A Bayesian Nonparametric Framework For Learning Disentangled Representations
Authors: Vaishnavi S Patil
Siddhi Patil
Matthew Evanusa
Amit Kumar Kundu
Cornelia Fermuller
Joseph JaJa
Venue: ICLR 2026 Poster (2026)
Keywords: representation learning
disentangled representations
unsupervised learning
nonparametric methods
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: A nonparametric variational inference framework for the unsupervised learning of disentangled representations

Abstract: Disentangled representation learning aims to identify and organize the underlying sources of variation in observed data. However, learning disentangled representations from observational data alone without any additional supervision necessitates inductive biases to solve the fundamental identifiability problem of uniquely recovering the true latent structure and parameters of the data-generating process. Existing methods address this by imposing heuristic inductive biases that typically lack these theoretical identifiability guarantees. Additionally, these methods rely on strong regularization to impose these inductive biases, creating an inherent trade-off in which stronger regularization improves disentanglement but limits the latent capacity to represent underlying variations. To address both challenges, we propose a principled generative model with a Bayesian nonparametric hierarchical mixture prior that embeds inductive biases within a provably identifiable framework for unsupervised disentanglement. Specifically, the hierarchical mixture prior imposes the structural constraints necessary for identifiability guarantees, while the nonparametric formulation allows the latent representation to scale with infinite capacity to faithfully represent the complete set of underlying variations without violating these structural constraints. To enable tractable inference under this nonparametric hierarchical prior, we develop a structured variational inference framework with a nested variational family that both preserves the hierarchical structure of the identifiable generative model and approximates the expressiveness of the nonparametric prior. We evaluate our proposed probabilistic model on standard disentanglement benchmarks, 3DShapes and MPI3D datasets characterized by diverse source variation distributions, to demonstrate that our method consistently outperforms strong baseline models through structural biases and a unified objective function, obviating the need for auxiliary regularization constraints or careful hyperparameter tuning.

Published as a conference paper at ICLR 2026
A BAYESIAN NONPARAMETRIC FRAMEWORK FOR LEARNING DISENTANGLED REPRESENTATIONS
Vaishnavi S Patil 1,4
Siddhi Patil
Matthew Evanusa'
Amit Kumar 2
Cornelia Fermüller
Joseph JáJá
'Department of Computer Science 'Department of Electrical and Computer Engineering University of Maryland Institute for Advanced Computer Studies *Institute for Health Computing University of Maryland, College Park (vspatil, scpatil, mevanusa, amit314, fermulem, jo @umd. edu
ABSTRACT
Disentangled representation learning aims to identify and organize the underlying sources of variation in observed data. However, learning disentangled represen- tations from observational data alone without any additional supervision necessi- tates inductive biases to solve the fundamental identifiability problem of uniquely recovering the true latent structure and parameters of the data-generating process. Existing methods address this by imposing heuristic inductive biases that typi- cally lack these theoretical identifiability guarantees. Additionally, these methods rely strong regularization impose these inductive biases, creating inher- ent trade-off in which stronger regularization improves disentanglement but limits the latent capacity represent underlying variations. To address both challenges, we propose principled generative model with Bayesian nonparametric hierar- chical mixture prior that embeds inductive biases within a provably identifiable framework for unsupervised disentanglement. Specifically, the hierarchical mix- ture prior imposes the structural constraints necessary for identifiability guaran- tees, while the nonparametric formulation allows the latent representation to scale with infinite capacity faithfully represent the complete of underlying varia- tions without violating these structural constraints. To enable tractable inference under this nonparametric hierarchical prior, we develop a structured variational inference framework with nested variational family that both preserves the hi- erarchical structure of the identifiable generative model and approximates the ex- pressiveness of the nonparametric prior. We evaluate our proposed probabilistic model on standard disentanglement benchmarks, 3llkss 3DShapes and MP13D datasets characterized by diverse source variation distributions, to demonstrate that method consistently outperforms strong baseline models through structural biases and unified objective function, obviating the need for auxiliary regularization constraints or careful hyperparameter tuning.
INTRODUCTION
primary objective of representation learning is not merely to perform density estimation or gen- crate realistic samples, but to discover and characterize the latent structure inherent in observational data. This notion formalized by disentangled representations that aim to separate the distinct, dependent, and informative generative factors of variation in the data such that each variable of the representation sensitive to changes in exactly one underlying factor while being relatively invari- ant to changes in others Bengio (2013). Disentangled representations have been shown to improve robustness and out-of -fttoo--dsttibution generalization (Träuble et al., 2021 et al. 2024). sample ef-
ficiency in few-shot learning (Van Steenkiste al.) 2019: Cheng et al. 2024), domain adaptation
via separation of transferable and domain-specttn features
(Tan & Huang 2019 Cai et al.
2019
controllable and interpretable generation (Zhu et al.
2021 Wang al. 2023) Zhou et al.
2025

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and causal inference and faimess through explicit separation of sensitive and task-relevant factors CCenng et al. 2024 Locatello et al. (019a).
However, unsupervised learning of disentangled representations is fundamentally challenged by identifiability which refers to whether the true generative factors and their structure can be uniquely inferred from observed data alone. Without identifiability, different parameterizations of the gener- ative model can produce identical distributions over observed data, making it theoretically impossi- ble recover the true generative factors. Prior work in nonlinear independent component analysis (Hyvärinen & Pajunen 1999; Hyvärinen et al., 2019, Khemakhem et al. 2020), deep generative modeling (Wang et al. 2021; Amour et al., 2022). and unsupervised disentanglement (Locatello) et al. (019bb has shown that enforcing the commonly used simple isotropic Gaussian prior in com- bination with a nonlinear generative function is generally insufficient to recover the true sources of variation. Without additional inductive biases, the model can learn infinitely many, potentially entangled representations that satisfy the marginal prior distribution yet fail to align with the true data-generating factors.
Moreover, prior works primarily impose heuristic inductive biases and typically rely on strong reg- ularization to enforce them inducing an inherent trade-off whereby stronger regularization enhances disentanglement but simultaneously restricts the representation capacity. Consequently, this mis- specification of the latent capacity either under-represents all relevant modes of variation forces encoding of the data in a manner that conflicts with the natural structure, leading to systematic constraints.
violation of the dissetanglemeentiidduciii
To address both these limitations we build upon the theoretical framework of Kivva et al. 2022), who prove that mixture priors provide sufficient inductive bias for identifiability in deep generative models with piecewise affine data-generating functions. We propose a Bayesian nonparametric hier- archical mixture prior that inherits these identifiability guarantees lacking in simple Gaussian priors while simultancously addressing the representation capacity mis-specification problem through its nonparametric formulation. The nonparametric formulation allows the complexity of factor-specific mixtures to remain unspecified a priori, a characteristic analogous to species discovery in unexplored ecosystems, where the number and types of species present cannot be predicted in advance. This formulation endows model with universal approximation capabilities, ensuring that the identi- fiable architecture is, in principle, expressive enough to recover the natural underlying structure of the data. To specifically learn disentangled representations, under which a nonparametric hierarchical mixture prior is placed over the space of each generative factor independently, such that mixture components correspond discrete variations of the respec- tive factor. Consistent with the principles of classical factor analysis, this factorized structure entails that observations are generated through the combinatorial composition of factor-specific mixture components, with each observation determined by unique combination of components across all generative factors Hsu et al. (2024a)
define a factorized prior structure
For tractable inference under this nonparametric hierarchical prior, we develop a structured varia- tional inference framework with nested variational family. The structured inference framework preserves the hierarchical structure of the identifiable generative model thereby enabling joint op- timization of the prior and deep generative model parameters within a unified objective function. The nested formulation enables the variational distribution to approximate the expressiveness of the nonparametric prior while maintaining computational tractability.
