content representations across models, quantiflyng the similarities with SMCC, we consider this measure to be invariant SMCC (INV-SMCC), and finally, (iii) for subspace equivariance, we consider a trained model with a view subset A C [V] and compute MCC of view information v by applying ran- dom homeomorphic transformations on samples x x (which can also be done by considering samples xB ~ X where cameras relative position satisfy the required con- straints 5.5, and analyse p(v^) and p(v
+
3. Transformer decoders:
Input/Output- transformer consists of linear transformers encoder output (N x d) and extracted slots (K x d) as input, while returning the slot conditioned feature as output with a dimension of (N x d). Transformer architecture: is made up of 4 transformer blocks, where each transformer block con- sists of a self-attention on input tokens, cross-attention with the set of slots, and residual two-layer MLP with hidden size 4 x d. Before the Transformer blocks, both the initial input and the slots are linearly transformed to d, followed by a layer norm.
Models & baselines. We consider two ablations with two types of decoders: (i) additive with MLPs and spatial broad- casting CNNs and (ii) non-additive decoders, which include transformer models. In all cases, we use LeakyReLU acti- vations to satisfy the weak injectivity conditions (Assump- tion F.4). In terms of object-centric learning baselines, we compare with standard additive autoencoder setups follow- ing (Brady et al., 2023), slot-attention (SA) (Locatello et al., 2020b), probabilistic slot-attention (PSA) (Kori et al., 2024),
CASE STUDY 1: ILLUSTRATION OF IDENTIFIABILITY To definitively show the validity of our claims about iden- tifiability (Thm 5.3, Thm 5.4, and Thm 5.5), we created a synthetic unconfounded scenario for modelling. This pro- vides us with two data modalities, Fig. 7 (i) projected point cloud data, and (ii) corresponding imagery data, we detail point cloud illustrations in appendix G.I. Additionally, this dataset also provides us with the ground truth object and viewpoint features for evaluation. To visualise the aggregate mixture, following Lemma F.2, we use the projected GMM to interpret the distribution of random variables in Rdd.
MulMON (L.i et al., 2020), and OCLOC (Yuan et al., 2024).
Architectures: As detailed in the paper we use two dif- ferent classes of decoder architectures: (i) additive and (ii) non-additive; within the additive architecture we use both spatial broadcasting and MLP decoders, for the non-additive architecture we use transformer decoders. Concretely, we follow SA (Locatello et al., 2020b) for spatial broadcasting decoders and DINOSAUR (Seitzer et al., 2022) for both MLP and transformer decoder. In detail we use:
The data-generating process is thoroughly explained in the App. D.1. In Fig. 5, we display the distributions of marginal- ized aggregate content distribution g(c), comparing individ- ual features and a mean feature across different runs that are either scaled, shifted, or split (increase in number of modes), which is reflective of affine transformation of fea- tures across runs. To quantitatively measure the same, we
1. Spatial broadcasting decoders - Input//utput: The generated slots are s E IR xd each slot representation is broadcasted onto a 2D grid of dimension 8 x 8 x d and augmented with position embeddings. Similar to slot attention, each such grid is decoded using a
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Identifiable Object Representations under Spatial Ambiguities
Run 40
Run #1
Run #2
Run #3
Run 46
Bar W0 Comb ned
Pun #11 Combend
Puch #2 Combined
Pun a Combtred
Purn Co-omiined
Figure 5. Identifiability of q(c). The top row indicates individual feature distribution across five different runs. The bottom row reflects the feature feature distribution, which use proxy for calti-dimensional features given Lemma F.2. observed, mean feature distribution across runs is either scaled, shifted, or split (increase in number of modes); this provides strong evidence of recovery of the latent space up affine transformations, empirically verifying our claims Thm. 5.3.
Table Comparing identifiability of g(s), q(c), and p(v) scores wrt existing OCL methods
METHOD
CLEVR-MV
GQN
GSO
SMCC
INCC
1CC
SMCC
INCC
MCC
INV-SMCC1
MCC 1
AE
0.32 .02
0.29 .02
0.24 + 0.08
SA
0.47 .03
0.38 .02
0.28 1. 0.06
PSA
0.49 :::::00 .02
0.38 + .02
0.30 + 0.04
MulMON
0.61 .03
0.62
06
0.61 +
0.56 + 0.04
0.48 0.06
OCLOC
0.63 .22
0.61 + .01
0.48 04
0.60 + .03
0.60 + .01
0.42 = .08
0.58 + 0.01
0.3-1 0.03
0.46 + 0.04
VISA
0.67 1:
0.66 .01
0.60 +: .4
0.39 1: 001
0.63 +
0.52 1: 0.60 +: .03
0.61 1
0.58 .03
Table 2. Identifiahility and generalisahility analysis Mv-MOVIC dataset.
METHOD
IN-DOMAIN
OUT-OF-DOMNIN
mB0 f
SMCC
INV-MMCC
MCC1
mB0
SMCC
INV-MMCC
MCC
SA-MLP
0.28 1: 0.091
0.36 +: 0.004
0.26 + 0.08
0.38 +: 0.006
PSA-MLP
0.30 1. 0.022
0.38
0.002
0.30 1: 0.03
0.40 1. 0.005
VISA-MLP
0.28 4: 0.021
0.52 =: 0.021
0.61 + 0.023
0.54 + 0.026
0.27 = 0.02
0.51 + 0.029
0.58 + 0.031
0.52 0.021
TA-RRRSSFRREER
0.34 1: 0.014
0.36 d. 0.016
0.33
0.041
0.36 d: 0.043
TRANSSRSSRR
0.37 0.021
0.38
0.007
0.37
0.033
+ 0.016
VISA-TRANSFORMER
0.38 1: 0.008
0.44 0.003
0.46 0.001
0.53 0.011
0.36 0.017
0.46 1: 0.033
0.46 :: 0.018
0.55 0.082
Run 80
Run #1
Run 02
the invariance of distribution q(c) across viewpoints (Thm. 5.4), we consider three different viewpoints. We use all possible pairs to learn g(c) distributions as illustrated in Fig. 6, where the distributions are described w.r.t view- points described by {g. r). fr. b). and {g. b), respectively. These distributions were also found have similar prop- erties as before, with an observed SMCC of 0.71 = 0.11. further confirming the claims in Thm. 5.4. Additionally, Fig. 2 demonstrates the improvement in identifiability as the number of viewpoints increases.
Min POLOmoncc
Rum CombEic
50n Combined
Figure 6. Viewpoint invariance for g(c). The top and bottom row indicates individual feature levels mean feature distributions, respectively. Each columns reflect marginalised aggregate content distribution q(c) when trained with different view pairs {(bluc, red), (green, blue), and (green, red)), respectively. As the resulting distributions with different datasets only vary by an affine transfor- mation, providing strong evidence for Thm. 5.4.
CASE STUDY 2: IMAGING APPLICATIONS. We first evaluate the framework on standard benchmarks, specifi- cally focusing on CLEVR-MV, CLEVR- GQN, and GSO with simple objects. Given the true generative factors are unobserved, derive our quantitative assessments from multiple runs. The results shown in Table confirming the validity of our theory on imaging datasets. Regarding the baseline comparisons that utilize single viewpoint, the INV-SMCC mirrors the SMCC due to its inherent design (i.e., aggregation of set with a single element is the same element). Moreover, in the case of AE, SA, PSA, and
computed SMCC and observed it to be 0.72 + 0.04, em- pirically verifying our Thm. 5.3. Furthermore, illustrate
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Identifiable Object Representations under Spatial Ambiguities
MULMON, the models do not estimate view information but either treat them independently use the observed view conditioning, rendering the MCC metric inapplicable. Fig. 12 showcases how the number of viewpoints impacts the iddetifiability of the s, v, and c variables; the involved exper- iments reflect the increase in performance with an increase in the number of views, across all benchmark datasets.
sumptions, particularly regarding the viewpoint sufficiency, are strong and may not always hold in practice. However, we did not observe limiting effects of this assumption on the proposed MV-MoVID dataset. A more extensive anal- ysis of this assumption and its implications in real-world applications is left for future work. We would also high- light that the weak injectivity of the mixing function may not always hold for different types of architectures. While generally applicable, the piecewise-affin functions may not always capture valid assumptions for real-world problems, e.g., when the model is misspecified. Neverthe- less, to the best of our knowledge, our theoretical results on multi-object, multi-view identifiability are unique and cap- ture key concepts object-centric representation learning, opening various new avenues for future research along the lines of generalisabiliyy world-modelling, and planning.
Additionally, we demonstrate our methodology on proposed complex datasets, MV-MOVIC and MV-MoVID, the latter dataset enables us to examine the model performs when the assumption 5.1 is not satisfied. To evaluate model behaviour in an environment with consistent objects but with different viewpoints, we conducted in-domain and out-of-domain (OOD) evaluations. For in-domain analysis, the model is trained and assessed on the same viewpoint group A [1, 3). Conversely, for OOD evaluation, consider the previously trained model but test it against a new set of viewpoints B- [3, 4, 5]. The findings presented in Table 2 regarding the MV-MOVIC dataset reveal that the SMCC, INV-SMCC, and MCC metrics show similar performance across both domains. This indicates that the distributional characteristics remain unchanged when both the training and testing environments contain the same objects. The MV-MoVID dataset analysis can be found in App. G.
Impact Statement
This paper proposes a view invariant slot attention algorithm, addressing spatial ambiguities with identifiability guaran- tees. The work extends theoretical advancements in the field of OCL, and such, it has little immediate societal or ethical consequences. Our method might be a step towards interpretable, equivariant, and aligned models, which are desired properties of trustworthy AL.
