Latent Variable Models

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Overview

Latent variable models represent observed data as arising from unobserved factors, classes, states, or embeddings, helping explain structure when measurements are incomplete, noisy, or high-dimensional. Common approaches include factor analysis and matrix factorization, finite mixtures and topic models, probabilistic graphical models, and deep generative models such as variational autoencoders; they are fit with EM, Bayesian or variational inference, and spectral or gradient-based methods. Applications range from clustering, recommendation, and multimodal integration to causal discovery, time-series forecasting, and personalized clinical prediction. A central challenge is identifiability: different hidden representations can explain the same observations, so work seeks assumptions and diagnostics that make latent structure recoverable while balancing flexibility, interpretability, computation, and calibrated uncertainty.

Latest papers

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153 papers14 papers in the last four weeks, level with the four weeks before. 0.1% of all new papers.

Mar 27, 2026

Automatic Laplace Collapsed Sampling: Scalable Marginalisation of Latent Parameters via Automatic Differentiation

We present Automatic Laplace Collapsed Sampling (ALCS), a general framework for marginalising latent parameters in Bayesian models using automatic differentiation, which we combine with nested sampling to explore the hyperparameter space in a robust and efficient manner. At each nested sampling likelihood evaluation, ALCS collapses the high-dimensional latent variables zz to a scalar contribution via maximum a posteriori (MAP) optimisation and a Laplace approximation, both computed using autodiff. This reduces the effective dimension from dθ+dzd_θ+ d_z to just dθd_θ, making Bayesian evidence computation tractable for high-dimensional settings without hand-derived gradients or Hessians, and with minimal model-specific engineering. The MAP optimisation and Hessian evaluation are parallelised across live points on GPU-hardware, making the method practical at scale. We also show that automatic differentiation enables local approximations beyond Laplace to parametric families such as the Student-tt, which improves evidence estimates for heavy-tailed latents. We validate ALCS on a suite of benchmarks spanning hierarchical, time-series, and discrete-likelihood models and establish where the Gaussian approximation holds. This enables a post-hoc ESS diagnostic that localises failures across hyperparameter space without expensive joint sampling.

Toby Lovick, David Yallup, Will Handley

Institute of Astronomy, University of Cambridge · Kavli Institute for Cosmology, University of Cambridge · Institute of Astronomy & Kavli Institute for Cosmology, University of Cambridge

Bayesian InferenceLatent Variable Models
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Mar 24, 2026

Conditionally Identifiable Latent Representation for Multivariate Time Series with Structural Dynamics

We propose the Identifiable Variational Dynamic Factor Model (iVDFM) for multivariate time series. The model combines variational inference with partial identifiability guarantees up to a known ambiguity class. Innovations follow a conditional exponential-family prior over observed auxiliary context and regime embeddings. Linear diagonal dynamics map innovations to factors and preserve that class, so factors are partially identifiable up to permutation and component-wise affine maps. We train by maximizing the ELBO and estimate uncertainty in both latent trajectories and forecasts. Under explicit assumptions, we prove partial identifiability up to the stated class and provide the full proof in the appendix. We evaluate factor recovery on synthetic DGPs, intervention behavior on synthetic SCMs, and forecasting on real benchmarks with CRPS and MSE.

Minkey Chang, Jae-Young Kim

Latent Variable ModelsLatent Dynamics ModelingRepresentation Learning
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Mar 23, 2026

Rule-State Inference (RSI): A Bayesian Framework for Compliance Monitoring in Rule-Governed Domains

Compliance monitoring in rule-governed domains (tax administration, clinical protocol adherence, environmental regulation) faces three structural obstacles that standard machine learning does not simultaneously address: the absence of labeled outcomes at deployment, strategically missing observations where non-compliant entities selectively withhold evidence, and a regulatory environment that changes faster than any supervised model can be retrained. We introduce Rule-State Inference (RSI), a Bayesian framework that reverses the usual paradigm. Rather than learning rules from data, RSI treats an authoritative, formalized rule set as structured Bayesian priors and infers the latent compliance state of a population through mean-field variational inference with exact coordinate-ascent updates. The central modeling object is a joint latent state per regulatory period: a global compliance-culture factor eta and per-rule components for activation, population compliance level, and parametric drift. RSI delivers three formal guarantees: O(n_k + K) regulatory adaptability per rule update; Bernstein-von Mises consistency for the identifiable continuous components; and monotone ELBO convergence at every iteration. We instantiate RSI on the Togolese fiscal system on a benchmark of 2,000 synthetic enterprises grounded in official regulatory law; full numerical validation is forthcoming. The framework is designed for direct extension to Sequential RSI, a state-space formulation where the posterior from one regulatory period becomes the prior for the next, yielding an exact Kalman filter for compliance-trajectory tracking and entity-level Bayesian scoring.

Abdou-Raouf Atarmla

Institut National des Postes et Télécommunications (INPT), Rabat, Morocco · Togo DataLab, Ministry of Digital Economy, Lomé, Togo

AI Regulatory ComplianceBayesian InferenceLatent Variable Models
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Mar 20, 2026

Var-JEPA: A Variational Formulation of the Joint-Embedding Predictive Architecture - Bridging Predictive and Generative Self-Supervised Learning

The Joint-Embedding Predictive Architecture (JEPA) is often seen as a non-generative alternative to likelihood-based self-supervised learning, emphasizing prediction in representation space rather than reconstruction in observation space. We argue that the resulting separation from probabilistic generative modeling is largely rhetorical rather than structural: the canonical JEPA design (coupled encoders with a context-to-target predictor) mirrors the variational posteriors and learned conditional priors obtained when variational inference is applied to a particular class of coupled latent-variable models, and standard JEPA can be viewed as a deterministic specialization in which regularization is imposed via architectural and training heuristics rather than an explicit likelihood. Building on this view, we derive the Variational JEPA (Var-JEPA), which makes the latent generative structure explicit by optimizing a single Evidence Lower Bound (ELBO). This yields meaningful representations without ad-hoc anti-collapse regularizers and allows principled uncertainty quantification in the latent space. We instantiate the framework for tabular data (Var-T-JEPA) and achieve strong representation learning and downstream performance, improving over T-JEPA across real-world tabular benchmarks while remaining competitive with strong raw-feature baselines.