Empirically, we show that this hierarchical mixture prior provides substantially stronger inductive biases enabling the learning of modular and compact disentangled representations that enhance in- terpretability. Our results on two image datasets with distinct factor distributions further demon- strate that the nonparametric hierarchical mixture prior and the corresponding inference framework provide sufficient inductive bias without additional computationally expensive auxiliary inductive biases or careful manual tuning of regularization hyperparameters
NONPARAMETRIC BAYESIAN PRIORS FOR AUTOENCODERS
The theoretical framework of Kivva et al. 2022 establishes identifiability of deep latent variable models by rigorously delineating the conditions under which such models are uniquely recoverable from data. Specifically, identifiability of the latent variable distribution up to affine transformations
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requires two critical conditions: (i) the marginal distribution of the latent variables follows a Gaus- sian mixture model (GMM), and (ii) the corresponding getarrenng function is piecewise affine and satisfies weak injectivity. Building on these conditions, the authors further demonstrate that stronger identifiability, up to permutation, scaling, and translation, necessitates the introduction of additional discrete latent variables which index the mixture components of the GMM, inducing hierarchical structure in which the continuous latent variables are conditionally independent given these discrete variables. Furthermore, imposing an additional maximality condition (P3 in Kivva et al. (2022) on this hierarchical structure yields identifiability of the complete latent representa- tion, including the latent dimensionality, the cardinality and distribution of the discrete variables, and the conditional distribution of the continuous latent variables.
To further bias this identifiable framework toward learning disentangled representations, we impose an additional structural constraint on the discrete latent space. Specifically, we factorize the multi- variate discrete latent variable into statistically independent components, where each component en- codes a distinct factor of variation and indexes set of mixture components representing the discrete modes of variations of its respective factor. Critically, this factorized prior enforces an orthogonal partitioning of the latent space, such that factor-specific variations are encoded in non-overlapping subspaces of the continuous latent representation, thereby promoting the independent encoding of factor-specific variations. Moreover, this factorized latent space induces generative process in which observations are generated through the combinatorial composition of factor-specific mixture components, with each observation generated by a unique combination of component assignments.
This inductive bias of combinatorial composition toward learning disentangled representations is closely related the works of Hsu et al. (2024a b). which similarly structure the latent space as a product of independent discrete sets to promote disentangled representations. In their framework, the discrete latent space is formed from continuous latent representations that are discretized via vec- tor quantization, wherein each latent dimension is independently mapped to discrete codes through distinct, learnable codebooks. For this architecture to serve as an effective inductive bias toward disentanglement, the size of each codebook, and thus the cardinality of the corresponding discrete latent variable indexing its embeddings, is kept small and fixed, constraining each dimension to capture only limited set of discrete modes. While this design choice encourages parsimonious representations, it imposes a rigid upper bound on the expressive capacity of each factor: factors whose discrete modes of variations exceed the cardinality of single codebook must either be dis- tributed across multiple independent codebooks or be only partially captured, in both cases reducing the interpretability and semantic coherence of the learned latent factors.
To address the expressive limitations of cardinalty discrete variables, we introduce a nonpara- metric Dirichlet Process mixture (Ferguson) 1973 Sethuraman 1994 Antoniak 1974) (see Section A.I.I for a detailed description) prior over the latent space, allowing the support of each latent vari- able to grow adaptively with the intrinsic complexity of its associated factor of variation. By placing nonzero probability mass over a countably infinite support, the DP prior equips each discrete latent variable with a universal approximation property over discrete distributions, enabling to represent an arbitrary discrete distribution over the modes of its associated factor. Consequently, this univer- sal approximation property reinforces the statistical independence of the joint latent prior, ensuring that the complete set of discrete modes of variation associated with each independent factor is repre- sented entirely by a discrete latent variable. Critically, because the DP prior is applicd independently to each discrete latent variable, each factor can adaptively expand its variation space without disrupt- ing the orthogonal partition structure induced by the remaining factors, thereby preserving statistical independence of the representation.
In the following sections, we formalize this framework by deriving a Bayesian nonparametric hi- erarchical generative model that is jointly identifiable, universally expressive through the nonpara- metric prior, and structurally biased toward disentangled representations with a particular emphasis on completeness and interpretability. We then formalize the structured variational family, which is specifically designed to accurately approximate the posterior distribution under the hicrarchical factorized structure of the prior. Finally, we present a nested extension of this structured varia- tional family that enables principled incremental expansion of representational capacity, with the structured component preserving the inductive biases of identifiability and factorial independence, and the nested component approximating the universal approximation capacity of the nonparametric prior, jointly ensuring that the learning algorithm remains faithful to the theoretical properties of the generative model throughout inference.
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2.1 GENERATIVE MODEL WITH NONPARAMETRIC PRIOR
We consider a hierarchical latent variable generative model in which each observation x E X' is generated via structured latent representation comprising continuous latent variable E IR and a multivariate discrete latent variable z = (Z1,..., E 2 with 2 denoting the corresponding discrete latent space. To impose an inductive bias toward disentangled representations, we assume a factorial structure on the discrete space 2, defined as the Cartesian product Z = Z1 X ... x Za, mutually independent and captures distinct factor of variation. To ensure identifiability of the continuous latent variable e up to affine transformations, we follow the conditions of Kivva et al. (2022) and specify the marginal distribution of e as Gaussian mixture model (Assumption PI). To further satisfy the conditions for stronger identifiability (Assumption P2), we require that, conditioned on the discrete latent variable x, the components of e are mutually independent, e; TT ej for all * j enforcing an orthogonal partitioning of the continuous latent space into d non-overlapping factor-specific subspaces.
where each 2; E 2, = {1,... (1....) }
To allow unbounded number of mixture components, model each e, under an independent Dirichlet Process mixture prior, inducing a countably infinite Gaussian mixture over the marginal p(e.). Formally, place DP prior G; (a(a, Gg) over the component parameters, where > O is the concentration parameter controlling the richness of the induced partition, and Go is the base distribution over the parameter space θ₁ Under the stick-breaking representation of the Dirichlet Process, we define an infinite sequence of stick breaking proportions 3; {Br k } k= with 3.,a Beta(1, a) and draw the corresponding component parameters independently from the base distribution, 0,,k (A as follows,
BE, p(3 a:) = Beta (1,a), 81 1~~ p(0) A) Go(A), Vi (1,... d}, Ak E From the stick-breaking proportions Gr we construct an infinite sequence of mixing weights nT. 3r, |= (1 ). satisfying Σx=1 TiKk almost surely, yielding the random discrete measure G. Ex i 60...- The discrete latent variable 2; then selects component index according to 2 (T). upon which the continuous component e, is drawn from the corresponding component-specifer distribution parameterized by 0,, E O, i.c. es | 4 = k, 02 ~ p(e; 02.k).
= k B ~ p(z; = k (:) Tikk, ~ p(e| 24. 0,)
II (p(e 0c,:)) A=1
We specify each component-specific distribution as Gaussian with unknown mean and precision, p(e: 0..a) = .N(e: Mi.k: ), where 6 = (Pi,k, st,x) denotes the mean and precision pa- rameters of the k-th component. Exploiting conjugacy, we set the base distribution Go to be a Normal-Gamma distribution, (A) ( Ko, Vo, Wo), which serves as the natural conjugate prior to the Gaussian likelihood with unknown mean and precision, enabling closed-form posterior updates over 0, Further, to enforce the maximality condition, Assumption P3 of Kivva et al. 2022), we place Gamma prior on a, biasing the model toward the minimal number of active dis- crete components consistent with the observed data, thereby selecting a unique representative within the equivalence class of paargina-preserving representations.