7 Conclusion & Discussion
Understanding when object-centric representations are both unambiguous and identifiable is essential for developing large-scale models with provable correctness guarantees. Unlike most existing work identifiability, which largely focuses single-view setups, we offer identifiability guar- antees in multi-view scenarios. We use distributional as- sumptions for latent slot and view representations, drawing inspiration from mixture model-based structures. To achieve this, we propose a model that viewpoint- wagnostt- and does not require additional view-conditionin information.
Acknowledgements
A. Kori is supported by UKRI (grant number EP/S023356/1), as part of the UKRI Centre for Doc- toral Training in Safe and Trusted AI. B. Glocker received support from the Royal Academy of Engineering as part of his Kheiron/RAEng Research Chair, and acknowledges the support of the UKRI AI programme, and the EPSRC, for CHAI - EPSRC Causality in Healthcare Al Hub (grant no. FPP/Y22885/1), Additionally, we thank Fabio De Sousa Ribeiro for insightful discussions and providing feedback the initial draft of the paper.
Our model specifically guarantees the identifiability of view- specific slot representations, niewpoint-invarial content representations, and view representations, all without the need for additional supervision (up to an equivalence re- lation). We visually validate our theoretical claims with unconfounded synthetic dataset with illustrative 2D data plots. We then empirically demonstrate the model's identi- fiability properties on multiple object-centric benchmarks, highlighting its ability to resolve view ambiguities in imag- ing applications. Furthermore, we showcase the scalability of our approach on large-scale datasets and more complex decoders using realistic datasets and transformer decoders, respectively, demonstrating its capacity to scale effectively with both data volume and decoder complexity.
Limitations & future work. We recognize that as-
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Identifiable Object Representations under Spatial Ambiguities
A Notations
O"
Abstract object set as observed from viewpoint v.
[V] [I
Exhaustive of viewpoints, representing all possible views.
A, [V]
Subsets of viewpoints, used for training.
X' = "
Data space, formed by the Cartesian product of data spaces for each view in subset A.
X = fx" : E A} Ex
Data sample, where x" is the data from view v, and x represents the set data across views in A.
fe
Encoder model maps input data a latent space or feature representation.
Spatial latent features, representing inferred spatial properties from the data.
s
View-specific slot space, a space for features that are tied to particular viewpoints.
View-invariant content space, representing features that are constant across different viewpoints.
SES
Samples from the view-specific slot space, representing view-dependent latent features.
CEC
Samples from the view-invariant content space, representing features that remain consistent across views.
S
Slot attention module, responsible for attending to and disentangling different parts of the input related to different views.
fa
Mixing function, which combines view-specific and iew-iivariaant features into a unified representation.
v
View information space, space that encodes information specific each viewpoint (e.g., angle, position).
VEV
A sample from the view information space representing a specific view or camera configuration.
fo
View extractor function, which extracts viewpoint-related information from the data.
Pe, Us ps
Mean of invariant content, view-specific slots, and view distributions.
Standard deviation of invariant content, view-specific slots, and view distributions.
Ga: 0s Do
Te: T.. T
Mixing coefficients of invariant content, view-specific slots, and view distributions.
Ank
Assignment confidence of a slot getting mapped to token n.
PEPC kxxK
Permutation matrix.
m,
Matching function, used to align object representations across views.
AK
Simplex in the space of dimension K.
Hy, H.
Space homeomorphic transformation.
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Identifiable Object Representations under Spatial Ambiguities
B Extended Related Works
Identifiable representation learning. Learning meaningful representations from unlabeled data has long been a primary objective of deep learning (Bengio et al., 2013). Several approaches, such as those proposed by (Higgins et al., 2017; Kim & Mnih, 2018; Eastwood & Williams, 2018: Mathieu et al., 2019), relied on independence assumptions between latent variables to learn disentangled representations. However, (Hyvärinen & Pajunen, 1999; Locatello et al., 2019) demonstrated the provable impossibility of unsupervised methods for learning independent latent representations from i.i.d. data. Which is tackled by restricting mixing functions to conformal maps (Buchholz et al., 2022) or volume-preserving preserving transformations (Yang et al., 2022), or with additional data assumptions (Zimmermann et al., 2021; Locatello et al., 2020a; Brehmer et al., 2022; Ahuja et al., 2022; Von Kügelgen et al., 2021). or by imposing structure in the latent space as in nonlinear Independent Component Analysis (ICA) (Hyvarinen et al., 2019; Khemakhem et al., 2020a;b), resulting in identifiable models. In the context of nonlinear ICA, (Dilokthanakul et al., 2016) introduced VAE model with a GMM prior, and (Willetts & Paige, 2021) empirically demonstrated the effectiveness of the GMM prior, which was later rigorously proven by (Kivva et al., 2022). (Kori et al., 2024) use this notion of latent GMM in the context of OCL, achieving identifiability guarantees for object-centric representations. Here, we use this notion in the context of multi-view object-centric representations, tackling the issues with spatial ambiguities and uncertainties in bindings.
Multi-view Object-centric learning. Recent progress in multi-view object-centric learning has seen notable contributions from methods like MULMON (Li al., 2020), ROOTS (Chen et al., 2021), SLOTLIFTERLLiu et al., 2025), and uOuo al., 2024), each offering distinct approaches compositional representation learning. However, these methods rely heavily on viewpoint annotations, which limit their applicability in fully unsupervised settings. MULMON refines object representations iteratively using annotated viewpoint-image pairs, while ROOTS, SLOTLIFTER, UOCF estimates 3D object positions performing an inverse rendering operation within grid and projects them into image space via viewpoint transformations. In contrast, we deal with fully unsupervised framework without the need of viewpoint annotations while providing approximate viewpoint equivariance for object representations.
Temporal Object-centric learning. An alternative approach to bypass the need for viewpoint annotations leverages tem- poral information. Methods for learning from single-viewpoint video sequences, such as Relational N-EM (Van Steenkiste et al., 2018), SQAIR (Kosiorek et al., 2018), SILOT (Crawford & Pineau, 2020), and SAVI (Kipf et al., 2021), focus on modeling object motion, interactions, and identity tracking across frames, even under occlusion. However, these methods assume fixed viewpoints, making them unsuitable for multi-view scenarios where objects appear in different spatial configurations. Additionally, object motion affects individual objects independently, unlike viewpoint changes, which influence the entire scene. Recent advances such as DYMON (Li et al., 2021) extend multi-view approaches like MULMON (Li et al., 2020) to dynamic scenes by disentangling object motion and viewpoint changes, assuming one dominates in adjacent frames. However, DYMON relies on viewpoint annotations, limiting its utility in unsupervised settings. Temporal methods such as SIMONE (Luo al., 2024) address this by leveraging temporal coherence across multi-view videos, using spatial and temporal positional embeddings to disentangle object and viewpoint representations. Yet, SIMONe's reliance on temporal continuity restricts its generalizability to scenarios where such coherence is absent. In contrast, our framework does not assume temporal dependencies.
C Algorithm
Here illustrate all the steps involved in the of proposed method VISA, refer 1.
D Datasets
D.I Illustrative dataset
To visually illustrate the effectiveness of our theory we experiment with two dimensional illustrative dataset. For this, similar to (Kori et al., 2024), we defined K = 5 component GMM, with differing mean parameters μ4 {f1,.... Ms J. and shared isotropic covariances, which we use to sample locations for an object. For a given location, we randomly select one object from { "cube', 'cylinder', 'pyramid', , 'sphere') } and generate 1000 random points on the surface of the selected shape uniformly covering it. To create a single data point, we randomly select three of the five locations and place a randomly selected object at the location. To include multiple viewpoints, we consider V camera location and project the objects, creating V different scenes. We then fill this by considering convex hull operation resulting in projected images
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Identifiable Object Representations under Spatial Ambiguities
Algorithm 1 View Invariant Slot Attention VISA
1: Input: A E [V] 2 (fx(x)) Ve E A) E RlAIxNxd
D. input representations
2: View: v = (v" ~ p(v", oz) VU E A} E RlAlad
D view representations P transformation parameters optional value key
3: View Transformation: 04 =1000 STN(V") Vo E A] E BlAIx2x3
4: key" +- Ta (z RlaxxNxd value" To (z Rl
5: #
6: for A do
7: Ak, (0); 4- K, 110);: NN Ia). o + 1a
8: fort = 0 + -o
9:
Ank* -- ΣK { (ko), (t)
> compute attention
10:
Ank +- Σi
Ank
b normalize attention
11:
mlt 1)x= +- >x==
N
Ank value,
update slot mean
12:
ot + 1) < "x=1 Ant (value, - p(t + 1)x)
> update slot variance
13:
Tt + 1)x- f N >
D= Ank
P update mixing coefficient
14:
end for
15:
s s S (()) o (T)) U (TTK
b> slot collection
16: end for
17: return ConvexCombination(s:: #)
K view invariant content
as illustrated in Fig. 7. To maintain uniformity, we only use imaging modality in the main paper while also demonstrating point cloud illustrations here in the appendix. We use different colours representing different objects in Fig. 8, ?? and used 10, 000 data points in total to train our toy models. Unlike existing benchmark datasets, here we remove all the confounding effects caused by lighting and depth. This provides an ideal test bed to validate all our theoretical claims.
D.2 Proposed dataset
In this work, we introduce the MV-MOVI datasets, created using Kubric (Greff et al., 2022), which feature multi-view scenes with segmentation annotations. We propose two variants of the dataset: MV-MOVIC, where the camera locations for every viewpoint remain fixed across all scenes, and MV-MOVID, where the camera locations dynamically change for each scene.
Both MV-MOVIC and MV-MoVID primarily consist of scenes generated by randomly selecting a background from a set of 458 available options and choosing K objects, where K S 6, from pool of 930 objects. In total, significantly high number of images can be generated in general. In contrast, for this work, we generate 72,000 scenes, each captured from 5 different viewpoints, with object segmentation masks for every view to facilitate the evaluation of model performance. In the case of MV-MOVIC, the locations of all five cameras are fixed across the 72, 000 scenes, while in MI,,,VID, the camera positions are dynamically sampled and vary across scenes.