Moritz Gögl, Christopher Yau

University of Oxford, UK · Health Data Research UK

Joint-Embedding Predictive ArchitectureSelf-Supervised LearningLatent Variable ModelsVariational Inference
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Mar 20, 2026

Hybrid topic modelling for computational close reading: Mapping narrative themes in Pushkin's Evgenij Onegin

This study presents a hybrid topic modelling framework for computational literary analysis that integrates Latent Dirichlet Allocation (LDA) with sparse Partial Least Squares Discriminant Analysis (sPLS-DA) to model thematic structure and longitudinal dynamics in narrative poetry. As a case study, we analyse Evgenij Onegin-Aleksandr S. Pushkin's novel in verse-using an Italian translation, testing whether unsupervised and supervised lexical structures converge in a small-corpus setting. The poetic text is segmented into thirty-five documents of lemmatised content words, from which five stable and interpretable topics emerge. To address small-corpus instability, a multi-seed consensus protocol is adopted. Using sPLS-DA as a supervised probe enhances interpretability by identifying lexical markers that refine each theme. Narrative hubs-groups of contiguous stanzas marking key episodes-extend the bag-of-words approach to the narrative level, revealing how thematic mixtures align with the poem's emotional and structural arc. Rather than replacing traditional literary interpretation, the proposed framework offers a computational form of close reading, illustrating how lightweight probabilistic models can yield reproducible thematic maps of complex poetic narratives, even when stylistic features such as metre, phonology, or native morphology are abstracted away. Despite relying on a single lemmatised translation, the approach provides a transparent methodological template applicable to other high-density literary texts in comparative studies.

Angelo Maria Sabatini

The BioRobotics Institute, Scuola Superiore Sant’Anna Pisa, Italy

Computational Literary StudiesTopic ModelingLatent Variable Models
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Mar 19, 2026

Hierarchical Latent Structure Learning through Online Inference

Learning systems must balance generalization across experiences with discrimination of task-relevant details. Effective learning therefore requires representations that support both. Online latent-cause models support incremental inference but assume flat partitions, whereas hierarchical Bayesian models capture multilevel structure but typically require offline inference. We introduce the Hierarchical Online Learning of Multiscale Experience Structure (HOLMES) model, a computational framework for hierarchical latent structure learning through online inference. HOLMES combines a variation on the nested Chinese Restaurant Process prior with sequential Monte Carlo inference to perform tractable trial-by-trial inference over hierarchical latent representations without explicit supervision over the latent structure. In simulations, HOLMES matched the predictive performance of flat models while learning more compact representations that supported one-shot backward transfer to higher-level latent categories. In a forward transfer task, HOLMES additionally achieved above-chance outcome prediction for stimuli with never-before-seen feature combinations, by exploiting abstract shape-level representations learned across diverse training instances. These results provide a tractable computational framework for discovering hierarchical structure in sequential data.

Ines Aitsahalia, Kiyohito Iigaya

Center for Theoretical Neuroscience and Zuckerman Institute, Columbia University, New York, NY 10027 · Department of Psychiatry, Columbia University Irving Medical Center, New York, NY 10032 · New York State Psychiatric Institute, New York, NY 10032 +1

Hierarchical Representation LearningLatent Variable ModelsHierarchical Bayesian ModelingOnline Continual Learning
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Mar 14, 2026

Exploring the Dimensions of a Variational Neuron

We introduce EVE (Elemental Variational Expanse), a variational distributional neuron formulated as a local probabilistic computational unit with an explicit prior, an amortized posterior, and unit-level variational regularization. In most modern architectures, uncertainty is modeled through global latent variables or parameter uncertainty, while the computational unit itself remains scalar. EVE instead relocates probabilistic structure to the neuron level, making it locally observable and controllable. In this paper, the term dimensions refers primarily to the neuron's internal latent dimensionality, denoted by k. We study how varying k, from the atomic case k = 1 to higher-dimensional latent spaces, changes the neuron's learned operating regime. We then examine how this main axis interacts with two additional structural properties: local capacity control and temporal persistence through a neuron-level autoregressive extension. To support this study, EVE is instrumented with internal diagnostics and constraints, including effective KL, a target band on mu^2, out-of-band fractions, and indicators of drift and collapse. Across selected forecasting and tabular settings, we show that latent dimensionality, control, and temporal extension shape the neuron's internal regime, and that some neuron-level variables are measurable, informative, and related to downstream behavior. Overall, the paper provides an experimentally grounded first map of the design space opened by a variational neuron.