Completing the generative hierarchy, the observation x is then generated via a piecewise affine deep neural network (with Rectified Linear Unit (ReLU) nonlinearities) data-generating function 90, .dd >> X' satisfying the weak injectivity conditions (F1 and F2 of Kivva et al. (2022),, with
additive observation noise: likelihood family.
pa, (x e) = Pgge, (e)) where P denotes an appropriate
Together, these modeling choices yield a generative model with three key properties: identifiabil- ity of the latent space up permutation, scaling, and translation; nonparametric expressiveness through the DP prior; and an explicit factorial structure promoting disentangled, factor-specific rep- resentations. The joint distribution over observations v, continuous variables e, discrete variables z, stick-breaking proportions B, and component parameters 0 factorizes according to the hierarchical generative process as
p (z, e, 2, a. A) Pa, (x
II p (c: 2,, (2) (.) II p (3r. p 1))
(1)
where the outer product over i encodes the factorial independence across latent dimensions, and the inner product over l reflects the infinite mixture structure induced by the Dirichlet Process prior.
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2.2 STRUCTURED NESTED VARIATIONAL FAMILY
To enable tractable posterior approximation under the nonparametric DP prior (see Section A. for a primer on variational inference for DPMMs), existing methods Blei & Jordan (2006); Hoffman et al. (2013) rely two simplifying assumptions. First, they approximate the infiiitedimmnniinn stick-breaking process using a truncated variational family by introducing an explicit truncation level T, fixing luy, (BiT 1) 1 for each dimension thereby forcing all subsequent stick- breaking weights (BAA and their corresponding mixing weights zero, limiting the mixture components to T. Second, they adopt mean mean-field variational family, imposing full factorization across all latent variables, thereby severing the hierarchical dependencies induced by the generative model.
We address the second limitation by adopting the structured variational framework of Hoffman & Blei (2015), which relaxes the mean-field assumption and retains the conditional dependencies in- duced by the generative model, yielding more accurate posterior approximations with reduced bias and lower sensitivity to initialization and hyperparameters. Specifically, for each dimension i, we preserve the hierarchical dependencies between the stick-brees boeckbreaking proportions Br and the discrete variable along with the component parameters 0,, and the continuous latents e; with the following structured variational family.
T-1 qu(e,, 24, B, = g(es 2s, 0,)((1 [.) II WeLL (3
T II que (0,,))
Truncated variational families render inference infinite-mixture models tractable by restricting the variational distribution to fixed number T' of mixture components. Intuitively, one might expect the ELBO to increase monotonically with T', additional components allow the variational posterior to better approximate the true nonparametric posterior, thereby motivating the use of large truncation levels. However, when the data are generated by a finite mixture, there can exist an optimal truncation level beyond which additional components do not improve the approximation, because the truncated families are not nested: the variational family with truncation level T is not subset of the family with truncation level T + 1. Consequently, increasing T beyond the optimal level does not necessarily improve the approximation, and crucially, undermines the inductive bias toward parsimonious representations Hsu et al. (2024a) that is essential for learning disentangled factors.
To address this, employ the nested variational family of Kurihara et al. (2006), which supports an unbounded number of components in the variational family by tying the variational parameters of all components beyond an implicit truncation level T' their prior values:
D
Luo (B,)
II qun,, (Bi,k) II p( a)), k=T
4u (0,)
II quo,, (0t,)) II k=T
p(O., 12)
(3)
Under this formulation, only the first T components have free variational parameters, ensuring that despite the variational distribution supporting an unbounded number of components only T sets of parameters need to be represented and optimized, rendering inference computationally tractable. Crucially, under this nested variational family, data points that are not explained by the first T components with free variational parameters can still be assigned to components beyond T, whose parameters remain tied to the prior. This enables a greedy inference procedure that begins with == 1 and incrementally adds components only when they yield a significant improvement in the ELBO, continuing until all data are assigned to components within the active truncation level and mass remains on prior-tied components. Notably, under this nested formulation, components that fail to encode meaningful variations collapse back to their prior distribution during training.
Given the generative model in equation T and the hierarchical structured nested variational family qu. the evidence lower bound (ELBO) for single data point, estimated via Monte Carlo, can be written
log p(3 ax) qu, (3) + Enguo
P(0 p(012) qu,, (f)
Po, (x e) p(e 2. p( q(e z, 0) q(z (
C Euga
log
+ Eng. log
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2.3 THE INFERENCE ALGORITHM
Efficient inference in hierarchical models relies on conjugate exponential family likelihoods, which preserve tractable conditional posterior distributions within the same exponential family the prior admitting closed-form updates. For general neural network observation models such as Pa,- the absence of conjugacy makes inference over latent variables substantially more expensive and often requires multiple passes through the generative network for single update. To retain the computa- tional advantages of conjugate structured inference while using flexible non-conjugate likelihoods, we follow Johnson et al. (2016) and use deep amortized recognition networks h(z; +) that output local conjugate likelihood potentials P& (Eq. 4), rather than directly parameterizing the variational distributions in standard variational autoencoders. These conjugate potentials serve amortized, data-dependent surrogates for the intractable likelihood terms during inference and are combined with the structured latent prior via message passing, leaving the generative model itself unchanged.
(e x) = io ()) = II exp (hi(1; の), (e(e))))
This factored structure of the recognition network mirrors the factorial prior c₁, allowing in- ference for the pair of local latent variables (ei, zi) associated with each dimension to be inferred independently. Consequently, the data-dependent term of the ELBO in Eq. 2.2 decomposes into sum of local contributions L,, one per dimension:
E Auy (8.))(x; (3.) log (= 21 (3))
p(e: 2i, 0.)
L
Byy (0))g(eslss..)) log
+ log P (es x))
(=2 (.))
g(e; 2,, 0,)
(5)
conditioning on the global variables B; and 0,, for any given value of the local variable 2, k, the inner expectation over e, admits the form of a variational lower bound on the conditional marginal likelihood of the conjugate potential:
Following Hoffman & Blei 2015),
B (x))) log
p(e;
zi, 02)
+ (a(e; 1x)
g(e: zi, 0z)
= -DLL (g(es 2s, 0,) (e Ii 24, 0,)) + log ( 2;, 0,) s log (
21, 0.)
(6)
where the local posterior and conjugate marginal likelihood are defined as
(e(es 24 zx)p(e, 24, O.) (or Zi, 0.) / po(e: xx)) xx) 21, O:)de; (7)
Po(es Jii zi, 0.)
(x Zi, 0.)
Since the Kullback-Leibler (KL) divergence non-negative, setting q(e; 2, 0) equal to the local posterior (e; I4, 0) minimizes the KL divergence to zero and yields the tightest lower bound on the local ELBO. Under conjugacy, this optimal variational distribution takes the exponential family form:
ale 2,,0,) 0:) = po(es Fi, 2i, 0.) exp (n (zi, no (O)), Φ), Le(e)) A.(1n((z,
(8)
(, (0,). = Σ 1 no ( ( hx(1;))
k=1
with natural parameters which explicitly combines the structured prior through ne (0,..)) with the data-dependent signal from the recognition network h,; ø). With this optimal choice of q(e, | zi, 0:), the local ELBO reduces to a variational lower bound on the marginal likelihood of the data conditioned on the global variable {3, follows:
log p(=; B:)
L; y (a.a3))))
+ B. (0.) (log zi, 0:)]
q (3)
- DKL - (g(=; (,)1 v., (.)) + to (3,) S log (0; (.G,)
and similar to the variational distribution of e, the locally optimal variational distribution over 22 for k < T' is given by:
log q(2 lo B.) log p(2, k 1,))) + E,,, (0,) log log (2 0a.b)) log Fi Eg,, (o.,)) A (10(0 hi(1R Q)) As (0.,)) (10)

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While the nested variational family supports unbounded number of components, naively com- puting the variational assignment g(2; = k G for all k > T would require sampling infinitely many stick- ttick-breaking proportions to evaluate the corresponding mixing weights Ri,A- However, two properties of the nested family together enable a tractable closed-form estimate. First, since the parameters of all components k > T are tied to the prior p(0.,: A), the expected log-likelihood Ep(o., (A) llog P (x; 0g,,)] is identical across all components beyond ' and can be factored out of the infinite sum. Second, since the mixing weights are normalized and satisfy ΣE= Rish =
almost surely, the total weight assigned to all components beyond T reduces to (1 Ek=1 Ti.k), which depends only on the T free variational parameters. Together, these two properties yield the following tractable expression for the total assignment probability to all prior-tied components:
log (z; k| (.)) log k B:) + =P( log Po 0s,z))
<(2 = (B:) x Tisk
exp Ep(ala) log ( 0)}
g(2 > T 11) Σ
g(2s = (3,))
k=T-1
(11)
Σ
A exp EACCIA) log (s 10)
k=T-1
(1
Ti,k) exp( Ep(ors) log po 10)
This quantity serves the stopping criterion for the greedy component-expansion procedure where a new component is added to the active truncation level when q(z; > T B:) exceeds a threshold, indicating that residual data cannot be explained by the current T components.
