Toward the Identifiability of Comparative Deep Generative Models (2401.15903v1)

Abstract: Deep Generative Models (DGMs) are versatile tools for learning data representations while adequately incorporating domain knowledge such as the specification of conditional probability distributions. Recently proposed DGMs tackle the important task of comparing data sets from different sources. One such example is the setting of contrastive analysis that focuses on describing patterns that are enriched in a target data set compared to a background data set. The practical deployment of those models often assumes that DGMs naturally infer interpretable and modular latent representations, which is known to be an issue in practice. Consequently, existing methods often rely on ad-hoc regularization schemes, although without any theoretical grounding. Here, we propose a theory of identifiability for comparative DGMs by extending recent advances in the field of non-linear independent component analysis. We show that, while these models lack identifiability across a general class of mixing functions, they surprisingly become identifiable when the mixing function is piece-wise affine (e.g., parameterized by a ReLU neural network). We also investigate the impact of model misspecification, and empirically show that previously proposed regularization techniques for fitting comparative DGMs help with identifiability when the number of latent variables is not known in advance. Finally, we introduce a novel methodology for fitting comparative DGMs that improves the treatment of multiple data sources via multi-objective optimization and that helps adjust the hyperparameter for the regularization in an interpretable manner, using constrained optimization. We empirically validate our theory and new methodology using simulated data as well as a recent data set of genetic perturbations in cells profiled via single-cell RNA sequencing.

• The paper introduces a theoretical framework showing that DGMs achieve identifiability through piece-wise affine mixing functions, enabling interpretable latent representations.
• It proposes a multi-objective constrained optimization method that enhances model stability and robust hyperparameter tuning across diverse datasets.
• Empirical tests with single-cell sequencing and noise models validate the approach, confirming strong latent variable disentanglement and practical applicability.

Overview of "Toward the Identifiability of Comparative Deep Generative Models"

This paper explores the theory and application of identifiability within the domain of comparative Deep Generative Models (DGMs), especially focusing on scenarios involving multiple data sources. The authors aim to establish a theoretical foundation for the identifiability of DGMs used in comparative analysis and introduce methodologies to enhance their applicability in practical settings such as single-cell RNA sequencing data.

Theoretical Contributions

The paper makes significant theoretical advancements by extending recent non-linear independent component analysis findings to address the identifiability of comparative DGMs. Specifically, it is demonstrated that while DGMs are typically unidentifiable under general settings, they achieve identifiability when the mixing function is piece-wise affine, as can be parameterized using ReLU neural networks. This is a pivotal finding as it provides a clear path towards interpretable and modular latent representations, which are necessary for meaningful comparative analysis across data sets.

In more detail, the theoretical analysis distinguishes between subspace identifiability and component identifiability, focusing on ensuring that different subsets of latent variables can be accurately separated and identified when the model's mixing functions conform to specific structural constraints. This is particularly relevant in contrastive settings, where the aim is to separate shared patterns from group-specific patterns in the latent space.

Practical Implications and Methodological Innovations

On the practical side, the paper discusses the challenges of model misspecification, particularly when the number of latent variables is not predetermined. The authors empirically show that regularization techniques, which are often used to aid in identifiability, are indeed beneficial but must be chosen with care.

To address potential shortcomings in the practical application of DGMs, the paper introduces a novel methodology based on multi-objective constrained optimization. This approach considers the goal of maximizing likelihoods across multiple data sets as a multi-objective optimization problem, rather than a single-objective one. It also proposes an interpretable hyperparameter tuning mechanism with constraints on correlation metrics, which enhances the stability and interpretability of the learned latent spaces.

Empirical Validation

The empirical work validates the theoretical propositions using simulations and real-world single-cell perturbation data. The results confirm that the proposed method achieves strong disentanglement of latent variables when the assumptions about the data generator are correctly specified. Moreover, through extensive simulations with both Poisson and negative binomial observational noise, the methodology demonstrates robust performance, affirming the theoretical claims about identifiability.

Future Directions

This research contributes to a deeper understanding of DGMs in comparative analysis, paving the way for further exploration into more complex non-linear and potentially non-parametric models. Future work could expand upon these findings by exploring additional real-world applications and further refining the constraints and assumptions needed to maintain identifiability in even more nuanced settings. Moreover, the intersections between DGM identifiability and causal representation learning present a rich avenue for future research.

In conclusion, this paper provides a rigorous framework for understanding and improving the identifiability of comparative DGMs, offering both theoretical insights and practical tools that enhance the utility of these models in scientific domains where data comparability is key.