Yves Ruffenach

Conservatoire National des Arts et Métiers

Latent Variable ModelsVariational Inference
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Mar 13, 2026

I Know What I Don't Know: Latent Posterior Factor Models for Multi-Evidence Probabilistic Reasoning

Real-world decision-making, from tax compliance assessment to medical diagnosis, requires aggregating multiple noisy and potentially contradictory evidence sources. Existing approaches either lack explicit uncertainty quantification (neural aggregation methods) or rely on manually engineered discrete predicates (probabilistic logic frameworks), limiting scalability to unstructured data. We introduce Latent Posterior Factors (LPF), a framework that transforms Variational Autoencoder (VAE) latent posteriors into soft likelihood factors for Sum-Product Network (SPN) inference, enabling tractable probabilistic reasoning over unstructured evidence while preserving calibrated uncertainty estimates. We instantiate LPF as LPF-SPN (structured factor-based inference) and LPF-Learned (end-to-end learned aggregation), enabling a principled comparison between explicit probabilistic reasoning and learned aggregation under a shared uncertainty representation. Across eight domains (seven synthetic and the FEVER benchmark), LPF-SPN achieves high accuracy (up to 97.8%), low calibration error (ECE 1.4%), and strong probabilistic fit, substantially outperforming evidential deep learning, LLMs and graph-based baselines over 15 random seeds. Contributions: (1) A framework bridging latent uncertainty representations with structured probabilistic reasoning. (2) Dual architectures enabling controlled comparison of reasoning paradigms. (3) Reproducible training methodology with seed selection. (4) Evaluation against EDL, BERT, R-GCN, and large language model baselines. (5) Cross-domain validation. (6) Formal guarantees in a companion paper.

Aliyu Agboola Alege

Epalea

Latent Variable ModelsUncertainty Calibration
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Mar 12, 2026

Probabilistic Joint and Individual Variation Explained (ProJIVE) for Data Integration

Collecting multiple types of data on the same set of subjects is common in modern scientific applications including, genomics, metabolomics, and neuroimaging. Joint and Individual Variance Explained (JIVE) seeks a low-rank approximation of the joint variation between two or more sets of features captured on common subjects and isolates this variation from that unique to eachset of features. We develop an expectation-maximization (EM) algorithm to estimate a probabilistic model for the JIVE framework. The model extends probabilistic principal components analysis to multiple data sets. Our maximum likelihood approach simultaneously estimates joint and individual components, which can lead to greater accuracy compared to other methods. We apply ProJIVE to measures of brain morphometry and cognition in Alzheimer's disease. ProJIVE learns biologically meaningful courses of variation, and the joint morphometry and cognition subject scores are strongly related to more expensive existing biomarkers. Data used in preparation of this article were obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database. Code to reproduce the analysis is available on our GitHub page.

Raphiel J. Murden, Ganzhong Tian, Deqiang Qiu +1

Department of Biostatistics and Bioinformatics, Rollins School of Public Health, Emory University, 1518 Clifton Rd NE, Atlanta, GA 30322, USA · Department of Radiology and Imaging Sciences, Emory University School of Medicine, 1364 Clifton Rd NE, Atlanta, GA 30322, USA

Latent Variable ModelsPrincipal Component AnalysisMulti-View Learning
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Mar 12, 2026

Slack More, Predict Better: Proximal Relaxation for Probabilistic Latent Variable Model-based Soft Sensors

Nonlinear Probabilistic Latent Variable Models (NPLVMs) are a cornerstone of soft sensor modeling due to their capacity for uncertainty delineation. However, conventional NPLVMs are trained using amortized variational inference, where neural networks parameterize the variational posterior. While facilitating model implementation, this parameterization converts the distributional optimization problem within an infinite-dimensional function space to parameter optimization within a finite-dimensional parameter space, which introduces an approximation error gap, thereby degrading soft sensor modeling accuracy. To alleviate this issue, we introduce KProxNPLVM, a novel NPLVM that pivots to relaxing the objective itself and improving the NPLVM's performance. Specifically, we first prove the approximation error induced by the conventional approach. Based on this, we design the Wasserstein distance as the proximal operator to relax the learning objective, yielding a new variational inference strategy derived from solving this relaxed optimization problem. Based on this foundation, we provide a rigorous derivation of KProxNPLVM's optimization implementation, prove the convergence of our algorithm can finally sidestep the approximation error, and propose the KProxNPLVM by summarizing the abovementioned content. Finally, extensive experiments on synthetic and real-world industrial datasets are conducted to demonstrate the efficacy of the proposed KProxNPLVM.

Zehua Zou, Yiran Ma, Yulong Zhang +5

the Hangzhou International Innovation Institute, Beihang University, Hangzhou 311115, China · the State Key Laboratory of Industrial Control Technology, College of Control science and Engineering, Zhejiang University, Hangzhou 310027, Zhejiang, China · the College of Electrical Engineering and Automation, Fuzhou University, Fuzhou 350108, China +4

Latent Variable ModelsVariational InferenceVirtual Sensing
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Mar 9, 2026

Beyond ReinMax: Low-Variance Gradient Estimators for Discrete Latent Variables

Machine learning models involving discrete latent variables require gradient estimators to facilitate backpropagation in a computationally efficient manner. The most recent addition to the Straight-Through family of estimators, ReinMax, can be viewed from a numerical ODE perspective as incorporating an approximation via Heun's method to reduce bias, but at the cost of high variance. In this work, we introduce the ReinMax-Rao and ReinMax-CV estimators which incorporate Rao-Blackwellisation and control variate techniques into ReinMax to reduce its variance. Our estimators demonstrate superior performance on training variational autoencoders with discrete latent spaces. Furthermore, we investigate the possibility of leveraging alternative numerical methods for constructing more accurate gradient approximations and present an alternative view of ReinMax from a simpler numerical integration perspective.