As in structured variational inference more broadly, restoring dependencies between global vari- ables and local variables along with amortized recognition networks enables the local variational distributions to set to their locally optimal values in closed form. At each iteration, the global parameters (3 and 0, are sampled from gua (B.) and qu, (0,) respectively using the reparameteriza- tion trick, and used to compute the local variational distributions. Since exact inference over the global variables remains intractable due the hierarchical dependencies (Hoffman & Blei 2015), and since the use recognition networks precludes analytic gradient computation, we optimize the ELBO using low-variance Monte Carlo estimators, yielding unbiased gradient estimates for both global and recognition network parameters. Step sizes are selected following the principled stabil- ity and convergence guidelines Mandt et al. (2017), with optimization carried out using adaptive preconditioned gradient methods, specifically Adam (Kingma & Ba 2017)
3 EXPERIMENTS
In this section, we present experiments designed to empirically assess whether the hierarchical Bayesian nonparametric approach to latent quantization provides effective inductive biases for learning interpretable disentangled representations. Specifically, we evaluate whether our approach achieves comparable performance relative prior work which impose equivalent inductive biases through multiple, distinct regularization terms.
Datasets. Our experimental framework systematically addresses these questions through compre- hensive guantitative evaluations conducted on two benchmark datasets labeled with ground-trute source information. Each dataset constructed from mutually independent sources through deter- ministic data generation process. In particular, we use the 3DShapes (Burgess & Kim 2018) dataset of 3D shapes generated from six ground-truth independent latent factors with approximately uni- form and small number of variations. Additionally, we use the MPI3D dataset (Gondal et al. 2019), collected from a real-world robotic environment, which exhibits a power-law distribution across the number of variations of different factors. Specifically, few factors contain extensive variations (e.g.. 40 discrete values for each rotational degrees of freedom), while the majority possess substan- tially fewer variations (e.g., 2-6 values for object properties).
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Prior Methods. We evaluate proposed approach against several state-of-the-ar methods that incorporate distinct inductive biases for unsupervised disentanglement. Specifically, we compare to B- VAV Higgins et al. (2017) and (3TCVVE Chen et al. (2018) which enforce discntanglement through information-theoren
regularization encouraging independence across latent dimensions. We further consider BioAE Whittington et al. (2022), which introduces biologically inspired con- ttraints-named nonnegativity and energy efficiency- ing neurons to become selective for single factors of task variation, together with grid-like struc- tural constraint as an architectural inductive bias. In addition, we examine QLAE et al. (2024a), which introduces an architectural bias based on latent quantization, and subsequently Tripod Hsu et al. (2024a), which combines latent quantization with additional inductive biases enforcing inde- pendence among latent variables as well as constraining the functional mapping from latent repre- sentations to the data space. For a concise primer on the disentanglement metrics and the different properties measured please refer to Section A.44 A.4
promote compact representations enfore-
Quantitative Comparison with Prior Methods. The experimental evaluation demonstrates that the proposed Bayes-QLAE consistently outperforms most baseline methods across both datasets in terms of modularity metrics (InfoM and D), with the notable exception of achieving competi- tive performance relative to QLAE and Tripod (Table I and Table ()). The observed improvements in compactness (InfoC) particularly pronounced when compared to the baseline QLAE, demon- strating the effectiveness of the nonparametric prior in adapting to the complexity of underlying gen- erative factors while maintaining consistency in modularity. Contrary the position advanced by the authors of QLAE, who prioritize modullrityylisetaalllnee through specific architectural design choices, we argue that achieving interpretable representations that faithfully capture mutually independent generative factors requires balanced weighting of both modularity and compactness metries. With competitive explicitness and informativeness measures, Bayes-QLAE demonstrates performance consistent with QLAE and Tripod while substantially out- performing alternative approaches, reinforcing the efficacy of latent quantization for disentangled representation learning.
over compactnees/coopllteeess
is worth noting that Tripod achieves its superior modularity and compactness performance through the application of a Normalized Hessian Penalty, which necessitates multiple forward passes through the generative network, thereby incurring additional computational overhead. In contrast, Bayes- QLAE achieves competitive performance through architectural inductive biases alone, without quiring additional regularization terms. Furthermore, Tripod's 's disentanglement performance, par- ticularly in modularity and compactness dimensions, exhibits sensitivity quantization level hy- perparameters, which must be specified a priori. Conversely, Bayes-QLAE demonstrates adaptive behavior that automatically learns quantization levels from the data while maintaining robustness across evaluation metrics.
We observe that the performance improvement of Bayes-QLAE is notably more pronounced on the 3DShapes dataset compared to the MPI3D dataset, where factor variations are characterized by pawer-law distribution. This differential performance suggests that the underlying distribu- tional properties of the generative factors significantly influence the efficacy of the nonparametric prior. We hypothesize that replacing the Dirichlet Process prior with a more flexible, generalized prior such the Pitman-Yor process-whice allows for richer clustering structure and can model power-law behaviors- yield further performance gains. We perform detailed ablation studies to systematically isolate and quantify the inductive biases contributed by each component of our hierarchical Bayesian nonparametric framework in Section A.3
Qualitative Comparisons. We conduct qualitative assessments of proposed method on each dataset to evaluate both sample reconstructions and latent traversals in Section [A.5] Furthermore, also assess the evolution of the latent traversals for the 3DShapes dataset with increase in the number of components in Figure 2
RELATED WORKS
The challenge of separating mutually independent sources in data traces back to the classical statis- tical problem of Independent Component Analysis (ICA) Comon [1944); Hyvärinen & Oja (2000). This core problem was later reinterpreted in the context of modern machine learning as disentangle- ment, formally articulated by Bengio/Bengio (2013) and formalized by Higgins al. (2018). When
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model
InfoM
InfoC
InfoE
D
3-VAE
0,62 + .00
0.44 .03
0.93 = .02
0.58 - 022
0,42 .02
0.97 02
3-TCVAR
0.65 + .03
0.56 + .02
0.91 + .02
0.56 .02
0.46
0.95 02
BioAE
0.58 .02
0.42 + .02
0.90 + .01
0.48 = .01
0 .02
0.91 + 02
QLAE
0.84 + .02
0.49 - .01
0.97 +
0.79 = .01
0.56
0.97 = 01
Tripod
0.91 + .03
0.58 - .03
0.96 1 .02
0.80 = .03
0.63
.03
0.97 = .02
Bayes-QLAE
0.91 + .03
0.61 .22
0.95
= .02
0.84 .03
0.65 = .03
0.97 = .02
Table 1: Disentanglement metrics measured in InfoMEC and DCI for 3DShapes dataset. For each metric a higher score is better. The scores for all the models were averaged across 5 runs with differ- ent random seeds with intervals denoting 95% confidence intervals of the mean estimated assuming a t-distribution. The results for the VQE-based and QLAE-based models are obtained using the hy- perparameter settings and experimental conditions described in Locatello et al. (2019b) and Hsu 2024a respectively.
