Compositional Latent Variables

• Compositional latent variable models are statistical frameworks that structure latent space as a composition of interrelated factors, enabling modular data representation.
• They leverage techniques like composite likelihood and variational inference to efficiently capture complex, multi-scale relationships within data.
• Their applications span psychometrics, generative modeling, and deep learning, offering improved interpretability and generalization across diverse domains.

A compositional latent variable refers to a latent structure in statistical and machine learning models in which the latent space is explicitly or implicitly organized as a composition of multiple interrelated parts, factors, or modules. This compositional architecture is designed to capture the modular organization of complex systems, enabling models to represent data as combinations of simpler, interpretable latent components—such as factors, parts, or attributes. Compositional latent variables are foundational in many domains, including measurement models, probabilistic generative models, structured prediction, and deep representation learning. Across approaches, compositional structure provides a mechanism for modularity, improved generalization, interpretability, and scalability.

1. Conceptualization and Mathematical Foundations

Compositional latent variables formalize the idea that complex phenomena are generated not by a single monolithic latent entity but by a set of (potentially structured) interdependent latent factors.

Latent Space Composition

The typical latent variable model expresses the joint distribution as

$p(x, z) = p(z) p(x|z)$

where $z$ is the latent variable and $x$ is the observed data. In compositional models, $z$ is itself partitioned or structured, e.g., $z = (z_1, z_2, \ldots, z_s)$, with each $z_i$ encoding a primitive factor (e.g., attributes, parts, tasks, or subspaces) (Farouni, 2017).

Notably, in models such as the structured canonical correlation model (Silva, 2012), $z$ is decomposed into "target" latent variables of direct interest ($X_s$) and nuisance/confounding latents ($X_x$), yielding a "compositional" latent structure:

• $X_s$: Pre-specified, interpretable sources (e.g., "job satisfaction")
• $X_x$: Unanticipated, confounding sources

Compositionality in the Measurement Model

In multivariate measurement settings (e.g., psychometrics, econometrics), compositional latent structures arise naturally. For example, each observed item may be associated with a subset of one or more latent factors, and the measurement model may compose these contributions:

$y_i = \sum_{k} \lambda_{ik} f_k + \epsilon_i$

with $f_k$ denoting compositional factors and $\lambda_{ik}$ loadings.

The compositional nature is also exploited in canonical correlation-inspired models, where observed variables are grouped, and each group is linked to a particular latent cause—together with bi-directed edges to account for unexplained associations (Silva, 2012).

Hierarchical and Modular Architectures

Compositionality extends to hierarchical models, where global and local latent variables combine to represent heterogeneity at different scales (Farouni, 2017):

$$p(y_{1:N}, z_{1:N}, \theta) = p(\theta) \prod_{n=1}^N p(y_n | z_n, \theta) p(z_n| \theta)$$

Here, the latent structure composes dataset-level and data-point-level factors, and deep latent variable models further arrange these hierarchically through multiple abstraction layers.

2. Methodological Paradigms

Compositional latent variable models are realized through a variety of methodological frameworks, each prescribing a distinct way of composing and inferring latent structure.

Mixed Graphical Models and Structure Learning

In latent composite likelihood learning (Silva, 2012), the dependence structure is modeled as a mixed graph $G_m$ comprising:

• Directed edges from each $X_s$ to its associated observed set
• Bi-directed edges among observed nodes to capture residual dependencies due to $X_x$

Efficient structure learning is achieved via composite likelihood optimization, enabling the model to detect and modularly fit confounding latent influences without explicit prior specification of their number or behavior.

Hierarchical and Surrogate Latent Variables

In variational composite autoencoders (Yao et al., 2018) and related hierarchical VAE frameworks (Berger et al., 2020), compositionality is operationalized by inserting intermediate surrogate latents, such as

$$q_{\theta_1}(s|x), \quad q_{\theta_2}(z|x), \quad p_\psi(s|z), \quad p_\phi(x|s)$$

effectively decomposing the representation and amortizing inference across sequential layers.

Linear and Geometric Decompositions

Compositionality also manifests as the imposition of linear or geometric structure on the latent space. For example, in compositional augmentation models (Pooladzandi et al., 2023), each transformation is realized as a linear operator in latent space, designed to be composable and invertible:

$$z_{\text{aug}_i} = z_{0} L_{\text{aug}_i}, \qquad z_{\text{comp}} = z_{0} L_{\text{aug}_1} L_{\text{aug}_2}$$

For models operating on manifolds (e.g., unit spheres in CLIP-like vision-LLMs), geodesically decomposable embeddings (Berasi et al., 21 Mar 2025) optimize for compositionality in the tangent space:

$$u_z = \mathrm{Exp}_\mu(z_1^\dag + \ldots + z_s^\dag)$$

where $\mathrm{Exp}_\mu$ is the exponential map at the intrinsic mean $\mu$.

Sequential Construction and Amortization

Compositional latent spaces with high cardinality (e.g., non-factorized discrete structures) are addressed with algorithms such as GFlowNets, which sequentially build the latent $z$ (Hu et al., 2023), efficiently sampling multimodal, compositional posteriors by leveraging policies that construct latent objects piecewise.