Daniel Wang, Thang D. Bui

School of Computing, College of Systems and Society, Australian National University

Straight-Through EstimatorVariance ReductionLatent Variable Models
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Mar 5, 2026

Random Dot Product Graphs as Dynamical Systems: Limitations and Opportunities

Can we learn the differential equations governing the evolution of a temporal network? We investigate this within Random Dot Product Graphs (RDPGs), where each network snapshot is generated from latent positions evolving under unknown dynamics. We identify three fundamental obstructions: gauge freedom from rotational ambiguity in latent positions, realizability constraints from the manifold structure of the probability matrix, and trajectory recovery artifacts from spectral embedding. We develop a geometric framework based on principal fiber bundles that formalizes these obstructions. We characterize invisible dynamics as exactly the skew-symmetric generators, and show the realizable tangent space has dimension nd−d(d−1)/2nd - d(d-1)/2. An holonomy dichotomy emerges: polynomial dynamics have commuting generators, stationary eigenvectors, and trivial holonomy, making gauge alignment purely statistical; Laplacian dynamics satisfy a non-commutativity criterion producing nontrivial holonomy, with curvature weighted by 1/(λι+λγ)1/(λ_ι+ λ_γ) linking gauge sensitivity to the spectral gap. In d=2d=2 this yields full restricted holonomy SO(2)\mathrm{SO}(2); for d≥3d \ge 3 generic full SO(d)\mathrm{SO}(d) remains conjectural. Cram'er--Rao lower bounds reveal that the same spectral gap controlling curvature and injectivity simultaneously controls Fisher information, so geometric and statistical difficulty are inextricable. We prove an identifiability principle: symmetric dynamics cannot absorb skew-symmetric gauge contamination, so dynamics structure can resolve gauge ambiguity. We demonstrate this constructively with anchor-based alignment and a UDE pipeline recovering vector fields from noisy graph sequences. Yet finite-sample interactions between noise, gauge, and dynamics expressiveness remain beyond the asymptotic theory. We frame this gap as an open challenge.

Giulio Valentino Dalla Riva

Latent Dynamics ModelingDynamic Graph LearningSystem IdentificationLatent Variable Models
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Mar 1, 2026

Probabilistic Learning and Generation in Deep Sequence Models

Despite exceptional predictive performance of Deep sequence models (DSMs), the main concern of their deployment centers around the lack of uncertainty awareness. In contrast, probabilistic models quantify the uncertainty associated with unobserved variables with rules of probability. Notably, Bayesian methods leverage Bayes' rule to express our belief of unobserved variables in a principled way. Since exact Bayesian inference is computationally infeasible at scale, approximate inference is required in practice. Two major bottlenecks of Bayesian methods, especially when applied in deep neural networks, are prior specification and approximation quality. In Chapter 3 & 4, we investigate how the architectures of DSMs themselves can be informative for the design of priors or approximations in probabilistic models. We first develop an approximate Bayesian inference method tailored to the Transformer based on the similarity between attention and sparse Gaussian process. Next, we exploit the long-range memory preservation capability of HiPPOs (High-order Polynomial Projection Operators) to construct an interdomain inducing point for Gaussian process, which successfully memorizes the history in online learning. In addition to the progress of DSMs in predictive tasks, sequential generative models consisting of a sequence of latent variables are popularized in the domain of deep generative models. Inspired by the explicit self-supervised signals for these latent variables in diffusion models, in Chapter 5, we explore the possibility of improving other generative models with self-supervision for their sequential latent states, and investigate desired probabilistic structures over them. Overall, this thesis leverages inductive biases in DSMs to design probabilistic inference or structure, which bridges the gap between DSMs and probabilistic models, leading to mutually reinforced improvement.

Wenlong Chen

Imperial College London Department of Computing

Bayesian InferenceSequence ModelingBayesian Neural NetworksLatent Variable ModelsGaussian Processes
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Feb 27, 2026

Hierarchical Action Learning for Weakly-Supervised Action Segmentation

Humans perceive actions through key transitions that structure actions across multiple abstraction levels, whereas machines, relying on visual features, tend to over-segment. This highlights the difficulty of enabling hierarchical reasoning in video understanding. Interestingly, we observe that lower-level visual and high-level action latent variables evolve at different rates, with low-level visual variables changing rapidly, while high-level action variables evolve more slowly, making them easier to identify. Building on this insight, we propose the Hierarchical Action Learning (HAL) model for weakly-supervised action segmentation. Our approach introduces a hierarchical causal data generation process, where high-level latent action governs the dynamics of low-level visual features. To model these varying timescales effectively, we introduce deterministic processes to align these latent variables over time. The HAL model employs a hierarchical pyramid transformer to capture both visual features and latent variables, and a sparse transition constraint is applied to enforce the slower dynamics of high-level action variables. This mechanism enhances the identification of these latent variables over time. Under mild assumptions, we prove that these latent action variables are strictly identifiable. Experimental results on several benchmarks show that the HAL model significantly outperforms existing methods for weakly-supervised action segmentation, confirming its practical effectiveness in real-world applications.