et al.
model
InfoM
InfoC
InfoE
D
C
I
8-VAE
0,41 = .03
0,40 .03
0.68 + .03
0.24 = .03
0.19 + .03
0.80 + .03
8-TCVAE
0.48 + .03
0.46 .03
0.62 + .03
0.27 = .03
0.24 + .03
0.79 + .03
BioAE
0.44 .03
0.38 = .02
0.61 + .03
0.26 = .02
0.14 022
0.77 = .02
QLAE
0.52
0.2
0.68 - .04
0.38 = .04
0.34 .04
0.81 04
Tripod
0.59 = .05
0.54 .05
0.74 .06
0.47 = .04
0.45 - .05
0.84 = .05
Bayes- QLAE
0.60 + .0
0.56 + .03
0.71 + .04
0.48 + .03
0.47 + .03
0.81 + .03
Table 2: Disentanglement metrics measured in InfoMEC and DCI for MPI3D dataset. For each metric a higher score is better. The scores for all the models were averaged across 5 runs with different random seeds with intervals denoting 95% confidence intervals of the mean estimated assuming a t-distribution.
the data generating process is governed by nonlinear transformations Hyvärinen & Pajunen (1999)
the task of learning disentangled representations becomes theoretically umidentifiable Hy Hyvärinen & Ojal (2000); Khemakhem et al. (2020): Locatello al. (2019b). Consequently, the incorporation of
auxiliary data Hyvärinen & Pajunen 1999); Hyvärinen et al. (209); Khemakhem et al. (2020) or weak supervision (Shu et al. (2019): Locatello al. (2020) necessary to achieve identifability in disentanglement. A distinct line of research focuses on the incorporation of inductive biases either
in the model, training objective, or the data (Locatello et al. 2019b) for identifiability.
Information-theoreh Regularization Biases. Many early and influential works leverage information-theoreet constraints on the latent space to encourage factorization. The 3-VAE vari- ants Higgins et al. (2017): BBurgess al. (2018) introduces scalar multiplicative factor the KL divergence penalty with isotropic Gaussian priors, forming an information bottleneck, limiting the amount of information each latent can capture. Extensions like Factor Kim & Mnih (2018) and 3-TCVAE/Chen et al. (0118) further refine these constraints by explicitly penalizing total correlation enforce statistical independence between dimensions. BioAE (Whitlington al., (202)) demon- strates that biologically inspired constraints, specifically, minimizing latent activity and weight en- ergy while promoting latent non-negativity encourage more factorized representations. In a similar vein, temporal sparsity is used to encourage the learning of factors varying independently across sequences (Sprekeler et al. 20144 Klindt al. 2020).
Architectural and Structural Biases. Structural inductive biases embedded directly into model architectures have proved powerful. Vector quantization in models like QLAE (Hsu al. 2024a) and the recent Tripod framework (Hsu et al.) 2024b) induce grid-like latent spaces that simplify factor separation. FactorQLAE (Baykal et al. 2024) combine scalar quantization the latent vari- ables with total correlation term in the optimization an inductive bias. On the theoretical front, Barin-Pacela et al. (2024) establish identifiability for quantized factors under nonlinear mappings. Further, Leeb et al. 2020) demonstrate that restricting different latents enter the decoding com- putation graph at different points can enable disentanglement. Diffusion-bused architectural biases have emerged particularly effective inductive structures. Yang al. (2023) introduce the first un-

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supervised framework for disentangling pre-trained diffusion models by automatically discovering latent factors and decomposing gradient fields into conditiond sub-gradients. Further, Yang et al. 2024) show that diffusion models with cross-attention mechanisms serve strong inductive biases, relying on the inherent information bottlenecks in the diffusion process and cross-attention mechanisms. Dynamic Gaussian Anchoring in Jun et al. (2025) bias towards cluster structure in the latent space of diffusion models with cross-attention mechanisms for better separability between factor variations. Compositional constraints offer another structural approach, where maximizing the validity of composite images generated through stochastic mixing operators between latent rep- resentations enforces meaningful factor recombination without factor-specific architectural biases Jung et al. 2025).
Recent work emphasizes the incorporation of multiple, complementary inductive biases, for exam- ple, Tripod integrates quantization, statistical independence, and inter-latent influence minimization into a single framework Hsu et al. (2024b). Similarly, our work combines complementary inductive biases derived from nonparametric priors, structured variational inference, and stochastic quanti- zation, within a principled Bayesian framework with a unified objective that provides theoretical grounding for their integration.
5 CONCLUSION
In this paper, we introduce novel approach that incorporates Bayesian nonparametric priors into the latent space of autoencoders. By leveraging the flexibility of nonparametrie Bayesian methods, our approach enables the model to adaptively partition the latent space in accordance with the underlying data complexity, promoting more interpretable and structured latent encodings. This prior biases the learned representations toward capturing the underlying structure inherent in the data, thereby facilitating the learning of disentangled representations.
To enable accurate posterior inference under this flexible and hierarchical prior, we introduce a tai- lored nested and structured variational family. This variational family is specifically designed to preserve both the hierarchical structure of the prior and the inductive bias imposed by discrete latent space, ensuring that the inference procedure remains expressive enough capture complex dependencies while maintaining the structural properties essential for effective representation learn- ing.
Our ablation studies systematically isolate and quantify the inductive biases contributed by each component of our hierarchical Bayesian nonparametrie hamnwoork-named the nested variational family, structured variational inference, and stochastic quantization. Bayes-QLAE consistently out- performs all ablated variants across disentanglement metrics, demonstrating that each component provides complementary inductive biases that, when combined, enhance distinct aspects of disen- tanglement. Our empirical results demonstrate the effectiveness and generalizability of the pro- posed approach across image datasets characterized by diverse factor variation distributions. Bayes- QLAE consistently achieves superior or competitive performance relative to bascline methods on both 3DShapes and MPI3D, particularly in terms of both modularity and compactness-based disen- tanglement metrics. Importantly, this performance is attained solely through architectural inductive biases, without reliance on additional computationally expensive regularization.
The differential in performance on the two datasets suggests that the underlying distributional prop- erties of generative factors significantly influence the efficacy of the nonparametric prior. We hy- pothesize that replacing the Dirichlet Process prior with more flexible prior such as the Pitman-Yor process -whch allows for a richer clustering structure and can model pawer-law rehaviors-mat yield further performance improvements. These findings highlight the potential of our framework for interpretable and structured representation learning in varied settings.
ACKNOWLEDGMENTS
This work partially supported by the University of Maryland Institute for Health Computing (UM-IHC) and the University of Maryland Institute for Advanced Computer Studies (UMIACS). We thank Professor Yiannis Aloimonos for insightful discussions and helpful feedback.
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A APPENDIX
A.1 PRELIMINARIES
For theoretical foundation for our proposed approach, we first review the Dirichlet Process and its role as nonparametric prior in mixture models, with particular emphasis on the natural cluster- ing structure that emerges from this formulation. Subsequently, we discuss the caantization-basel generative model, the YQ-VAE, which utilizes discrete latent representation spaces and review prior work leveraging this crntiizatiionsstuuutut for learning disentangled representations.