E
Mask Generation
In the case of additive decoders, the decoder outputs K three channelled tensors along with K single channelled mask. We consider normalising these masks with softma fmax transformation along slot dimension, ensuring that each pixel only contributes to a single slot. The resulting softmaxed masks are used in composing (image = Σx mask; image,) the slots to reconstruct an image for training. During inference, we normalise masks with s sigmoid transformation, allowing us to estimate occluded objects visually, resolving the spatial ambiguities with occluded objects. In a later section, we illustrate the results with both so softmm ftmax and s immoid transformations.
E.1 Additivity Implications
Definition E.1. (Additive models) Function f is considered to be an additive decoder if, for any object decoders foby and masking mechanism Mobj. if they can be expressed as:
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Identifiable Object Representations under Spatial Ambiguities
Figure Data generating process: The figure illustrates point cloud data first row, with camera location highlighted red, blue, and green arrow. Following rows indicates projected images and point cloud as observed from red, blue, and green cameras, respectively.
f(z) Σ mobj(Zk) fobj(Za)
(11)
KE[K]
As pointed out in (Lachapelle et al., 2023), softmax-based masks do not truly fall under the category of additive decoders due to the competition between masks for groups of pixels. This implies that the additive decoders studied in (Lachapelle et al., 2023) are not expressive enough to represent the "masked decoders" typically employed in object-centric representation learning. The issue arises from the normalization alpha masks, and care must taken when extrapolating the findings from (Lachapelle al., 2023) to the models used in practice.
Although sigmoi d-based masks satisfy the condition of additivity during inference, it is important note that the model is still trained using soft normalization in our setting. The effect of using si gmoid masks during inference can be visually observed in App. G.
F Proofs
Lemma F.1 (ELBO). With prior distributions p(v) and p(c) for view and content latent random variables, the likelihood p(x) can be maximised by maximising the following expression:
logp(x) 2 Elogp(x (e=(c), (L x) p(V)) ELBO(x)
(12)
Proof. Considering the generative model Eqn. respecting the graphical model in Fig. 4, we get:
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Identifiable Object Representations under Spatial Ambiguitics
p(x) 11 p(x* Tem(c), v²) p(c) p(vA) dv de
(13)
logp(x)= log /f p(x* Su p(c1.)) p(v g(v, x²) q(v, x4) dv dei.x
(14)
/ʃ a(v x()(( (x=1(x))) logp(x' To (c)::K, vn) p(ví-k) p(cf-x)
g(v xx q(c.KK xx1 dv der. (15)
-E g(v" x")g(cn.k Tg (x")) To- (c)1., v") p(v1K) p(c "ЕΛ
dv" de1.K (16)
g(v x*) q(c1.x x²)
Given the iterative update for with EM algorithm, ideally expect posterior converge to prior, which results in:
Σ lf q(v" xx))(cnk TT- log (ox v") p(v(.k) q(v xx)
log pxx)
dv" de
(17)
"EΛ"
Σ
Be,v logp(x" To-(c), KL (g(v x") || p(v))
(18)
Given the subscript notation, the above expression can also be expressed as:
Bey log To. (c), v) (T x) p(v)) = ELBO(x)
(19)
Lemma F2 (Mean GMM). Lef z E B.N xd be random variable drawn from GMM with K components:
K
Z
N2; pkk, Ex),
(20)
Am
where &u are the mixture weights, fAx. E Rdd the mean vectors, and Σx BRdxd the covariance matrices. Assuming the mixture satisfies the ICA assumption, such that the components of statistically independent. A projected random variable E as the average over the dimensions of z:
1
Zjj
(21)
d
j=1
is also distributed according to GMM with K components, with appropriately transformed means and variances.
Proof. Given the random variable follows GMM, so its density can be expressed as:
p(z)
TNN(2; pki, Ex); Σ
(22)
1=1
where:
[4 = [Jk,,,, Mk,2,............
Σx = diag(lo2,1,0e
(23)
Considering, the projection of z onto defined as:
Zj-
(24)
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Identifiable Object Representations under Spatial Ambiguities
Given the ICA assumption, the components 2.,j are independent. For fixed component the projected mean and variance of can be derived
d
1
1
Var ()
(25)
d
RJ
Since the projection is a linear combination of independent Gaussian variables, remains Gaussian for each component k. Thus, the overall distribution of is also a GMM:
K
p(2)
Σ
Th (E:
(26)
where:
Mg,k d
MAJi
2
d
ki
(27)
j=1
==1
This concludes the proof.
Lemma F.3 (Convex Combination of GMMs). Let 8² = (sl...,,kk and 8² = = ,8)) be two sets of K random
vectors in Rd, each distributed according to GMMs:
K NPNN Σid)i S²~
Σ
T2 T,NNN Σ2,e)
(28)
where Pi, E ERR, Di, E Radd and T are the means, covariances, and mixing coefficients respectively.
Then for any weights W1, W2 E R such that w1 + w2 = the convex combination =wis wiss + w2s² is also distributed according GMM with K components.
Proof. Without loss of generality, assume the components of both GMMs are aligned. For each component we derive the parameters of the resulting mixture:
The mixing coefficients of the resulting GMM are weighted combinations of the original coefficients:
W 11 - W2722
(29)
For each component k the convex combination of Gaussians results a Gaussian distribution. The mean of the resulting Gaussian
f Wi + 42
(30)
T
The covariance of the resulting Gaussian for each component & can be derived as follows. Firstly, recall that for a random variable X, the covariance is:
Var(X) = E[ - (X E(X]) 1 EXXX E
(31)
First lets compute E[sus]] )
Wi T1, + 22.
W11 (WITTI,S) + W2 A
S = E
(32)
7
/ .: E[s | (s)) 1 + 3 E|s (s)
(33)
(T&))
W0 122 21 1 k
Els
E
((
(34)
( 2
19

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We further extend this q(c), with the result from Lemma F.3, we know that the g(c sA) is GMM with same number of Identifiable Object Representations under Spatial Ambiguitics
Then, substitute known expectations:
Els (s = E + MLRMi
(35)
E|s (s)) 11,,,2
(36)
Finally, by subtract E[st 1 i we get the covariance:
w 21.6 + w 22
Σg
(37)
[ )
To verify this forms non degenerate GMM, we show the mixing coefficients sum 1:
K
K
( 11 +
22
(38)
k=1
W1
11,
W2
2
(39)
W1
+ W2
1=1
(40)
Therefore, the convex combination results in valid Gaussian mixture model with K components, where each component has mean igg covariance Σx. and mixing coefficient πg.
Lemma 5.2. (Optimal Content Mixture) For A [V], given the local content distribution q(c, s x²) (per-scene x MM ) which can be expressed as GMM with K components, the aggregate posterior q(c) is obtained by marginalizing out is non-degenerate global Gaussian mixture with M K components:
M
p(c) - c(c)
NaN (c; flik: oiee)
(41)
M
Proof. We extend the proof in (Kori et al., 2024), by incorporating hierarchical slot to aggregate content formalisation. For which, we begin by noting that the aggregate posterior q(c) is the optimal prior p(c) so long as our posterior approximation a(c S AA, xA) is close enough to the true posterior p(c s xA), since for a dataset x E {x1² A AM Ji=1 for which we start with
g(s x)), wlog, given view point transformation deterministie, we consider x Toa (xx) we have that:
f ppss I x xxx ddx
p(s^) =
(42)
=xxxx)())) [p(s xd]]
(43)
M
p(s xx²1
(empirical approximation)
(44)
M
1
q(s
xx1)
(posterior approximation)
(45)
M
= q(s )
(46)
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Identifiable Object Representations under Spatial Ambiguities
components as g(s" 8²)) for any v [V] as follows
p(c) f ple s (*)p(s dss
(47)
Agpp(())) [p(c s*)]
(48)
M
p(c (s())
(49)
M
g(c s
(50)
M
= (()),
(51)
where we approximated p(x) using the empirical distribution, then substituted in the approximate posterior, marginalizing x to get p(s), we later consider the distributional form of p(s) and marginalise 5 to get p(c). This observation was first made by (Hoffman Johnson, 2016) and was used in (Kori et al., 2024) we use to motivate our setup. Given our model fits a local GMM to each latent variable sampled from the approximate posterior: zd g(z xx.), for i 1,..., M. Let fa (zA) denote the (local) the product of GMM density, its expectation is given by:
/I p(x p(x" )((=" xx (x")fo(z") Jdx dz
(p(x)))g(aadee (xd) [fa(x*)]
(52)
||
M Σ 8(x* xz)) x ) dz
(53)
1=1
M
= M Σ ( (x))(()))
(54)
/ M Σ N(z uxx 0,
K
Σ (x(xx)) JN (z:: (x(x)),, of dz
Σ 8(z^ u(x²)) Σ
n Tx (x("N IN (zA: xx(x))),(((( dza
(55)
K
T& (x
(56)
q(2
(57)
where we again used the empirical distribution approximation of p(x). and the following basic identity of the Dirac delta to simplify: f 5(x (x)xxx fx)).
For the general case, however, we must instead compute the product of g( x*) and (z rather than use a Dirac delta approximation as in Eqn. 55. To that end we may proceed as follows w.r.t. to each datapoint x;"
q(zA x(a) fo(zA) N( (((x))(())) of (xx+))
Σ (x()x) (M (8
(58)
(x/t [W u(x ) (x4)) N (x un (xx) [x({)
(59)
Given that means across all views are aligned, similar to Lemma F33, we know the resulting combined GMM has same
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Identifiable Object Representations under Spatial Ambiguities
number of components:
[A| K g(z") xxx) fa(z)) = II N N Plok, Piok).