Junxian Huang, Ruichu Cai, Hao Zhu +5

Guangdong University of Technology · Carnegie Mellon University · Mohamed bin Zayed University of Artificial Intelligence +1

Temporal Action SegmentationLatent Action LearningHierarchical Representation LearningLatent Variable Models
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Feb 25, 2026

Goodness-of-Fit Tests for Latent Class Models with Ordinal Categorical Data

Ordinal categorical data are widely collected in psychology, education, and other social sciences, appearing commonly in questionnaires, assessments, and surveys. Latent class models provide a flexible framework for uncovering unobserved heterogeneity by grouping individuals into homogeneous classes based on their response patterns. A fundamental challenge in applying these models is determining the number of latent classes, which is unknown and must be inferred from data. In this paper, we propose one test statistic for this problem. The test statistic centers the largest singular value of a normalized residual matrix by a simple sample-size adjustment. Under the null hypothesis that the candidate number of latent classes is correct, its upper bound converges to zero in probability. Under an under-fitted alternative, the statistic itself exceeds a fixed positive constant with probability approaching one. This sharp dichotomous behavior of the test statistic yields two sequential testing algorithms that consistently estimate the true number of latent classes. Extensive experimental studies confirm the theoretical findings and demonstrate their accuracy and reliability in determining the number of latent classes.

Huan Qing

School of Economics and Finance, Chongqing University of Technology, Chongqing, 400054, China

Latent Variable Models
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Feb 25, 2026

Fair Model-based Clustering

The goal of fair clustering is to find clusters such that the proportion of sensitive attributes (e.g., gender, race, etc.) in each cluster is similar to that of the entire dataset. Various fair clustering algorithms have been proposed that modify standard K-means clustering to satisfy a given fairness constraint. A critical limitation of several existing fair clustering algorithms is that the number of parameters to be learned is proportional to the sample size because the cluster assignment of each datum should be optimized simultaneously with the cluster center, and thus scaling up the algorithms is difficult. In this paper, we propose a new fair clustering algorithm based on a finite mixture model, called Fair Model-based Clustering (FMC). A main advantage of FMC is that the number of learnable parameters is independent of the sample size and thus can be scaled up easily. In particular, mini-batch learning is possible to obtain clusters that are approximately fair. Moreover, FMC can be applied to non-metric data (e.g., categorical data) as long as the likelihood is well-defined. Theoretical and empirical justifications for the superiority of the proposed algorithm are provided.

Jinwon Park, Kunwoong Kim, Jihu Lee +1

Graduate School of Data Science, Seoul National University · Department of Statistics, Seoul National University

ClusteringGroup FairnessLatent Variable Models
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Feb 19, 2026

Characterizing the Predictive Impact of Modalities with Supervised Latent-Variable Modeling

Despite the recent success of Multimodal Large Language Models (MLLMs), existing approaches predominantly assume the availability of multiple modalities during training and inference. In practice, multimodal data is often incomplete because modalities may be missing, collected asynchronously, or available only for a subset of examples. In this work, we propose PRIMO, a supervised latent-variable imputation model that quantifies the predictive impact of any missing modality within the multimodal learning setting. PRIMO enables the use of all available training examples, whether modalities are complete or partial. Specifically, it models the missing modality through a latent variable that captures its relationship with the observed modality in the context of prediction. During inference, we draw many samples from the learned distribution over the missing modality to both obtain the marginal predictive distribution (for the purpose of prediction) and analyze the impact of the missing modalities on the prediction for each instance. We evaluate PRIMO on a synthetic XOR dataset, Audio-Vision MNIST, and MIMIC-III for mortality and ICD-9 prediction. Across all datasets, PRIMO obtains performance comparable to unimodal baselines when a modality is fully missing and to multimodal baselines when all modalities are available. PRIMO quantifies the predictive impact of a modality at the instance level using a variance-based metric computed from predictions across latent completions. We visually demonstrate how varying completions of the missing modality result in a set of plausible labels.

Divyam Madaan, Sumit Chopra, Kyunghyun Cho

New York University. · New York University Grossman School of Medicine · Genentech +1

Missing-Modality LearningMultimodal LearningMultimodal ClassificationLatent Variable ModelsMultimodal Model Interpretability
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Feb 18, 2026

Knowledge-Embedded Latent Projection for Robust Representation Learning

Latent space models are widely used for analyzing high-dimensional discrete data matrices, such as patient-feature matrices in electronic health records (EHRs), by capturing complex dependence structures through low-dimensional embeddings. However, estimation becomes challenging in the imbalanced regime, where one matrix dimension is much larger than the other. In EHR applications, cohort sizes are often limited by disease prevalence or data availability, whereas the feature space remains extremely large due to the breadth of medical coding system. Motivated by the increasing availability of external semantic embeddings, such as pre-trained embeddings of clinical concepts in EHRs, we propose a knowledge-embedded latent projection model that leverages semantic side information to regularize representation learning. Specifically, we model column embeddings as smooth functions of semantic embeddings via a mapping in a reproducing kernel Hilbert space. We develop a computationally efficient two-step estimation procedure that combines semantically guided subspace construction via kernel principal component analysis with scalable projected gradient descent. We establish estimation error bounds that characterize the trade-off between statistical error and approximation error induced by the kernel projection. Furthermore, we provide local convergence guarantees for our non-convex optimization procedure. Extensive simulation studies and a real-world EHR application demonstrate the effectiveness of the proposed method.