A.1.1 DIRICILET PROCESS MIXTURE MODELS
We begin by considering nonparametric models, defined by an -imeniiadi parameter space that fundamentally allows the model's complexity to adapt and grow with the data. These models are typically used priors over distributions with broad support that encompasses the entire space of all possible distributions. The Dirichlet Process (DP), in particular, is a stochastic process whose realizations are discrete probability distributions, thereby defining valid nonparametric prior prob- ability distribution over the space of discrete probability measures. /Sethuraman (1994) constructive definition of the DP represents each discrete distribution drawn from the DP as a weighted sum of countably infinite atomic measures sampled from a continuous base distribution. This definition uses the stick-breaking construction, where the infinite sequence of weights for the atomic measures of the discrete distribution is generated by iteratively partitioning unit-length stick. In the first step, segment of length 31 is broken off the stick, where 31 drawn from a Beta distribution, 31 ~ Beta(1, at), parameterized by a, ensuring 31 E (0, 1). This segment is assigned as the weight of the first atomic measure 01, which is independently sampled from a base distribution (A) with parameters A. The remaining portion of the stick, with length 1 31, is then recursively partitioned the same manner: at each step, a segment of length 3, ~ Beta(1, ox), scaled to the length of the re- maining stick given by [[; (1 - 3,), is broken off and assigned as the weight of an atomic measure 8, drawn independently from the base distribution. This explicit stick- breaking construction gen- erates random discrete distribution G DP(a, Go) over the countably infinite atomic measures 0 drawn from the base distribution Go- Building on this, the DP serves as the nonparametric prior over the mixture components in the Dirichlet Process Mixture Models (DPMMs) Antoniak (1974). DPMMs generate data by first sampling discrete distribution G from the prior Da(a, Go) using the breaking construction and then using the set of atomic measures sampled from the base distribution Go to parameterize a Lata-generating distribution F. To generate each data point, we first sample a latent variable = from the discrete distribution defined by the stick biek-brea weights, then use the corresponding atomic measure 02 to parameterize the data-generating distribution F. which is further used to draw the observation xt:
3x a
p( 35 a:) Beta a),
0K | p(Ox ( Go(A), Wk E
(12)
k-1
z = k B p(= (3) 8x
II 3z), x [2,0] ~ p(x 22 F(O.)
(13)
1=1
Because each realization G drawn from the DP is a discrete distribution over the atomic measures 8, the above data-gennn tataa-geneating process results in repeated parameter values for the data- settarennr function F. This effectively induces a partitioning of the data, where each partition or component corresponds the data points generated with identical parameter values, allowing the generative
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process to be interpreted mixture model. Consequently, this results in hierarchical Bayesian framework, where the parameters of the data-generating distribution F are sampled from a discrete probability distribution drawn from the DP. The joint distribution of the data (x1, (. .xx and the latent variables: stick-breaking lengths / = { ....... J. component parameters 0 = (01, .......
and assignment variables {21,-..> 2N ). factorizes hierarchically follows:
p (x, 2, 3,0 a. X) = p (B p (z pr 2.
(14)
The primary objective of the learning process is to infer the posterior distribution of the latent vari- ables B. 0 and 2 conditioned on the observed data x and the hyperparameters a. X, denoted by 0(3,0,2 =| x. ax A).
Computing the exact posterior over the latent variables given the observed data introduces dependen- cies among the variables. As consequence evaluating the marginal likelihood of the data requires integrating over every possible latent configuration, making it intractable. In the nonparametric set- ting, such as under a Dirichlet Process (DP) prior, the posterior cannot be computed exactly and must be aproximatee/Wannnrrhht et al. 2008 introduce deterministic approach to approximate the intractable posterior with a simpler. tractable family of distributions by breaking certain depen- dencies among latent variables. They define variational family qu. parameterized by free parameters v, and optimize v to minimize the Kullback-Leible divergence between qu and the true posterior. Equivalently, this corresponds to maximizing the evidence lower bound (ELBO) on the log marginal likelihood of the data, as defined below:
logp(z: A) 2 Eg. [logp(z, e, 2, 3,0 A) log qu z, (,
(15)
A. 1.2 VECTOR QUANTIZATION FOR DISENTANGLEMENT
Next, we discuss the Vector Quantized- Quantized-Variational AutoEncoder (VQ-VAE) Van Den Oord et al. (017), a generative model which learns a discrete latent representation via vector quantisation (VQ). The VQ- (QE model discretizes the continuous encoder outputs (z) by mapping them to a discrete latent space consisting of a codebook with a finite set of K embedding vectors {ex A The posterior distribution q(> .) of the latent variable 2 is categorical over the embedding space, with probabilities determined by the Euclidean distances between the encoder output and the embedding vectors in the codebook. Samples drawn from this distribution index the set of embedding vectors, which are then passed as input to the decoder Zy as follows:
a(= k 12)
{
for l arg miny |Ize(x) Ezll2, otherwise,
(16)
~~ q(> 2), 2²(x) = e₂2 = ek
(17)
To enable gradient propagation through the non dineeifferet quantization step, a straight-through estimator used, wherein gradients from the decoder are directly propagated back to the encoder output (x)). The loss function used to train the VQ-VAE, defined in equation [8] consists of the re- construction loss. jointly optimizing the encoder and decoder to maximize the evidence lower bound (ELBO) the data likelioel.oood. Assuming uniform prior p(z) and deterministic posterior as in equation 16] the KL divergence of the ELBO simplifies to the constant log K and is ignored. The second term corresponds to the vector quantization loss, which updates the embedding vectors by moving them toward the encoder outputs. The third term is the commitment loss, encouraging the encoder outputs to remain close to the selected embeddings and thereby ensuring alignment between the encoder space and the embedding space.
L log p( (z)())) + (1((((2))) ell2 (z) - sglelle.
(18)
where sg stands for the stop-gradient operator which blocks the gradient from propagating through the computational branch of the operand, treating it as a constant. While standard VQ-VAE ap- proaches discretize the latent representations using single codebook of high-dimensional em- bedding vector and optimize primarily for reconstruction fidelity, learning disentangled represen- tations necessitates strong inductive biases Locatello et al. (2019b). To structure the latent space such that distinct dimensions capture independent generative factors, the approach of Hsu et al. (2024a) instead propose latent quantization, which enforces structural regularity in the latent space by quantizing each latent dimension using separate learnable scalar codebooks. Specifically, the pro- posed quantized latent autoencoder (QLAE) parameterizes the latent space as the Cartesian product
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= X Ca. where each codebook Cj contains scalar embeddings. This clement-win quantization enforces a combinatorial factorized encoding, allowing the decoder to learn consistent interpretations for each latent dimension. Furthermore, a higher weight decay is used to regular- ize the model to encourage reliance on the discrete codebook structure. Collectively, these design choices promote disentangled representations through explicit architectural and regularization bi- ases.
A.2 NETWORK ARCHITECTURES AND HYPERPARAMETERS
Recognition Network (or Encoder) The encoder is a feedforward convolutional network. Each convolutional block consists of a downsampling convolution (kernel size stride 2) followed by batch normalization, and each block is followed by a Leaky ReLU activation (slope 0.3). Four such blocks with channel widths 32, 64, 128, and 256 produce 256 x 4 feature map. This feature map is flattened and passed through two fully connected layers of width 256 with Leaky ReLU activations. final linear layer outputs the mean and precision parameters of the approximate conjugate likelihood, using a Softplus transformation on the precision to enforce positivity.
Date-Gnner Function (or Decoder). The sampled latent representation first passed through two fully connected layers of width 256 with ReLU activations and no normalization, preserving the piecewise affine structure of the mapping. The resulting feature vector is reshaped to 256 x x and processed by four wansposed-convolutio blocks. Each block applies upsampling transposed convolution (kernel size 4, stride 2) followed by a Leaky ReLU activation (slope 0.3). The blocks channel widths 128, 64, 32, and 3, yielding final observation of shape 3 x 64 x 64. For binary cross entropy loss function the output is followed by sigmoid activation function.
hyperparameter
value
d
10
o
1 (with a Gamma(1, 1) prior)
3 (scaling of KL div.)