(60)
Given the product of GMM is a GMM with the number of components equal to the product of a number of components in individual GMM, however in our setting we consider all the components in individual GMM across viewpoints to be aligned resulting GMM with number of components equal the sum of individual components which in our case correspond to K. The posterior parameters of the resulting mixture given in closed form by:
-1
F tokk
(x
Pink
Fiek
(61)
(x
(xx")
The resulting GMM is still on the view-specific slots, the aggregation of these slots to obtain content vectors marginalises the viewpoint-level information with convex combination of parameters across all the viewpoints considered as described in cf. F.3, results in:
M K
K RnN (z; Tates o
n NyN (zi Piuks Pux)) Σ
(62)
M
2
0% = g0(0kk Tike
ik
oiko
(63)
π
[A
7
Pick
-( Tttk)
P.kk
(64)
Tikk
Now to show that the resulting GMM is non-degenerate need to show Σk the = 1, for i = 1,2,.... ..... M. Based on Eqn. 56:
M
K
T.
Σ
A
1
M|A|= 11
(65)
M
A
MA
M|A
1=1
M
K
m =1
(66)
M
k=
based on the above equation we can say that the scaled sum of the mixing proportions of all K components in all M GMMs when the components are aligned must equal 1, show that the resulting aggregate posterior non-degenerate and valid probability distribution.
Assumption F.4 (Weak Injectivity). Let f : 3 -> X' be a mapping between latent space and image space, where dim(Z) S dim(x). The mapping fa is weakly injective if there exists Xo E X' and 0 such that [/-1 = Vx 5) n f(Z), and (x E X |f= ( = oc) C f(Z) has measure zero w.r.t. to the Lebesgue measure on f(2) (ef. (Kivva et al., 2022)).
Remark F.5. In words, Assumption F.4 says that mapping fa is weakly injective if: (i) in small neighbourhood around specific point Xo E X the mapping is injective meaning each point in this neighbourhood maps to exactly one point in the latent space 2; and (ii) while fa may not be globally injective, the set of points in X' that map back to an infinite number of points in Z (non-injective points) is almost non-existent in terms of the Lebesgue measure on the image of Z under fa-
Theorem F6 (Mixture of Concatenated Slots). Let fa denote a permutation equivariant probabilistic slot attention function such that fa(z", PP"') = Pf.(z",s") where P E {0,1)Kxt Kxk is an arbitrary permutation matrix. Let c = (g(s(),), ........)....... .)) E kR be the concatenation of K individual content vectors, where each vector is an aggregate of all the slots obtained from considered viewpoints in viewpoint-set A C [V] (cf. (Kori et al., 2024)). Due to the
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Identifiable Object Representations under Spatial Ambiguities
nature of the aggregator function, the individual content vector is Gaussian distributed within a K K-component mixture: Cj: (HUK, mx}
E TRd vk
E K))...K.. Then, e. is also GMM distributed with K mixture components:
k1
p(c) Σ क NcN Ep), where E EAK-1 Hp E RR Ep RdRdddd
(67)
p=1
We additionally borrow some theorems and definitions from (Kivva et al., 2022) which are essential for our proofs. First, we restate the definition of a generic point as outlined by (Kivva et al., 2022) below.
Definition E.7. (Generic point) A point x E Ja(R")) C IR" is generic if there exists > O, such that fa Bss - Ra is affine for every s E fā ({x})
Theorem F.8 (Kivva et al. (Kivva et al., 2022)). Given fa Rm + T is a piecewise affine function such that (x E R" : '15 ((x)) = ]o} S fa(R"") has measure zero with respect to the Lebesgue measure on fa((B""), this implies (((()((RR)) ) = m and almost every point in [a(("") (with respect to the Lebesgue measure on [a(()")) generic with respect fa.
Theorem F,9 (Kivva et al. (Kivva et al., 2022)). Consider a pair of finite GMMs TR.
y y' T Nyy' uzj. Ej).
n Ny: Ej),
and
(68)
Assume that there exists a bull Bxx, 5) such that y and y' induce the same measure on B 8). Then y = y', and for some permutation T we have that πT; [()) and (pt, Σ,) (m/=(s)> Ez()))
Theorem F.10 (Kivva et al. (Kivva et al., 2022)). Given ~ ~E' NTN p., E.) and z' ~ N/='' Ej) and fa(z) and fa(z') are equally distributed. We can assume for E R" and >> O, fa is invertible on B(x, 26) n fF("))). This implies that there exists X1 E B(x, 5) and ón > such that both fa and fa are invertible on B(x1, 0 fR
(Affine Equivalence of aggregate content) For any subset A S [V]. such that [A] > 0, given a set images A A and a corresponding aggregate content e E C and a non-degenerate content posterior g(c s considering two mixing function fa Ja satisfying assumption F.4, with shared image, then e are identifiable up to ~ equivalence.
Theorem 5.3
Proof. Based on the results of (Kori et al., 2024) we know that when p(s) is aggregate posterior of q(s x),, p(s) is identifiable up to ~ equivalence. Additionally, based on lemma 5.2 we know that both q(s x) and q(c s) are non-degenerate GMM with valid probability distribution. Using similar arguments in (Kori al., 2024; Kivva al., 2022) we show that p(c) and p(s) are identifiable up to ~ equivalence. W. w.l.o.g. given view point transformation is deterministie, we consider x Txx
We know that
p(81) = / a( xa )xx )dx
(69)
/ II g(s" xx))(x""ddx"
(70)
vEA
K
II
Σ
TEN (s" (x(xxx)))))))))))))))) o((()))) p(x )dxa
(71)
uEA
Σ TEN (c': (ex xxxx)) 8(x" )x
(72)
X
uEA
II AEA
A (s": Pheeet ozokk)
(73)
X
A=1
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Identifiable Object Representations under Spatial Ambiguities
Change of variables from get prior over random variable with matching function y, results in:
p(c):)) = p(s(:)) (si:x - g(si. t(x)) def.
(74)
Given the transformation is linear, resulting us with the distribution with mean given by:
Eg (C1:K) = E. (g(s TA,LK))
(75)
9 (Bs A k) }TT1k
(76)
π TiK
Bs (s(x))
(77)
DEA
and the covariance follows the diagonal structure as in p(c), which can be described as follows:
Var(ci:k) Σ
iiK EvEA Ti:K
Vara
(78)
Finally, the mixture components can be expressed as:
T1K EVEA T.K
(79)
[A|
With distribution parameters described in equations 77, 78, and 79, define the aggregate content distribution as GMM expressed as follows:
p(c)
Σ
TWN
E()),, Var(c)))
(80)
X
Validity of p(c): The outer summation in equation 80 be split into for image samples and other for original mixing coefficients, which results in the equation:
X
K
p(c)
N
(v; E(c)a) Var(c) Var())))
(81)
X
Based this observe the each component in our GMM corresponds to particular slots for given image in given viewpoint, triple describing each component is:
T Porks 0 }
for V == 1..., |A| i == 1, 1,2,...,
| and k= 1,2, K.
(82)
To verify that p(c) is non-degenerate mixture, we observe the following implication:
x K
E
म = 1,
(83)
AA|
k=
K
1
T
[A X 191-|4|-1
11
(84)
X
A
X
k=1
similar to lemma 5.2, this says that the scaled sum of the mixing proportions of all components all |X GMMs must equal 1, proving that the associated aggregate posterior mixture p(c) is a well-defined and non degenerate probability distribution.
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Identifiable Object Representations under Spatial Ambiguities
Invertibility restrictions: Given two piece-wise affine compositional functions fas Sa C V. + X, for a given set of views VA, let (CI....,,Ck) Cj [c Pk; Σx} and c' (ci,....;Cks) .x,,, c N(cc; 1',, 2n} be a pair of aggregate content representations, result of sampling a concatenated higher dimensional GMM distribution in Rg as shown in Theorem F.6, (Kori et al., 2024). In the case when, faz (C, {v^}) and fag(C', {v² are equally distributed. Now assume that there exists x E X' and 8 such that fa and fa are invertible and piecewise affine B(xA, n fa(S), for given set of views vA, which implies dim fa(C, {v |C|.
Affine subspace: We now restrict the space Bxx 5) to a subspace (xx'A, 8')) where x E B(x'A. .8) such that fa and fa are now invertible and affine on Bxx'A, fu {v With L S X be an |C| (CI-dmmnsia affine subspace (assuming | 2 (C]), such that Bxx'A, 8')n (C, (vd1) ((xA,, 8') We also define hy. h C + L to be pair of invertible affine functions where h (B( 8') n L) Saz (B( n L; A and h (B(x'A 88) L) faz (B(x', ,8')
L; A Implying (c) and h (c') are finite GMMs that coincide with B(x"A, 88)) n L and hy(c) hy(c'), theorem 1.9, (Kivva al., 2022). Given, f 1 b and hy(c) and hc(c") then h is an affine transformation such that lx(e) = c'.
~ equivalence: Given Theorems F.8 and F.10, there exists a point x E Sas(C, {vA]) that is generic with respect fa and fa and invertible on Bxx, 6) n (as (). Having established that there is an affine transformation h(c) = c' and invertiblility of piece-wise affine functions fa and Ja on B(x 6) n fay(C, {vA]), this implies that e is identifiable up an affine transformation and permutation of C₁ E c, concluding our proof.