Weijing Tang, Ming Yuan, Zongqi Xia +1

Department of Statistics and Data Science, Carnegie Mellon University · Department of Statistics, Columbia University · Department of Neurology, University of Pittsburgh +1

Representation LearningLatent Variable ModelsElectronic Health Records
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Feb 17, 2026

Quantifying construct validity in large language model evaluations

The LLM community often reports benchmark results as if they are synonymous with general model capabilities. However, benchmarks can have problems that distort performance, like test set contamination and annotator error. How can we know that a benchmark is a reliable indicator of some capability that we want to measure? This question concerns the construct validity of LLM benchmarks, and it requires separating benchmark results from capabilities when we model and predict LLM performance. Both social scientists and computer scientists propose formal models - latent factor models and scaling laws - for identifying the capabilities underlying benchmark scores. However, neither technique is satisfactory for construct validity. Latent factor models ignore scaling laws, and as a result, the capabilities they extract often proxy model size. Scaling laws ignore measurement error, and as a result, the capabilities they extract are both uninterpretable and overfit to the observed benchmarks. This thesis presents the structured capabilities model, the first model to extract interpretable and generalisable capabilities from a large collection of LLM benchmark results. I fit this model and its two alternatives on a large sample of results from the OpenLLM Leaderboard. Structured capabilities outperform latent factor models on parsimonious fit indices, and exhibit better out-of-distribution benchmark prediction than scaling laws. These improvements are possible because neither existing approach separates model scale from capabilities in the appropriate way. Model scale should inform capabilities, as in scaling laws, and these capabilities should inform observed results up to measurement error, as in latent factor models. In combining these two insights, structured capabilities demonstrate better explanatory and predictive power for quantifying construct validity in LLM evaluations.

Ryan Othniel Kearns

St Catherine’s College

LLM EvaluationLatent Variable Models
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Feb 16, 2026

Traceable Latent Variable Discovery Based on Multi-Agent Collaboration

Revealing the underlying causal mechanisms in the real world is crucial for scientific and technological progress. Despite notable advances in recent decades, the lack of high-quality data and the reliance of traditional causal discovery algorithms (TCDA) on the assumption of no latent confounders, as well as their tendency to overlook the precise semantics of latent variables, have long been major obstacles to the broader application of causal discovery. To address this issue, we propose a novel causal modeling framework, TLVD, which integrates the metadata-based reasoning capabilities of large language models (LLMs) with the data-driven modeling capabilities of TCDA for inferring latent variables and their semantics. Specifically, we first employ a data-driven approach to construct a causal graph that incorporates latent variables. Then, we employ multi-LLM collaboration for latent variable inference, modeling this process as a game with incomplete information and seeking its Bayesian Nash Equilibrium (BNE) to infer the possible specific latent variables. Finally, to validate the inferred latent variables across multiple real-world web-based data sources, we leverage LLMs for evidence exploration to ensure traceability. We comprehensively evaluate TLVD on three de-identified real patient datasets provided by a hospital and two benchmark datasets. Extensive experimental results confirm the effectiveness and reliability of TLVD, with average improvements of 32.67% in Acc, 62.21% in CAcc, and 26.72% in ECit across the five datasets.

Huaming Du, Tao Hu, Yijie Huang +5

Southwestern University of Finance and Economics, Chengdu, Sichuan, China · Hunan University of Technology and Business, Xiangjiang Laboratory, Hunan, China · Emory University, Atlanta, Georgia, United States

Causal Discovery with Latent ConfoundingCausal Structure LearningLatent Variable ModelsLarge Language Model-Guided Scientific Discovery
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Feb 4, 2026

Hierarchical Latent Structures in Data Generation Process Unify Mechanistic Phenomena across Scale

Contemporary studies in mechanistic interpretability have uncovered many puzzling phenomena in the neural information processing of Transformer-based language models, such as induction heads, function vectors, and the Hydra effect. Some of these individual phenomena have been independently tied to different data distributional properties, while some have been loosely associated with model architecture and how Transformers process information. However, a unified understanding of the relationship between data, model architecture, and optimization remains lacking, failing to answer the fundamental question: why do these three phenomena appear universally across different model families and scales, despite their seeming disconnect? In this work, we answer this question by unifying these three phenomena as consequences of hierarchical latent structures in the data generation process, coupled with decorrelated gradients across additive model components and directional concavity in the representation geometry. We validate our theoretical results in a toy model regime and in a large-scale synthetic data regime, comparing them with language models trained on natural language data.

Jonas Rohweder, Subhabrata Dutta, Iryna Gurevych

UKP Lab, TU Darmstadt

Transformer InterpretabilityLatent Variable ModelsInduction HeadsFunction Vectors
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Feb 4, 2026

LLM surprisal is necessary but not sufficient to capture English garden-path effects: Evidence from joint latent modeling of reading paradigms

Temporarily ambiguous garden-path sentences ("While the team trained the striker wondered... ") are known to cause processing difficulty, which can manifest itself in a variety of reading behaviors (in-situ slowdowns, rereading), as well as in miscomprehension or outright rejection of the sentence as ungrammatical. Which types of reading behavior are observed critically depends on the experimental method used to collect the data, which makes comparing results between reading paradigms difficult. To address this problem, we present a latent-process multinomial processing tree (MPT) model of human reading and comprehension/judgment behavior in garden-path sentences that we fit to combined data from four different reading paradigms (eye tracking, uni- and bidirectional self-paced reading, Maze). The model distinguishes between the probability of adopting an incorrect initial analysis, the cost of encountering an incompatible continuation, and the cost of syntactic reanalysis. By taking into account trials with inattentive reading, more realistic estimates of the cost parameters are obtained. Cross-validation reveals that the MPT model has a better predictive fit to human reading patterns and end-of-trial task data than a model based solely on LLM-derived surprisal values. We also test several models that assume an influence of surprisal within the MPT architecture, and find that adding surprisal as an additional predictor or reading time and/or garden-path cost further improves predictive fit.