3. Inference, Estimation, and Uncertainty

Inference in compositional latent variable models must accommodate modularity, non-independence, and possibly high combinatorial complexity.

Composite Likelihood Approaches

In practical structure learning scenarios, full likelihoods are often intractable. Composite likelihoods—approximated by aggregating low-dimensional marginal or conditional likelihoods across all variable pairs or triplets—allow efficient and modular inference (Silva, 2012). The composite log-likelihood PCL is given by:

$$\text{PCL}(G_m, B, \Sigma) = F(B, \Sigma) + \log T(G_m)$$

Variational and Hybrid Inference

For models combining latent network estimation and variable selection involving compositional count data (Osborne et al., 2020), structured variational inference with EM steps allows for scalable posterior approximation over both compositional latent layers and network structure, often leveraging spike-and-slab priors to enforce sparsity and interpretability.

Uncertainty Quantification

A critical aspect—particularly in compositional models for physical sciences (e.g., mineral spectra, population proportions)—is the propagation and amplification of uncertainty. The uncertainty arising in estimated system operators (as in (Park, 2018)) directly impacts downstream latent inference, as quantified by explicit variance scaling factors and MCMC-based characterization of posterior uncertainty.

4. Empirical Results and Applications

Psychometric and Social Science Measurement

Compositional latent variable frameworks have proven effective in large-scale measurement settings, such as the NHS staff survey (Silva, 2012), where accounting for nuisance latent factors improved predictive latent embedding performance (as measured by AUC), enhanced interpretability, and reduced false positive edge inference rates in synthetic experiments.

Biostatistics and Microbiome Analysis

Hierarchical Bayesian compositional latent models have been successfully applied for network estimation and variable selection in high-dimensional microbiome data (Osborne et al., 2020). The explicit modeling of compositional constraints and latent layers results in improved recovery of both interaction networks and associations with external covariates.

Computer Vision, NLP, and Generative Modeling

In deep generative settings, compositional latent variables enable controllable and structured generation, such as:

• Compositional video prediction via trajectory-level latent variables supporting diverse and coherent future sampling (Ye et al., 2019)
• Modular latent spaces for disentangled image generation and editing, leveraging geometry-aware compositions in VLM representations (Berasi et al., 21 Mar 2025)
• Part-based mesh synthesis in 3D generative models using disentangled, part-specific latent tokens and hierarchical attention (Lin et al., 5 Jun 2025)

In NLP, compositional representations underlie advancement in topic modeling, sequence prediction, and unsupervised parsing, enabling models to represent structured syntactic and semantic phenomena hierarchically (Kim et al., 2018).

5. Theoretical Insights and Significance

Compositional latent variable models are theoretically motivated by their ability to mirror real-world hierarchical and modular organization. The decomposition:

$$p(y_{1:N}, z_{1:N}, \theta) = p(\theta) \prod_{n=1}^N p(y_n|z_n, \theta) p(z_n|\theta)$$

is not only mathematically convenient but reflects the cognitive principles of part–whole reasoning and modular abstraction (Farouni, 2017, Ren et al., 2023). Analyses based on Kolmogorov complexity demonstrate that compositional mappings have exponentially lower description length than arbitrary bijections, reinforcing compressibility as a statistical and cognitive advantage (Ren et al., 2023).

In probabilistic modeling, compositionality enables efficient inference via partitioning, facilitates parameter modularity, and supports the design of expressive yet interpretable models—often making inference scalable by localizing computation.

6. Extensions, Limitations, and Generalizations

The compositional latent variable paradigm extends beyond simple additive or hierarchical decomposition. Recent work generalizes compositionality to:

• Geodesic decomposition on manifolds for non-Euclidean latent geometries (Berasi et al., 21 Mar 2025)
• Sequential, non-factorized construction of complicated discrete structures for language and vision tasks (Hu et al., 2023)
• Joint modeling of reflective (latent variable) and formative (composite construct) effects in SEM, allowing seamless integration of different construct types within a unified estimation framework (Schamberger et al., 8 Aug 2025)

A possible limitation is that inference, estimation, or identifiability may become challenging as the complexity or flexibility of the compositional structure increases, particularly in the absence of strong structural or regularization assumptions. Empirical evidence, however, suggests that composite likelihood approximations, amortized inference, and discrete message-passing procedures can mitigate many practical challenges.

7. Implications and Outlook

Compositional latent variable models provide a unifying framework for representing, inferring, and understanding the modular structure of complex phenomena across statistical, probabilistic, and deep representation learning paradigms. Their modularity enables not only interpretability and adaptability to new tasks (e.g., zero-shot recognition, transfer learning) but also supports robust scientific and applied inference in high-dimensional measurement regimes.

Applications span a wide range—from improved measurement in the social sciences (Silva, 2012), network and covariate analysis in life sciences (Osborne et al., 2020), modular and controllable generation in computer vision and NLP (Kim et al., 2018, Berasi et al., 21 Mar 2025), to advances in structured equation modeling (Schamberger et al., 8 Aug 2025), zero-shot composition in recognition (Shi et al., 4 Jun 2025), and 3D scene decomposition (Lin et al., 5 Jun 2025). Recent innovations leveraging non-Euclidean geometry, iterated learning, and amortized inference indicate a rich and expanding field with substantial theoretical, methodological, and application-oriented significance.