Remark: Given Theorem F.9, we know that for each higher dimensional mixture component in p(c) induces the same measure on BxxA, 8) and hence for some permutation we have that (Mx(i)) Ext = (('()(())> (+() Therefore, each mixture component (x(s) is identifiable up affine transformation, and permutation of aggregate content representations in c. Now, given sampling C₃ is equivalent to obtaining K samples from the GMM, (z) and concatenating, this makes ((z) identifiable up to affine transformation, h and permutation of slot representations in c. It now trivially follows that each of the aggregate content representation C₁ ~ (cci μ4, Σx} E V E (l....A K} is identifiable up to affine transformation, h based on the following observed property of GMMs:
K
The (N(s us, Ex)) ~ h4
Σ TrN(s E%)):
(85)
Theorem 5.4 (Invariance of aggregate content) For any subset A, B S [V] such that |A] > [B > 0 and both A, B
satisfy an assumption 5.1, we consider aggregate content to be invariant to viewpoints if SA ~s [B for data X x 8B
Proof. Based on equation 80, (s) and Ps($) can be expressed as follows:
pa(c)= Σ
K
1
Σ
N
C:
π DUEA
Tuk
2
o
(86)
X
A
Σ
ELEA
1=1
VEA
BEA
pn(c)= = Σ
Σ
E B
N
C; Σ WEB
Tuk
u Mk
R 2
(87)
X
L
T
EVEB T
WEB
Given the assumption of viewpoint sufficiency 5.1 we know the objects observed in viewpoint set A are same as the object observed in set B. Following the results of Theorem 5.3, we know that both pA (s) and (s are independently identifiable up to ~ equivalence, which means fA and fa are invertible for given views 4 and B respectively.
'ay correspond to push forward operation, applying function fa on all the elements of the given set.
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Identifiable Object Representations under Spatial Ambiguities
Affine mapping. Without loss of generality, for given of views v there exists some L C X be an di-mennnsi affine subspace, such that (xx'A, 5) n TAp(C, {v4]) (Cy [v^]) B(x'A, 6) n L. This implies their exists affine map between с fr v and ė = fn f :x Let AA C + L to be invertible affine functions where h (B(x'A,
fAi (x(x .8') L show their exists an affine map between f f (xA:v and = fB (x B, vB), such that hgg (c) č'.
fB: (B(x 8') n L; v resulting in ha(c) = c'. Similarly, we can
Invariance setup. In the case when representations are invariant, (c(c) and pB(c) (c) are equally distributed, which means aggregate content domain both cases are same or similar C^ = Cn.
c' h(d') ha(c) (h o (C)
(88)
(89)
= c= (h o (c)
(90)
Given composition of affine maps affine, we can consider the mapping (ha² h hB) to be an affine, resulting in an ~M equivalence between SA and fn.
Theorem 5.5 (Approximate representational equivariance) For a given aggregate content c, for any two views V, V ~ (v()), resulting in respective scenes x ~ px | v, and x ~ px(x v. for any homeomorphic, monotonic transformation f E Hl such that (x) = x. their exists another homcomorphic and monotonic transformation h, E H.p such that Hy C H2 C [ddimx)) and v = hh,1 (fr (f=" ()))
Proof. For a given view v and mixing function fa that satisfy assumptions F.4 and is piecewise affine, from theorem 5.3 know the latent view representations identifiable up ~. equivalence for given aggregate content vector. We know that ((v) is expressed as GMM with a considered set of viewpoints, ideally learning each component for each viewpoint.
A Σ
pvv)
π r ((V;
Following similar arguments in Theorem 5.3 and (Kivva et al., 2022), we can show that for given content representation the view distribution p(v) is identifiable up to affine transformation. This means, for any two considered models fa, fa. such that (V; {c}) and (a;(V; {c}) are equally distributed, then for any x ~ X with the corresponding content representations given by the views = fā (x";c), v'= ffa fx";""";) c) are related in by an affine transformation h(v) = v', results in:
| Σ x h2 (N(v: (.)))) ~ b4
|
Σ n N((: o3)
(91)
Without loss of generality we can consider any function C x V - X' is identifiable up to affine transformation, with this for given views v, v ~ p(v) and for any object representations c, the resulting scenes are sampled by distributions learned with mixing function of is given by x ~ ( c, v). & ~ py c, v). As previously established for some affine transformation h,
h(v) = - = v h- (f-
(92)
Given (x) = I. when combined with above equation we know v = h- (x:c)) v== (fxxxx))):: for some invertible affine transformations h and M.
26

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Identifiable Object Representations under Spatial Ambiguities
Run #1
Run #2
Run #3
Run #
22
Figure 8. Identifiability of q()) and g(s). Estimated marginalised slot distribution (q(s)-blue contours) and marginalised content distribution (g(c)-orange contours, across 4 runs of VISA. This provides strong evidence of recovery of the latent space up to affinc transfommatios, empirically verifying our claims in Thm. 5.3.
Given h is homeomorphic and monotonic, and is piecewise linear, the inverse can be transferred resulting in hi-1 [fx c))), similarly can also swap h--1 with he, resulting in V Ty (N'² (x=1 ())). Additionally combining the results from theorem 5.3 and (Kivva et al., 2022), we know that h' h is an affine transformation h. This results in:
h = h1-1'~1p h
(93)
v hho h) (x: c))
(94)
v hv(v)
(95)
Given affine transformation preserves monotonicity and homeomophism, the resulting transformation hn Hyy and hy E Hz, concluding the proof.
G
Experiments
G.1
Toy Setting
Here, we repeat the experiments in CASE STUDY with point cloud giving us two dimensional distributions, which can analysed visually. In Fig. 8, we display the distributions of marginalized aggregate content distribution q(c), comparing different runs that are either rotated, skewed, or mirrored with respect to each other, indicating identifiability up to affine transformation. To quantitatively measure the same, we computed SMCC and observed it to be 0.95 0.01, empirically verifying our Thm. 5.3. Furthermore, to illustrate the invariance of distribution q(c) across viewpoints (Thm. 5.4), we consider three different views. We use all possible pairs learn q(c) distributions illustrated Fig. 9, where the distributions from second to last sub-figum sub-figures are learned wrt viewpoints described by {g. r). {r. b). and {g. b), respectively. Similar to our previous findings, these distributions were also found to be rotated, skewed, or mirrored relative to each other, with an observed SMCC of 0.87 = 0.11, further confirming the claims in Thm. 5.4.
G.2 Synthetic dataset results
Here, we illustrate visual results reflecting object binding in the case of view ambiguities. Table 3, demonstrates identifiability results on CLEVR-AUG datasets. In Fig. 10, we demonstrate the results of VISA across three different views. We additionally highlight some of the occluded regions which seem to be better captured by our proposed model, which can be attributed to the multi-view setting and the s i gmoid id mask.
Additionally, we also illustrate the results from CLEVR-MV dataset figure 11.
G.3 Influence of Number of Views
Here, we demonstrate the influence of the number of views on the overall identifiability of object- representations. Similar to Fig. 2, in Fig. 12, we observe an increasing number of views increase overall results.
27

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Identifiable Object Representations under Spatial Ambiguities
aA (c)
q [ (c)
22
2/
21
21
Figure 9. Viewpoint invariance for (()). Estimated marginalised aggregate content distribution q(c) when trained with different view pairs {(green, red), (red, bluc), (green, blue)} are illustrated in later figures. As the resulting distributions with different datasets only vary by affine transformation, providing strong evidence for Thm. 5.4.
Table Comparing identifiability of g(s), g(c), and p(v) scores wit existing OCL methods on CLEVR-AUG dataset.
METHOD
SMCC1
INV-SMCC
MCC1
AE
0.26 + .01
SA
0.45 1: .05
PSA
0.48 = .03
MulMON
0.56 + .04
0.57 + 001
OCLOC
0.58 = .02
0.60 = 01
0.48 .01
VISA
0.64 = .01
0.06 = 01
0.57 + .04
G.4 MvMoVI Results
Here, we discuss the results obtained from the proposed dataset. To reiterate, MVMOVI-C a variant where fixed camera positions are maintained for all viewpoints across all scenes the dataset. This setup helps assign fixed type of viewpoint conditioning for all images captured from a particular camera.
The detection and binding quality of different models are illustrated in Table 2. From these results, can clearly observe that while the model demonstrates similar binding capabilities, the identifiability of object representations is improved in our proposed model. This suggests that the use of fixed camera positions in MvMOVVIC enhances the consistency and quality of object representation learning, leading to better detection performance across different viewpoints.
Figure 13 & 14 showcases the object discovery capabilities of the VISA. the iteration of the MvMoVI-D dataset, we vary the camera position for each scene, making the dataset more dynamic and allowing for the potential violation of assumption 5.1 in certain cases. Table 4 presents the binding and identifiability results for both in-domain and out-of-domain data, following similar analysis in Table 2. We observe consistent trends and behaviours, suggesting that the impact of the assumption is minimal. A more detailed analysis of the assumption's 's effects will be left for future work.
G.5
View warm-up
Given the stochasticity during the initial phase of training, facilitate meaningful representation in content aggregator function, consider view warm-up strategy. For the initial 100,000 iterations, we randomly use the view-specific slots for reconstruction instead of invariant content with probability of 0.5.
This primary makes sure the feature extractor extracts meaningful representations before aggregation, which helps to stabilize the training process and allows the model to effectively bind and integrate information from different perspectives later stages of training.
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Identifiable Object Representations under Spatial Ambiguities
Orig.
Recon.
Stot-1
Slot-2
Slot-3
Slot-4
Slot-5
View!
View22
View-3
Orio.
Recon.
Slot-1
Slot-2
Slot-3
Slot-4
Slot-5
View1
View2
Vire-33
Slot-1
View!
View2
View-3
Figure 10. Visual illustrations of benchmark results CLEVR-AR dataset.
G.6 Hyperparameters
In Table 5, we detail all the hyper-parametee parameters used in our experiments. In the case of benchmark experiments, we use trainable CNN encoder as used in (Locatello et al., 2020b; Kori et al., 2023), while in the case of proposed MvMoVI datasets we use DINO (Caron et al., 2021) encoder to extract image features and change our objective reconstruct these
29

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Identifiable Object Representations under Spatial Ambiguities
Orig.
Recon.
Slot-1
Slot-2
Slot-3
Slot-4
Slot-5
View-1
View-2
View-3
Oria-
Recon.
Slot-1
Slot-2
Slot-3
Slot-4
Slot-5
View-2
View-
Recon.