Dario Paape, Tal Linzen, Shravan Vasishth

Department of Linguistics, University of Potsdam · Department Linguistik, Universität Potsdam, Karl-Liebknecht-Straße 24–25, D-14476, Germany. · Center for Data Science/Department of Linguistics, New York University

Latent Variable ModelsCognitive ModelingComputational Psycholinguistics
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Dec 25, 2025

Nonnegative matrix factorizations and related compositional models: Equivalence, identifiability, and an application on the grain-size analysis of sediments

Across fields such as machine learning, social science, and geology, considerable attention has been given to models that factorize a nonnegative matrix into the product of two or three matrices, subject to nonnegative or row-sum-to-1 constraints. Although these models are to a large extent similar or even equivalent, they are presented under different names, and their similarity is not well known. This paper highlights similarities among five models, latent budget analysis (LBA) and latent class analysis (LCA) from social science, end-member analysis (EMA) from geology, probabilistic latent semantic analysis (PLSA) and nonnegative matrix factorization (NMF) from machine learning. We focus on the identifiability of these models. We prove that the solution of LBA, EMA, LCA, PLSA is unique if and only if the solution of NMF is unique. Consequently, existing uniqueness theorems for NMF directly apply to LBA, EMA, LCA, PLSA, and vice versa. We also provide a brief review of algorithms for the estimation of these models. We illustrate NMF on a sedimentary grain-size distribution dataset from sedimentary geology, and end the paper with a discussion of closely related model: archetypal analysis.

Qianqian Qi, Peter G. M. van der Heijden, Maarten A. Prins

Hangzhou Dianzi University, China · Utrecht University, the Netherlands · University of Southampton, UK +1

Latent Variable ModelsParameter IdentifiabilityNon-Negative Matrix Factorization
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Dec 12, 2025

Emergence of Nonequilibrium Latent Cycles in Unsupervised Generative Modeling

We show that nonequilibrium dynamics can play a constructive role in unsupervised machine learning by inducing the spontaneous emergence of latent-state cycles. We introduce a model in which visible and hidden variables interact through two independently parametrized transition matrices, defining a Markov chain whose steady state is intrinsically out of equilibrium. Likelihood maximization drives this system toward nonequilibrium steady states with finite entropy production, reduced self-transition probabilities, and persistent probability currents in the latent space. These cycles are not imposed by the architecture but arise from training, and models that develop them reproduce the empirical distribution of data classes more faithfully, with a clear correlation between agreement with the data and entropy production. Compared with equilibrium approaches such as restricted Boltzmann machines, our model breaks the detailed balance between the forward and backward conditional transitions and relies on a log-likelihood gradient that depends explicitly on the last two steps of the Markov chain. Hence, this exploration of the interface between nonequilibrium statistical physics and modern machine learning suggests that introducing irreversibility into latent-variable models can improve the fidelity of the generated data distribution.

Marco Baiesi, Alberto Rosso

Department of Physics and Astronomy, University of Padova, Via Marzolo 8, I-35131 Padova, Italy · INFN, Sezione di Padova, Via Marzolo 8, I-35131 Padova, Italy · LPTMS, CNRS, Universit´e Paris-Saclay, 91405, Orsay, France

Latent Variable ModelsLatent Dynamics ModelingMarkov ModelsGenerative Modeling
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Dec 7, 2025

Latency-Response Theory Model: Evaluating Large Language Models via Response Accuracy and Chain-of-Thought Length

The proliferation of Large Language Models (LLMs) necessitates valid evaluation methods to provide guidance for both downstream applications and actionable future improvements. The Item Response Theory (IRT) model with Computerized Adaptive Testing has recently emerged as a promising framework for evaluating LLMs via their response accuracy. Beyond simple response accuracy, LLMs' chain of thought (CoT) lengths serve as a vital indicator of their reasoning ability. To leverage the CoT length information to assist LLM evaluation, we propose the Latency-Response Theory (LaRT) model, which jointly models both the response accuracy and CoT length by introducing a key correlation parameter between the latent ability and the latent speed. We derive an efficient stochastic approximation Expectation-Maximization algorithm for parameter estimation. We establish rigorous identifiability results for the latent ability and latent speed parameters to ensure the statistical validity of their estimation. Through both theoretical asymptotic analyses and simulation studies, we demonstrate LaRT's advantages over IRT in terms of superior estimation accuracy and shorter confidence intervals for latent trait estimation. To evaluate LaRT in real data, we collect responses from diverse LLMs on popular benchmark datasets. We find that LaRT yields different LLM rankings than IRT and outperforms IRT across multiple key evaluation metrics including predictive power, item efficiency, ranking validity, and LLM evaluation efficiency. Code and data are available at https://github.com/Toby-X/Latency-Response-Theory-Model

Zhiyu Xu, Jia Liu, Yixin Wang +1

Department of Statistics, Columbia University. · Department of Statistics, University of Michigan

LLM EvaluationLatent Variable ModelsCoT EvaluationItem Response Theory
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Nov 22, 2025

Improving Forecasts of Suicide Attempts for Patients with Little Data

Ecological Momentary Assessment (EMA) studies provide real-time data on suicidal thoughts and behaviors, but forecasting suicide attempts remains challenging: attempts are rare, and the pathways patients take to them are heterogeneous. Here, we investigate a cohort of patients from an EMA study with recorded suicide-related events. We show that a single model fit to all patients forecasts poorly, while idiographic (per-patient) models show improvement but overfit for those with little data. Based on this result, one may hypothesize that patients should be partitioned into subgroups---this way, similar patients' data can be pooled together to improve forecasts. However, we show that grouping patients at random already improves forecasts, with performance increasing monotonically with the number of groups. Moreover, we show that grouping patients by demographics yields worse forecasts than random groupings. From these results, we hypothesize that patient similarity is continuous, rather than discrete, and must be inferred from the data. This motivated us to use Latent Variable Multiple Output Gaussian Processes (LVMOGPs), adapted to our data. Preliminary results show that, even without careful kernel design, LVMOGPs already match the strongest baseline models on most metrics, and their latent spaces yield a similarity between patients that we can inspect directly. Because the cohort is conditioned on the outcome and the splits are not temporal, we read these results as evidence that idiographic structure exists and can be recovered, not as deployable forecasting performance---an area for future work.

