Given only observational data $X = g(Z)$, where both the latent variables $Z$ and the generating process $g$ are unknown, recovering $Z$ is ill-posed without additional assumptions. Existing methods often assume linearity or rely on auxiliary supervision and functional constraints. However, such assumptions are rarely verifiable in practice, and most theoretical guarantees break down under even mild violations, leaving uncertainty about how to reliably understand the hidden world. To make identifiability actionable in the real-world scenarios, we take a complementary view: in the general settings where full identifiability is unattainable, what can still be recovered with guarantees, and what biases could be universally adopted? We introduce the problem of diverse dictionary learning to formalize this view. Specifically, we show that intersections, complements, and symmetric differences of latent variables linked to arbitrary observations, along with the latent-to-observed dependency structure, are still identifiable up to appropriate indeterminacies even without strong assumptions. These set-theoretic results can be composed using set algebra to construct structured and essential views of the hidden world, such as genus-differentia definitions. When sufficient structural diversity is present, they further imply full identifiability of all latent variables. Notably, all identifiability benefits follow from a simple inductive bias during estimation that can be readily integrated into most models. We validate the theory and demonstrate the benefits of the bias on both synthetic and real-world data.

Nonlinear independent component analysis (ICA) is fundamental in unsupervised learning.

Linear causal models are important tools for modeling causal dependencies and yet in practice, only a subset of the variables can be observed. In this paper, we examine the parameter identifiability of these models by investigating whether the edge coefficients can be recovered given the causal structure and partially observed data. Our setting is more general than that of prior research--we allow all variables, including both observed and latent ones, to be flexibly related, and we consider the coefficients of all edges, whereas most existing works focus only on the edges between observed variables. Theoretically, we identify three types of indeterminacy for the parameters in partially observed linear causal models. We then provide graphical conditions that are sufficient for all parameters to be identifiable and show that some of them are provably necessary. Methodologically, we propose a novel likelihood-based parameter estimation method that addresses the variance indeterminacy of latent variables in a specific way and can asymptotically recover the underlying parameters up to trivial indeterminacy. Empirical studies on both synthetic and real-world datasets validate our identifiability theory and the effectiveness of the proposed method in the finite-sample regime.

Machine learning is increasingly used in the legal domain, where it typically operates retrospectively by treating past case outcomes as ground truth. However, legal outcomes are often shaped by human interventions that are not captured in most machine learning approaches. A final decision may result from a settlement, an appeal, or other procedural actions. This creates label indeterminacy: the outcome could have been different if the intervention had or had not taken place. We argue that legal machine learning applications need to account for label indeterminacy. Methods exist that can impute these indeterminate labels, but they are all grounded in unverifiable assumptions. In the context of classifying cases from the European Court of Human Rights, we show that the way that labels are constructed during training can significantly affect model behaviour. We therefore position label indeterminacy as a relevant concern in AI & Law and demonstrate how it can shape model behaviour.

Neural Information Processing Systems http://nips.cc/

Nonlinear independent component analysis (ICA) is fundamental in unsupervised learning.

This paper presents NeutroSENSE, a neutrosophic-enhanced ensemble framework for interpretable intrusion detection in IoT environments. By integrating Random Forest, XGBoost, and Logistic Regression with neutrosophic logic, the system decomposes prediction confidence into truth (T), falsity (F), and indeterminacy (I) components, enabling uncertainty quantification and abstention. Predictions with high indeterminacy are flagged for review using both global and adaptive, class-specific thresholds. Evaluated on the IoT-CAD dataset, NeutroSENSE achieved 97% accuracy, while demonstrating that misclassified samples exhibit significantly higher indeterminacy (I = 0.62) than correct ones (I = 0.24). The use of indeterminacy as a proxy for uncertainty enables informed abstention and targeted review-particularly valuable in edge deployments. Figures and tables validate the correlation between I-scores and error likelihood, supporting more trustworthy, human-in-the-loop AI decisions. This work shows that neutrosophic logic enhances both accuracy and explainability, providing a practical foundation for trust-aware AI in edge and fog-based IoT security systems.

Linear causal models are important tools for modeling causal dependencies and yet in practice, only a subset of the variables can be observed. In this paper, we examine the parameter identifiability of these models by investigating whether the edge coefficients can be recovered given the causal structure and partially observed data. Our setting is more general than that of prior research--we allow all variables, including both observed and latent ones, to be flexibly related, and we consider the coefficients of all edges, whereas most existing works focus only on the edges between observed variables. Theoretically, we identify three types of indeterminacy for the parameters in partially observed linear causal models. We then provide graphical conditions that are sufficient for all parameters to be identifiable and show that some of them are provably necessary.