Slot-1
Slot-2
Slot-3
Slot-4
Slot-5
View-1
View-2
View3
Figure II. Visual illustrations of benchmark results on CLEVR-M dataset.
features rather than the original image as proposed in (Seitzer et al., 2022). For most of hyperparameters we use the values suggested by (Locatello et al., 2020b; Seitzer al., 2022), based on their ablation results.
30

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Identifiable Object Representations under Spatial Ambiguities
Dataset-CL -URR-au
Datasat-CLEVR--mu
Datasel-GON
Numbar views
Numbar of Mews
Numbar of viows
Figure 12, Influence of Number of viewpoints on identifiability for synthetic datasets.
Orig.
Segmentation
Stot-1
Slot-2
Stot-3
Slot-4
Slot-5
View-1
(a)
Orig.
Sogmentation
Slot-1
Slot-2
Slot-3
Slot-4
Slot-5
View-1
(b)
Figure 13, Visual illustrations of benchmark results on MVMOVI-C datasct with 2 views.
G.7 Computational Resources
We run all our experiments on cluster with a Nvidia NVIDIA L40 48GB GPU cards. Our training usually takes between eight hours couple of days, depending the model and the dataset. is to be noted that speed might differ slightly with respect to the considered system and the background processes. All experimental scripts will be made available on GitHub at later stage.
31

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Identifiable Object Representations under Spatial Ambiguitics
Orig.
Segmentation
Stot-1
Slot-2
Slot-3
Slot-4
Slot-5
View-2
View-
Orig.
Segmentation
Slot-1
Slot-2
Slot-3
Slot-4
Slot-5
View-1
View-22
(b)
Figure 14. Visual illustrations of benchmark results on MVMOVI-C dataset with 3 views.
Table 4. Identifiability and generalisability analysis on MV-MoVID dataset.
METHOD
INDOMAIN ANALYSIS
OUT DOMAIN
mBO
SMCC1
INV-MMCC
MCC1
mB0
SMCC 1
INV.MMCC
MCC
SA-MLP
0.24 + 0.031
0.41 + 0.005
0.24 + 0.097
0.45 + 0.008
PSA-MLP
0.26 1. 0.022
0.44 0.006
0.25 0.012
0.42 1. 0.006
VIP
0.24 +: 0.099
0.45 d: 0.009
0.46 1: 0.054
0.57 0.021
0.25 1: 0.011
0.48 +: 0.006
0.51 :: 0.021
0.55 0.021
SA- TRANRFFRREERR
0.34 # 0.017
0.40 1: 0.041
0.34 0.066
0.38 + 0.031
PSA- TRANSFORMERR
0.37 + 0.021
0.38 +: 0.007
0.36 + 0.024
0.36 + 0.016
0.39 + 0.016
0.46 + 0.001
0.48 0.001
0.54 0.032
0.37 0.051
0.46 + 0.022
0.45 = 0.010
0.51 0.029
32

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Identifiable Object Representations under Spatial Ambiguitics
Table 5. Experimental details w.r. datasets
DATASETS())
PARAMETERS
VALUES
No. Layers
4
No. Views
10 (GSO: 8)
No. Slots
7
Training Epochs
5000
Batch Size
32
Optimizer
ADAM
CLEVR, GSO
Learning Rate
0.0002
Initial Slot f
N(O, 1) II
Initial Slot
Warmup Steps
10000
Decoder
SPATIAL BROADCASTING-CNN
X- likelihood
N( 1
No. Layers
4
No. Views
10
No. Slots
4
Training Epochs
5000
Batch Size
64
Optimizer
ADAM
GQN
Learning Rate
0.0002
Initial Slot μ
N( 1) II
Initial Slot
Warmup Steps
10000
Decoder
SPATIAL BROADCASTING-CNN
X- likelihood
N(p, 1
No. Layers
4
No. Views
5
No. Slots
7
Training Epochs
560
Batch Size
64
Optimizer
ADAMW
Learning Rate
0.0002
Initial Slot l4
N(0, 1) II
Initial Slot
Warmup Steps
10000
Pretrained Encoder
DINO DINO_VITBI6
Decoder
MLP, TRANSFORMER
x- likelihood
NV W( II)
33


Reviews:
[{"review_id": "LChIkvjdRe", "reviewer": "Reviewer_zW6Z", "summary": "The paper introduces View-Invariant Slot Attention (VISA), a probabilistic object-centric learning model designed to achieve identifiable object representations from multi-view images without explicit viewpoint annotations. VISA overcomes limitations of single-view methods by resolving spatial ambiguities like occlusions and viewpoint variations. The authors provide theoretical guarantees of identifiability using latent spatial Gaussian Mixture Models (GMMs) and empirically validate the approach on synthetic and newly proposed datasets (MVMOVI-C and MVMOVI-D).\n\n## update after rebuttal\nThe authors addressed my concerns in the rebuttal. I will keep my rating.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their detailed feedback and are glad that the reviewer found our claims were supported by clear and concise evidence with correct proofs and sound experiments.\n\n\n> more extensive details on computational complexity, model training time, parameter counts\n\n**VISA complexity:** VISA achieves $\\mathcal{O(VTNKd)}$ with the added complexity of $\\mathcal{2O(VNd)}$ for inverse and forward view point transformation given by $\\mathcal{T}_{\\theta}$, while it retains the complexity per view to be $\\mathcal{O(VTNKd)}$ which is the same as slot attention and probabilistic slot attention. \nAdditionally, the representation matching function contributes $\\mathcal{O(VK^3d)}$: this term does not alter the dominant term, in the general case when $K<<N$.\nSimilar to PS, when VISA is combined with an additive decoder, the complexity of the decoder can be lowered due to the property of automatic relevance determination (ARD), eliminating the need to decode inactive slots.\n\n\n**Decoder architecture:** As detailed in the paper we use two different classes of decoder architectures:  (i) additive and (ii) non-additive; within the additive architecture we use both spatial broadcasting and MLP decoders, for the non-additive architecture we use transformer decoders. Concretely, we follow SA(Locatello et al., 2020) for spatial broadcasting decoders and DINOSAUR(Seitzer et al., 2023) for both MLP and transformer decoder. In detail we use:\n\n- spatial broadcasting decoders: \n\nInput/Output: The generated slots are $\\mathbf{s} \\in \\mathbb{R}^{K\\times d}$, each slot representation is broadcasted onto a 2D grid of dimension $8 \\times 8 \\times d$ and augmented with position embeddings. Similar to slot attention, each such grid is decoded using a shared CNN to produce an output of size W × H × 4, where W and H are width and height of the image, respectively. The output channels encode RGB color channels and an (unnormalized) alpha mask. Further, we normalize the alpha masks with Softmax and perform convex combinations to obtain reconstruction.\n\n\tShared CNN architecture: 3 x [Conv (kernel = 5x5, stride=2), LeakyReLU(0.02)] + Conv (kernel = 3x3, stride=1), LeakyReLU(0.02) \n\n\n- MLP decoders:\n\nInput/Output: similar to the spatial broadcasting decoder, here, each slot representation is broadcasted onto N tokens (resulting in $N \\times d$) and augmented with position embeddings. Then the individual slot representation is transformed with a shared MLP decoder to generate a representation corresponding to feature dimension along with additional alpha mask, which is further normalised with Softmax and used in creating convex combinations to obtain reconstruction.\n\n\tShared MLP architecture: [Linear (d, d, bias = False), LayerNorm(d)] + 3 x [Linear (d, d_{hidden}), LeakyReLU(0.02)] +  Linear (d_{hidden}, d_{feature}+1) \n\n\n- Transformer decoders:\n\nInput/Output: transformer consists of linear transformers encoder output $(N \\times d_{feature})$ and extracted slots $(K \\times d )$ as input, while returning the slot conditioned feature as output with a dimension of  $(N \\times d_{feature})$. \n\nTransformer architecture: is made up of 4 transformer blocks, where each transformer block  consists of a self-attention on input tokens, cross-attention with the set of slots, and residual two-layer MLP with hidden size $4 x d_{feature}$. Before the Transformer blocks, both the initial input and the slots are linearly transformed to $d_{feature}$, followed by a layer norm. \n\n\n\n**Model training time:** As detailed in appendix G.7, our training usually takes between eight hours to a couple of days, depending on the model and the dataset. We run all our experiments on a cluster with a Nvidia NVIDIA L40 48GB GPU cards, we will add a pointer in the main text.\n\n\n> extensive ablations on larger real-world datasets would strengthen scalability claims.\n\nWe do agree experiments on large real-world datasets would be helpful, however to the best of our knowledge, there aren’t any real-world datasets with many multiple viewpoints: this was the main motivation for proposing the synthetic dataset which consists of  72,000 x 5 number of images with 72,000 scenes with 5 viewpoints; in terms of variation it consists of 930 objects variety with 458 complex backgrounds. \n\nAdditionally, the primary focus of the work is to provide theoretical guarantees for identifiability in multi-view scenarios, while requiring no viewpoint annotations, which builds upon the formalisms in a single view scenario in Kori et al. (2024); Brady et al. (2023); Lachapelle et al. (2023). To the best of our knowledge, this is the first work addressing explicit formalisations of assumptions and theory required for achieving this. \n\nHaving said that, if the reviewer can point to any large real-world dataset we would be happy to consider it for the final version of the paper.", "claims_and_evidence": "The claims made in the submission are supported by clear and convincing evidence and experiments.\n\nThe authors state their model \"demonstrates scalability on two new complex datasets (MV-MOVI-C and MV-MOVI-D)\". Although Table 2 provides evidence of good performance on these datasets, more extensive details on computational complexity, model training time, parameter counts, and extensive ablations on larger real-world datasets would strengthen scalability claims.", "methods_and_evaluation_criteria": "The methods and evaluation criteria proposed in this paper are mostly suited to the problem addressed.", "theoretical_claims": "I briefly check the correctness of the theoretical proofs presented in the paper.", "experimental_designs_or_analyses": "The experimental designs on synthetic datasets are sound, while short of real-world datasets.", "relation_to_broader_scientific_literature": "The paper positions itself clearly in relation to the broader literature on object-centric representation learning (OCL), nonlinear independent component analysis (ICA), and representation identifiability.", "essential_references_not_discussed": "Nil.", "other_strengths_and_weaknesses": "Although empirical evidence from synthetic data and benchmarks strongly suggests correctness, explicit verification for complex real-world cases remains a limitation.", "other_comments_or_suggestions": "L72-73, the set symbol may conflict with the object symbol.", "questions_for_authors": "Nil.", "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": "tgLiWnEcCi", "reviewer": "Reviewer_jmDg", "summary": "This paper focuses on object-centric learning and proposes View-Invariant Slot Attention (VISA). It extends the probabilistic slot attention (PSA) into multi-view scenarios. It introduces a content descriptor, learns identifiable object-centric representations from multi-view observations and accounts for occlusion and view ambiguity that emerges in multi-object settings. Theoretical analysis is provided. Empirical experiments on several datasets demonstrate VISA's good performance compared with other object-centric learning baselines.