0.005
X (muo. Ko, Wo)
(0., 1., 2., 0.1)
Autoencoder learning rate
I x 10-4
Latent learning rate
1 x 10-3
Adam 31
0.9
Adam 82
0.99
Adam updates
2 x 105
Grad clip norm
batch size
64
Table 3: Fixed Hyperparameters and initializations
A.3 ABLATION STUDIES AND DISCUSSION
Model
Info M
Info C
Info
Bayes-QLae QLAE
0.58 + 004
0.51 t.0
0.71
.04
T-QLAE (k=10)
0.54 + .04
0.40 +.0
0.68 - .04
T-QLAE (k=50)
0.51 + .06
0.48
0.62
MF-QLAE
0.49 + .04
0.49
0.76 + .04
DQ-QLAE
0.52 + 020
0.43 1
0.71 + .02
Table 4: Model performance comparison across different information metrics
We structure the experiments this section to isolate and quantify the specific inductive biases derived from each component of our hierarchical Bayesian nonparametric framework: the nested variational family, structured variational inference, and stochastic quantization. Specifically, we
The implementation is www.ah/wwwm////tiiioooooayaaiieg
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perform an ablation by replacing the nested variational family with truncated one (T-QLAE) with different truncation levels K. Similarly, to evaluate the role of the structured variational family, we substitute it with mean-field variational family (MF-QLAE), as detailed in (Johnson et al., 2016). Finally, to isolate the effect of stochastic quantization, we replace with a deterministic nearest- neighbor quantization, with the straight-through estimator used to propagate gradients through the quantization step.
From our experiments (as detailed in Table 4 Bayes-QLAE consistently outperforms its ablated variants across all disentanglement metrics, confirming that each component contributes an induc- tive bias which, when combined, enhances performance. For models based on truncated variational families, we observe a negative correlation between modularity and truncation level, while a positive correlation with compactness. This aligns with the intuition that representations obtained with fewer quantized values are biased toward modularity. Notably, the truncated model with k = 10 surpasses QLAE in modularity due to the benefits of stochastic quantization and structured variational infer- ence, though with slight reduction in compactness as a consequence of stochasticity. Removing structured variational inference and defaulting to deterministic quantization degrades both modular- ity and compactness, with the mean-field family exhibiting more severe decline in modularity, sug- gesting a bias towards representation which minimize the reconstruction cost over representations adhering to structured prior distribution. Finally, deterministic quantization exacerbates posterior collapse, leading to representations with a lower compactness metric.
We empirically demonstrate that small codebook is not prerequisite for disentanglement and therefore need not be fixed a priori. Critically, across all three axes of disentanglement assess- inrmmiinformaiivensss (reconstruction fidelity), modularity (independence), and compactness (one- to-one factor-dimension correspondence) observe that disentanglement quality remains stable or improves as the codebook size expands adaptively in response data complexity. These findings directly challenge the common assumption that small, fixed codebooks are necessary for learning disentangled representations.
Rather, the critical factors enabling disentanglement are two structural properties: (1) the implicit regularization effect induced by discrete latent encodings, and (2) the combinatorial composition of factor-specific codes encode representations. During early training stages, when the codebook size is small, the encoder operates under a representational bottleneck that necessitates the construction of latent representations through combinatorial composition of the restricted set of available codes. This bottleneck implicitly regularizes the learning process, strongly biasing the encoder toward al- locating disjoint, factor-specific codes to each factor-specific codebook. Consequently, the learned compositional structure mirrors the underlying generative process of the dataset, wherein the set of observations arise from the Cartesian product of discrete factor instantiations. This early-stage regu- larization effect establishes foundation for disentanglement by enforcing modular, compositional encoding scheme that respects the factorial structure of the data-generating distribution.
In our approach we initialize the nonparametric prior with single code per codebook. The nested variational family provides a principled mechanism to increase the number of codes: new codes are instantiated if and only if their inclusion yields an improvement in the variational lower bound. This criterion ensures that capacity expansion occurs only when statistically justified by the data. Consequently, the model inherits the inductive bias of sparse codebooks while avoiding any explicit hard constraint on the upper bound of the cardinality of the codebooks. This adaptive regularization mechanism resolves the tension between early-stage structural learning and asymptotic expressive- ness.
We validate this hypothesis through ablation studies comparing our nested variational inference framework against a truncated variant. In the truncated approach, we fix the number of mixture components at predetermined upper bound for each factor, effectively climinating the adaptive capacity of the nonparametric formulation. This modification results in measurable degradation across all disentanglement metrics relative to the nested variational inference approach. These sults demonstrate that the adaptive, data-driven discovery of codebook size-rather than absolute codebook cardinality-is the essential mechanism underlying successful disentanglement.
Moreover, the tendency toward cluster expansion is explicitly governed by the concentration param- eter a of the nonparametric prior, which itself assigned a Gamma hyperprior. This hicrarchical Bayesian formulation provides regularization of the cluster proliferation rate, enabling the model
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to infer from data the appropriate balance between model parsimony and representational capacity without manual specification.
A.4 DISENTANGLEMENT METRICS
For quantitative evaluation, we compute two complementary disentanglement metrics which com- prehensively measure disentanglement properties using different computational approaches. The InfoMEC metric (Hsu al. 2024a) relies on coommtiinntthoreede mutual information estimation computed from the empirical joint distribution between latent representations and ground-truth fac- tors. In contrast, the DCI metric (Fastwood & Williams 2018) trains predictive models to map the learned representations to the underlying factors of variation. Both metrics evaluate disentangle- ment quality across three fundamental dimensions, though with different terminology: InfoMEC measures InfoModularity (InfoM) while DCI measures Disentanglement (D) to quantify the extent to which sources are encoded in mutually disjoint subsets of representations, InfoExplicitness (In- foE) and Informativeness (I) measure the degree to which the relationship between the sources and representations can be characterized by simple functional or statistical dependency, and InfoCom- pactness (InfoC) and Completeness (C) quantifies the degree which latent variables encode infor- mation exclusively about mutually disjoint subsets of the sources. We train our proposed approach in a completely unsupervised manner on the entire dataset and evaluate the learned representations on subset of samples, using the open-source implementations of disentanglement metrics by Locatello 2019b); Hsu al. (002a)).
al.
A.5 QUALITATIVE EVALUATION
We conduct qualitative assessments of proposed method each dataset to evaluate both sam- ple reconstructions and latent traversals. For latent traversals, we encode a single image into the latent space and systematically visualize the effects of intervening Specifically, for each latent variable we vary its value across the range of values encoded in the representations (sample with replacement) while holding all other latent dimensions fixed, then de- code the resulting latent vectors to observe their effects in the data space. In the visualization, each row corresponds to interventions on single latent variable, while columns represent different val- ues sampled from the empirical distribution of that dimension. Well-disentangler representations should exhibit smooth, semantically meaningful changes along individual latent dimensions, with each dimension controlling a distinct generative factor independently of others.
individual latent dimensions.
Reconstruction Fidelity. We assess the informativeness of learned representations by examining reconstruction quality. High-fidelity reconstructions that faithfully preserve visual details of the original images indicate that the latent representations are sufficiently informative to capture the full range of variations present in the data. Conversely, poor reconstructions suggest that certain factors of variation have been inadequately encoded or lost during the encoding process.
Modularity. We evaluate the modularity, disentanglement, properties of learned representations through latent traversal analysis. A representation exhibits modularity when each latent dimension independently controls single underlying generative factor while remaining invariant to variations in other factors. Operationally, this is assessed by examining whether each row in the traversal visualization demonstrates isolated semantic changes corresponding without coupling to other factors. Such independence in the latent space reflects successful recovery of the true compositional structure of the underlying independent generative factors.
single factor of variation
Compactness. We further assess the compactness, or completeness, of the learned representations by determining whether all variations of single generative factor are captured within single latent dimension. Compact representations, wherein each factor is encoded by exactly one latent variable, are crucial for interpretability as they establish a one-to-one correspondence between latent dimen- sions and semantically meaningful generative factors. This property enables intuitive understanding and manipulation of specific attributes in the generated outputs.
