Constrained latent state modeling: A unifying perspective on representation learning under competing constraints

2026-05-15

AI Summary

Existing latent variable models often suffer from under-constrained objectives, leading to non-identifiable, ambiguous, and poorly interpretable representations. This work proposes the Constrained Latent State Modeling (CLSM) framework, which systematically integrates six core constraints—namely predictive sufficiency, minimality, temporal consistency, and others—for the first time. Grounded in information theory and dynamical systems theory, CLSM formally characterizes the intrinsic couplings and trade-offs among these constraints. By reframing representation learning as a constrained optimization problem, the framework unifies diverse approaches such as variational autoencoders and state-space models, revealing that non-identifiability stems from insufficient constraints rather than technical shortcomings. CLSM thus provides a principled foundation for designing latent variable models that are interpretable, robust, and aligned with downstream tasks.

Abstract

Learning latent representations from complex data is central to modern machine learning, spanning temporal, multimodal, and partially observed systems. In such settings, representations are better understood as latent states capturing underlying system dynamics, rather than as mere compressed summaries of observations. Yet current approaches remain fragmented, relying on distinct -- and often implicit -- assumptions about what these states should represent. We argue that this fragmentation reflects a more fundamental limitation: latent representations are typically learned from underconstrained objectives that fail to specify the properties that meaningful latent states should satisfy. As a result, multiple representations can satisfy the same objective, leading to ambiguity in their structure and interpretation. While many of the underlying principles have been explored in isolation, their interactions have not been explicitly formalized. In this work, we propose constrained latent state modeling (CLSM) as a unifying perspective. We identify a set of core properties -- predictive sufficiency, minimality, temporal coherence, observation compatibility, invariance to nuisance factors, and structural constraints -- and show that they are intrinsically coupled through fundamental trade-offs. Revisiting major modeling families through this lens, we show that existing approaches can be interpreted as enforcing different subsets of constraints, thereby occupying distinct regions of a common design space. This perspective reframes persistent challenges such as lack of identifiability as consequences of underconstrained formulations, rather than isolated technical limitations. More broadly, CLSM provides a principled framework to make design choices explicit, to analyze trade-offs, and to guide the development of more interpretable, robust, and task-aligned latent state models.

Problem

• latent state modeling
• representation learning
• identifiability
• constraints
• trade-offs

Innovation

• constrained latent state modeling
• representation learning
• identifiability
• trade-offs
• unifying framework