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their detailed feedback and are glad to see that the reviewer acknowledges our superior performance and believes the paper to address novel and important topic. \n\n\n\n> (W1)It is difficult to parse the curves in Figure 5 and 6. …\n\nWe do agree with complications in analysing higher dimensional latents: Figure 5 and 6 are an attempt to visualise the feature-wise aggregated distribution to capture the trend.  While we have described the behaviour of this distribution in the discussion, we will expand it for clarity in the final version. Additionally, we also have included a 2D-variant experiment in appendix  G.1 where the distributions can be seen to be equivalent up to an affine transformation validating our claims. We would also like to point to qualitative results included in appendix Figure 10-14. Depending on space availability we will bring the 2D experiments to the main paper.\n\n\n\n> (W2)It seems to be strongly connected to PSA while the writing of Section 4 …\n\nAs mentioned in a paragraph Viewpoint specific slots (L193-L214), the viewpoint specific slots are extracted with **EM algorithm as proposed in Kori et al., 2024 (which is same as PSA)** but applied on a transformed encoder output, with a transformation corresponding to a viewpoint specific inverse transformation. This is done to project all the slot representations for multiple views into a common vector space.\n\nAs pointed out by the reviewer and in the viewpoint specific slots paragraph, the algorithm here is the same as PSA with the only difference being with the transformation of inputs. We will cross reference section 3 in that paragraph to make this difference more explicit. \n\nAdditionally, the primary focus of the work is that it provides theoretical guarantees for identifiability in multi-view scenarios, and requires no viewpoint annotations, which builds upon the formalisms in a single view scenario in Kori et al. (2024); Brady et al. (2023); Lachapelle et al. (2023). To the best of our knowledge, this is the first work addressing explicit formalisations of assumptions and theory required for achieving this. \n\n\n\n> (W3) Meanwhile, the details of the model implementation are heavily missing…\n\nThanks for pointing this out, we will include them: \n\n**Decoder architecture:** As detailed in the paper we use two different classes of decoder architectures:  (i) additive and (ii) non-additive; within the additive architecture we use both spatial broadcasting and MLP decoders; for the non-additive architecture we use transformer decoders. Specifically, we follow SA (Locatello et al., 2020) for spatial broadcasting decoders and DINOSAUR (Seitzer et al., 2023) for both MLP and transformer decoder: for details about architectures please refer to the response of **reviewer zW6Z**\n\n\n\n> The K in equation 6 appears to be a hyperparameter of choice that …?\n\nThat's a valid point, however the dependency on K is inherent to SA and PSA, here we develop on these work to address spatial ambiguities by considering multiple viewpoints.\nSimilar to the ARD study in Kori et al., 2024, during inference we did observe when K is set to higher that the required number, the model ignores the additional slots by assigning the mixing coefficient to 0, however lower K affects performance, similar to the ablations in Locatello et al., 2020.\n\n> If some object is completely occluded, how do you identify the object and its visibility without knowing the 3D prior of the environment?\n\nThat’s a great point. One of our key assumptions is viewpoint sufficiency, meaning that each object in the environment is visible in at least one of the considered viewpoints. If an object is completely occluded across all viewpoints, identifying it falls beyond the scope of this work. We will make this explicit in the paper", "claims_and_evidence": "The review finds it difficult to collect enough empirical evidence to support the claims from the paper. The experiment results do not straightforwardly demonstrate identifiability, viewpoint invariance, and spatial ambiguity. It is difficult to parse the curves in Figure 5 and 6. There is no visualization of any scene to illustrate these results. (W1)", "methods_and_evaluation_criteria": "The reviewer finds it difficult to assess the novelty of the method. \n\nIt seems to be strongly connected to PSA while the writing of Section 4 fails to directly point out the connection and extension. It is difficult to associate the equations in Section 4 with equations in Section 3. For example, while the formalization in the beginning of section 4 is clear, the reviewer fails to see the similarity and difference between the \"Viewpoint specific slots\" and a single-view PSA. Since the PSA is introduced in Section 3 as preliminary, Section 4 should refer to it to help explain the new method (VISA) and where it differs from PSA. (W2)\n\nMeanwhile, the details of the model implementation are heavily missing.  The writing vaguely suggested use of MLP and Transformer without any details such as the format of inputs and outputs, or if there is any modification. (W3)", "theoretical_claims": "I did not rigorously check the correctness of any proofs.", "experimental_designs_or_analyses": "The experiments are conducted on several benchmarks, including public (CLEVR, GQN, GSO) and newly generated ones (mv-MoVIC, mv-MoVID). VISA's results appear to be superior. (S1)\n\nTable 2 aims to show that the proposed VISA generalizes to novel views, however, the performance gap is minimal for all the baselines as well. Perhaps more diverse viewpoints should be provided, but there is no clue as the test environments are not illustrated in the paper. (W4)", "relation_to_broader_scientific_literature": "The related work is sufficient. There is an additional section of related work in the appendix.", "essential_references_not_discussed": "None.", "other_strengths_and_weaknesses": "Strengths:\n- See S1 from the above discussions.\n- S2: the paper tackles multi-view object-centric representation learning, which the reviewer believes to be an important and novel topic.\n\nWeaknesses:\n - See W,1 W2, W3 and W4 from the above discussions.", "other_comments_or_suggestions": "- typo in line 253-254, there are two $\\pi^1$ and no $\\pi^3$.\n- typo line 340-341 \"feature feature\" distribution\n- For visualization of the mvMovi-C in figure 13, there is no visualization of any cluttered environments where objects are occluded.", "questions_for_authors": "- The K in equation 6 appears to be a hyperparameter of choice that controls the number of components in GMM. Is the performance sensitive to K?\n- How is The Transformer used for VISA and PSA? Is there any modification? What are the formats of the inputs and outputs?\n- If some object is completely occluded, how do you identify the object and its visibility without knowing the 3D prior of the environment?", "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": "00sHh3vlco", "reviewer": "Reviewer_oFKs", "summary": "The paper aims to learn identifiable object representations even under spatial ambiguities, i.e., occlusions and view ambiguities.\nThe authors propose View-Invariant Slot Attention (VISA), a probabilistic slot attention variant to learn such representations.\nTheoretic analysis is provided to prove identifiable under given assumptions.\nEmpirical results on synthetic datasets are shown to verify the model.\n\n## update after rebuttal\nPlease see the rebuttal comment below.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their detailed feedback and are glad that the reviewer finds our paper well written, clear, and with reasonable claims and proofs. We are also glad that intuitions aided in the understanding of theorems and our claims. \n\n\n\n> As mentioned in the weakness section on page 8, the viewpoint sufficiency assumption is strong. It is questionable whether proposed model could be applied on real-world data\n\nWe do agree experiments on large real-world datasets would be helpful, however to the best of our knowledge there aren’t any real-world datasets with many multiple viewpoints:  this is was the main motivation for proposing the synthetic dataset which consists of  72,000 x 5 number of images with 72,000 scenes with 5 viewpoints(in terms of variation it consists of 930 objects variety with 458 complex backgrounds). \n\nAdditionally, the primary focus of the work is that it provides theoretical guarantees for identifiability in multi-view scenarios, and requires no viewpoint annotations, which builds upon the formalisms in a single view scenario in Kori et al. (2024); Brady et al. (2023); Lachapelle et al. (2023). To the best of our knowledge, this is the first work addressing explicit formalisations of assumptions and theory required for achieving this. \n\nHaving said that, if the reviewer can point to any large real-world dataset we would be happy to consider it for the final version of the paper. \n\n\n> The evaluation metrics are not well-introduced. It is a little bit hard to understand how good the numbers represent.\n\nThanks for pointing this out.  We borrow the definition of SMCC from Kori et al., 2024. For two sets of slots $\\\\{\\mathbf{s}_i\\\\} _{i=1}^M$ and $\\\\{\\tilde{\\mathbf{s}}_i\\\\} _{i=1}^M$, where $\\mathbf{s}_i \\in \\mathbb{R}^{K \\times d}, \\tilde{\\mathbf{s}} _i \\in \\mathbb{R}^{K \\times d}$, extracted from $M$ scenes, the SMCC between any $\\mathbf{s}$ and $\\tilde{\\mathbf{s}}$ is obtained by matching the slot representations and their order. The order is matched by mapping slots in $\\tilde{\\mathbf{s}}$ w.r.t $\\mathbf{s}$ assigned by $\\tau$, \n\nfollowed by a learned affine mapping $\\mathbf{A}$ between aligned $\\tilde{\\mathbf{s}}_{\\tau(i)}$ and $\\mathbf{s}$: \n\n$ \\mathrm{SMCC}(\\mathbf{s}, \\tilde{\\mathbf{s}}) \\coloneqq \\frac{1}{K\\times d} \\sum _{i=0}^K \\sum _{j=0}^d \\rho(\\mathbf{s} _{ij}, \\mathbf{A} \\tilde{\\mathbf{s}} _{\\tau(i)j}).$\n\nBy design the SMCC metric is bounded between [0, 1], with the higher the value the better. We will add these details in the appendix.\n\n\n\n> For example, could you elaborate more on 'vary by an affine transformation' in the caption of Figure 6? How do you evaluate and compare the results?\n\nIn the context of Figure 6 the behaviour of affine transformation implies the distribution across view sets, only differ in scale and translation.  