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(a) Sample reconstructions: Original images (top row) and corresponding reconstructions (bottom row)
(b) Latent traversals: Each row shows the effect of systematically varying single latent dimension while holding all other dimensions fixed. Columns represent different values sampled from the distribution of that dimension. The model successfully disentangles six ground-truth factors of variation: object orientation (row 1). floor hue (row 3), wall hue (row object shape (row object hue (row 9), and object scale (row 12). Rows 2,
10, and 11 correspond to inactive latent dimensions that do not encode interpretable factors.
Figure I: Reconstructions and Latent traversals for the 3DShapes dataset: Reconstructions demon- strate high fidelity in capturing visual details demonstrating the model's ability to faithfully encode and decode the full range of variations in the data. Latent traversals illustrate that individual latent variables control distinct, interpretable factors of variation in the generated images. Moreover, the presence of inactive latent variables and the encoding of each factor in single latent variables indi- cates that the model has learned a compact representation recovering the true generative structure
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Figure 2: Evolution Latent Traversals for 3DShapes dataset: The figure illustrates the evolution of the number of clusters associated with each generative factor demonstrating the adaptive capacity of the nonparametric formulation. Each column in each row corresponds to a factor-specific mix- ture component, and the distinct components within row denote the clusters capturing the encoded variations of that factor. The vertical axis indicates cluster multiplicity, revealing how the model pro- gressively discovers and encodes additional variations for each factor. This dynamic cluster growth exemplifies the nonparametric property of the hierarchical mixture prior, which enables data-driven inference of latent capacity without manual specification. Notably, factors with higher contribution to the reconstruction objective-suce as floor hue, object hue, and wall hue hue-exhibit earlier cluster proliferation during training, suggesting the model prioritizes encoding variations that most signif- icantly impact reconstruction fidelity. In contrast, geometric factors such object orientation and shape undergo refinement later training stages, indicating hierarchical learning strategy wherein the model first captures high-vare variance attributes before refining This demonstrates that the nonparametric prior successfully balances model capacity across factors according to their respective complexities and contributions data likelihood
structural properties.
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(a) Sample reconstructions: Original images (top row) and corresponding reconstructions (bottom row)
(b) Latent traversals: Each row shows the effect of systematically varying single latent dimension while holding other dimensions fixed. Columns represent different values sampled from the distribution of that dimension. The model successfully disentangles the following ground-truth factors of variation: vertical axis (row 1). object color (row 3), object shape (row 4), background color (row 6), object size (row 9), horizontal axis (row 11) and camera height (row 12). Rows 5, 8, 10 correspond inactive latent dimensions that do not encode interpretable factors.
Reconstructions and Latent traversals for the MPI3D real dataset: Reconstructions demon- strate high fidelity in capturing visual details demonstrating the model's ability faithfully encode and decode the extensive range of variations in the data. Latent traversals illustrate that individual latent variables control distinct, interpretable factors of variation the generated images. is worth noting that, although the representations do not capture the full set of underlying variations, they remain both modular and compact, closely reflecting the true underlying structure of the generative process.
Figure 3:
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Reviews:
[{"review_id": "MSh0iAOHie", "reviewer": "Reviewer_5ePo", "summary": "This work replaces the gaussian prior of latent variables with discrete codebook with a nonparametric Dirichlet Process (DP). This process can adjust representation capacity according to the data complexity. The authors propose a hierarchical prior structure to capture complex dependences of latent variables.  The proposed method is verified on 3DShapes and MPI3D to show superior disentanglement scores.", "questions": "What are the advantages of Bayes-QLAE compared to Tripod?\nWhy do we need the hierarchical Bayesian nonparametric approach? Are there any special benefits beyond disentanglement?\nCan the experiments be added to demonstrate the advantage of the work for practical problems? \nAlso, adding training details would be beneficial. \nI case of deep encoders with a lot of non-linearities consisting of a high-dimensional latent space, would the encoder network not be able to learn to project the data into a space where Gaussian prior assumption would be enough? \nCan the principles laid in this work for learning disentangled representation be shown on some recent architectures (e.g., disentanglement in the latent space of diffusion models)?", "limitations": "", "rating": 4, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "This work aiming at removing assumption of the capacity of each factor could be an important step to practical applications of disentanglement learning.\n\nThe proposed solution, Dirichlet Process for a Bayesian nonparametric prior on latent variables seems technically sounds.\n\nThe proposed method, Bayes-QLAE, achieves good disentanglement metrics on both 3DShapes and MPI3D.", "weaknesses": "The proposed method still lacks practical proof in real situations where some combinations of generative factors are missing.\nExperiments did not verify how the DP adjusts the capacity of each factor, which is an important claim of this work.\nNo visualization to demonstrate the reconstruction quality.\nThe experimental results show that the proposed Bayes-QLAE did not surpass Tripod on 3Dshapes and MPI3D.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "fb7e2kTRhf", "reviewer": "Reviewer_EAMz", "summary": "This paper introduces Bayes-QLAE, an innovative framework for learning disentangled representations through a Bayesian non-parametric prior. Its core idea is to use a Dirichlet Process to allow the codebook for each latent dimension to adaptively adjust its capacity based on data complexity. Experiments show that the method achieves disentanglement performance comparable to strong baselines on standard benchmarks without the need for additional regularization.", "questions": "Please refer to Weakness section.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "1. Transforming the codebook size from a fixed hyperparameter into a quantity learned from data via a  Dirichlet Process is an elegant and significant advancement that directly addresses a key limitation of existing quantization methods.\n2. The paper provides solid and rigorous theoretical proofs, while also achieving competitive results on standard datasets. Furthermore, ablation studies analyze the contribution of the model's various components.", "weaknesses": "1. While making the codebook size adaptive is intuitively appealing, it contradicts findings from prior work (e.g., Tripod et al.), which suggests that maintaining a smaller codebook is beneficial for disentanglement. How does the proposed method balance the tendency of the codebook to grow with the need to maintain a compact, disentangled representation? A clear discussion or analysis of this trade-off is necessary.\n2. The current experimental results are limited to standard disentanglement metrics like InfoMCE and DCI. A more comprehensive evaluation is needed. For instance, a comparison of reconstruction accuracy against baseline methods would provide crucial insight into whether the gains in disentanglement come at the cost of representation fidelity.\n3. The paper hypothesizes that allowing adaptive codebook sizes helps align individual factors with single dimensions. However, the experiments fail to substantiate this claim. The final learned sizes of the codebook dimensions should be reported to demonstrate this adaptivity. Furthermore, visualizing the effect of a single latent variable is essential to qualitatively assess the smoothness and interpretability of the learned representations, which is a standard practice for evaluating disentanglement.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "72HD7pNmc0", "reviewer": "Reviewer_Avba", "summary": "This paper proposes a Bayesian nonparametric prior over the embedding space of latent-quantization autoencoders to determine the appropriate latent space complexity and the optimal strength of regularization constraints, and shows the effectiveness with experiments on two benchmark datasets. The proposed method produces competitive performance relative to some baseline methods.", "questions": "- What is the main difference of the proposed method compared with the recent methods for disentangled representation learning? Most of the methods in related works are out of dated.\n\n- Is the performance gain enough to argue as an achievement? Depending on the metrics, some of the baselines produce better performance. More in-depth discussion is required.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "- A sound theoretical derivation of the new framework for unsupervised learning of disentangled representation.\n\n- Experiments support the proposed framework with relative superiority to the baselines.", "weaknesses": "- Relatively weak novelty on the methods for disentangled representation learning. More recent methods should be compared and discussed.\n\n- The comparing methods should be updated with recent methods.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)