Genesis Hang, Annie Chen, Hope Neveux +2

Wellesley College · Harvard University

Latent Variable ModelsGaussian ProcessesMulti-Output Gaussian ProcessesClinical PredictionSuicide Risk ModelingClinical Risk Prediction
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Oct 14, 2025

Learning Latent Energy-Based Models via Interacting Particle Langevin Dynamics

We develop interacting particle algorithms for learning latent variable models with energy-based priors. To do so, we leverage recent developments in particle-based methods for solving maximum marginal likelihood estimation (MMLE) problems. Specifically, we provide a continuous-time framework for learning latent energy-based models, by defining stochastic differential equations (SDEs) that provably solve the MMLE problem. We obtain a practical algorithm as a discretisation of these SDEs and provide theoretical guarantees for the convergence of the proposed algorithm. Finally, we empirically validate the effectiveness of our method on synthetic and image datasets and demonstrate that using a particle based approach offers significant improvement in computational efficiency.

Joanna Marks, Tim Y. J. Wang, O. Deniz Akyildiz

Department of Mathematics, Imperial College London

Parameter EstimationLatent Variable ModelsEnergy-Based ModelsMaximum Marginal Likelihood Estimation
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Jun 20, 2025

Variational Learning of Disentangled Representations

Disentangled representations separate factors that are shared across conditions from those that are condition-specific. Such separation is needed for generalization to new domains, treatments, patients, or species. A dominant line of work pursues this goal through variational formulations. While these approaches achieve partial disentanglement, they often exhibit three common limitations: they either do not remove all condition-specific information from the condition-specific representation, allow the condition-specific representation to become uninformative, or impose independence assumptions that do not reflect the underlying generative process. In this work, we introduce DisCoVR, a variational framework that addresses these limitations. Its objective is aligned with the probabilistic structure of the data-generating process, and includes an adversarial term that prevents condition-specific information from being encoded in the condition-specific representation.DisCoVR reconstructs the data from both shared and condition-specific representations, ensuring that each remains informative, and uses a structured prior that further reinforces the informativeness of both representations. We show that across synthetic, image, and single-cell RNA-sequencing datasets, DisCoVR achieves stronger disentanglement compared to previous approaches.

Yuli Slavutsky, Ozgur Beker, David Blei +1

Department of Statistics, Columbia University, New York City, USA · Irving Institute for Cancer Dynamics, Columbia University, New York City, USA · Department of Computer Science, Columbia University, New York City, USA +1

Variational AutoencodersDisentangled Representation LearningLatent Variable Models
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May 14, 2025

Scalable Krylov Subspace Methods for Generalized Mixed-Effects Models with Crossed Random Effects

Mixed-effects models are widely used to model data with complex grouping structures and high-cardinality categorical predictor variables. However, for high-dimensional crossed random effects, current standard computations relying on Cholesky decompositions can become prohibitively slow. In this work, we present Krylov subspace-based methods that address existing computational bottlenecks, and we analyze them both theoretically and empirically. In particular, we derive new results on the convergence and accuracy of the preconditioned stochastic Lanczos quadrature and conjugate gradient methods for mixed-effects models, and we develop scalable methods for calculating predictive variances. In experiments with simulated and real-world data, the proposed methods yield speedups of several orders of magnitude and are more computationally robust than Cholesky-based computations, while maintaining essentially the same accuracy.

Pascal Kündig, Fabio Sigrist

Lucerne University of Applied Sciences and Arts · University of Basel · Seminar for Statistics, ETH Zurich

Latent Variable ModelsKrylov Subspace MethodsMixed-Effects Models
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Feb 11, 2025

The observational partial order of causal structures with latent variables

For two causal structures with the same set of visible variables, one is said to observationally dominate the other if the set of distributions over the visible variables realizable by the first contains the set of distributions over the visible variables realizable by the second. Knowing such dominance relations is useful for adjudicating between these structures given observational data. Here, we consider the problem of determining the partial order of equivalence classes of causal structures with latent variables relative to observational dominance. We provide a complete characterization of the dominance order in the case of three visible variables, and a partial characterization in the case of four visible variables. Our techniques also help to identify which observational equivalence classes have a set of realizable distributions that is characterized by nontrivial inequality constraints, analogous to Bell inequalities and instrumental inequalities. We find evidence that as one increases the number of visible variables, the equivalence classes satisfying nontrivial inequality constraints become ubiquitous. (Because such classes are the ones for which there can be a difference in the distributions that are quantumly and classically realizable, this implies that the potential for quantum-classical gaps is also ubiquitous.) Furthermore, we find evidence that constraint-based causal discovery algorithms that rely solely on conditional independence constraints have a significantly weaker distinguishing power among observational equivalence classes than algorithms that go beyond these (i.e., algorithms that also leverage nested Markov constraints and inequality constraints).

Marina Maciel Ansanelli, Elie Wolfe, Robert W. Spekkens

Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario Canada N2L 2Y5 · Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1

Latent Variable ModelsCausal DiscoveryCausal Structure LearningCausal Discovery with Latent Confounding
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