We do agree with complications in analysing higher dimensional latents. Figure 5-6 was our attempt on this visualising the feature-wise aggregated distribution to capture the trend. We have discussed the behaviour in the paper, but will expand it for clarity. \n\nAdditionally, we also have included a 2D-variant experiment in appendix  G.1 where the distributions can be seen to be equivalent up to an affine transformation, validating our claims. Depending on space availability we will bring the 2D experiments to the main paper.\n\n\n> do you have some insights on how to ensure this assumption if other researchers are going to apply your model? The reviewer addresses this as a limitation, independent of the technical details within this assumption.\n\nTheoretically, verifying the viewpoint sufficiency assumption is challenging. In some cases, as few as two viewpoints may be enough to satisfy this assumption, while in more complex scenes, adding additional views can improve visibility. However, beyond a certain number of viewpoints, we expect diminishing returns in performance, as the marginal gain from each additional view decreases.\nWhen we have control over the data generation process, we can adopt an adaptive viewpoint selection strategy. This could involve dynamically selecting views based on occlusion-aware heuristics or ensuring a larger set of viewpoints that are equidistant from the scene of interest while varying the angle and azimuth. This approach helps mitigate occlusion issues and ensures more robust object visibility, we will include this discussion in the main paper.", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "The proposed method is reasonable. \nThe synthetic datasets are newly-proposed but reasonable.\nThe evaluation metrics are referenced but no details.", "theoretical_claims": "The theoretical claims in the main paper are checked. To the best of my knowledge, the claims are reasonable and correct.", "experimental_designs_or_analyses": "The data generation process (Figure 7) and experimental results in the main paper are checked.\nThe patterns in the result are not fully-explained. For example, 'vary by an affine transformation' in the caption of Figure 6 could be further explained/illustrated.", "relation_to_broader_scientific_literature": "This work extends the field of object-centric representation learning. This work explicitly learn view-invariant object representations, rather than learning the object-consistency representations unconditional to views in the prior work.", "essential_references_not_discussed": "The reviewer is not aware of any missing essential references.", "other_strengths_and_weaknesses": "Strength:\n1. The paper is well-written, with a clear structure and thorough proof.\n2. The intuition and example sections are helpful for understanding the proof.\n\nWeakness:\n1. As mentioned in the weakness section on page 8, the viewpoint sufficiency assumption is strong. The experiments are conducted on the well-designed dataset. It is questionable whether proposed model could be applied on real-world data.\n2. The evaluation metrics are not well-introduced. It is a little bit hard to understand how good the numbers represent. For example, the computed SMCC is 0.72 (on page 7, line 379-right column), but hard for the readers to understand how good this number is.", "other_comments_or_suggestions": "Some typos:\n1.  In definition 3.1 (line 160-right column), there is a c before colon, but not showing after.\n2. The caption for figure 5 (line 341): feature feature distribution.", "questions_for_authors": "1. The patterns in the result are not clearly explained. For example, could you elaborate more on 'vary by an affine transformation' in the caption of Figure 6? How do you evaluate and compare the results?\n2. Can you elaborate more on the result that the computed SMCC is 0.72 (on page 7, line 379-right column). How would you interpret this value?\n3. In the real world settings, when applying your model, we need to ensure viewpoint sufficiency as an assumption. Do you have some experiment results on the real-world datasets? If not, based on your data generation process, do you have some insights on how to ensure this assumption if other researchers are going to apply your model? The reviewer addresses this as a limitation, independent of the technical details within this assumption.", "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": "KPdTI3cgjs", "reviewer": "Reviewer_jXho", "summary": "The paper presents a multi-view probabilistic approach aimed at learning modular object-centric representations that are essential for human-like reasoning. This paper introduces View-Invariant Slot Attention (VISA), which addresses spatial ambiguities caused by occlusions and view ambiguities. This method aggregates view-specific slots to capture invariant content information while simultaneously learning disentangled global viewpoint-level information. Unlike prior single-view methods, this approach resolves spatial ambiguities, provides theoretical guarantees for identifiability, and requires no viewpoint annotations.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their feedback and are glad that the reviewer found our experiments to be extensive and method to be robust.\n\n\nThe primary focus of the work is that it provides theoretical guarantees for identifiability in multi-view scenarios, and requires no viewpoint annotations, which builds upon the formalisms built in a single view scenario in Kori et al. (2024); Brady et al. (2023); Lachapelle et al. (2023). To the best of our knowledge, this is the first work addressing explicit formalisations of assumptions and theory required for achieving this. \nWe also provide empirical evidence with synthetic datasets, where the transformation in distribution clearly demonstrates our claims. \n\nIn order to make the paper more self contained, based on other reviewers feedback we plan to include complexity argument and metrics details, along with model architectures as below.\n\n\n**VISA complexity:** VISA achieves $\\mathcal{O(VTNKd)}$ with the added complexity of $\\mathcal{2O(VNd)}$ for inverse and forward view point transformation given by $\\mathcal{T}_{\\theta}$, while it retains the complexity per view to be $\\mathcal{O(VTNKd)}$ which is the same as slot attention and probabilistic slot attention. \nAdditionally, the representation matching function contributes $\\mathcal{O(VK^3d)}$: this term does not alter the dominant term, in the general case when $K<<N$.\nSimilar to PS, when VISA is combined with an additive decoder, the complexity of the decoder can be lowered due to the property of automatic relevance determination (ARD), eliminating the need to decode inactive slots.\n\n**SMCC details:** we borrow the definition of SMCC from Kori et al., 2024. For two sets of slots $\\\\{\\mathbf{s}_i\\\\} _{i=1}^M$ and $\\\\{\\tilde{\\mathbf{s}}_i\\\\} _{i=1}^M$, where $\\mathbf{s}_i \\in \\mathbb{R}^{K \\times d}, \\tilde{\\mathbf{s}} _i \\in \\mathbb{R}^{K \\times d}$, extracted from $M$ scenes, the SMCC between any $\\mathbf{s}$ and $\\tilde{\\mathbf{s}}$ is obtained by matching the slot representations and their order. The order is matched by mapping slots in $\\tilde{\\mathbf{s}}$ w.r.t $\\mathbf{s}$ assigned by $\\tau$, \n\nfollowed by a learned affine mapping $\\mathbf{A}$ between aligned $\\tilde{\\mathbf{s}}_{\\tau(i)}$ and $\\mathbf{s}$: \n\n$ \\mathrm{SMCC}(\\mathbf{s}, \\tilde{\\mathbf{s}}) \\coloneqq \\frac{1}{K\\times d} \\sum _{i=0}^K \\sum _{j=0}^d \\rho(\\mathbf{s} _{ij}, \\mathbf{A} \\tilde{\\mathbf{s}} _{\\tau(i)j}).$\n\nBy design the SMCC metric is bounded between [0, 1], with the higher the value the better. We will add these details in the appendix.\n\n\n**Decoder architecture:** As detailed in the paper we use two different classes decoder architectures  (i) additive and (ii) non-additive, within additive we use both spatial broadcasting and MLP decoders, for non-additive we use transformer decoders. In terms of architectural, we follow SA(Locatello et al., 2020) for spatial broadcasting decoders and DINOSAUR(Seitzer et al., 2023) for both MLP and transformer decoder, we will describe architectures in detail in the appendix as in the response of **reviewer zW6Z**:", "claims_and_evidence": "This paper highlights that while OCLOC focuses on achieving object-consistency unconditional to views, this approach explicitly learns view-invariant object representations. The paper provides theoretical guarantees for identifiability in cases of partial or full occlusions without additional view information, which advances beyond previous work in single-view OCL. The use of spatial Gaussian mixture models in latent distribution across viewpoints to encourage identifiability without auxiliary data is justified. The experimental results across multiple datasets provide convincing evidence for their theoretical claims.", "methods_and_evaluation_criteria": "The evaluation criteria focuses on three key claims: identifiability, invariance, and equivariance. The authors use appropriate metrics such as slot mean correlation coefficient (SMCC) and invariant SMCC (INV-SMCC) to quantify their results. The comparison with various baselines, including standard additive autoencoder setups, slot-attention (SA), probabilistic slot-attention (PSA), MulMON, and OCLOC, provides a thorough assessment of the performance.", "theoretical_claims": "was not reviewed in depth", "experimental_designs_or_analyses": "The paper includes extensive experimental validation on standard benchmarks (CLEVR-MV, CLEVR-AUG, GQN) and complex datasets (MVMOVI-C and MVMOVI-D), demonstrating the robustness of this method.", "relation_to_broader_scientific_literature": "I am not familiar with the literature in this area", "essential_references_not_discussed": "I am not familiar with the literature in this area", "other_strengths_and_weaknesses": "No", "other_comments_or_suggestions": "No", "questions_for_authors": "No", "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:

Meta-review: All the reviewers agree in the end that the paper is acceptable, the proposed method addresses an important problem for the object identification across different views with ambiguities. While the paper investigates theoretically the guarantees for the identifiability of the objects, various experimental investigations have also been carried out over both the standard and proposed custom datasets. The authors provided details in the rebuttal for all the concerns/comments, made various clarifications and explanations and provided more experimental results.
Decision: Accept (poster)