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1. Learning joint latent space EBM prior model for multi-layer generator

   https://doi.org/10.1109/CVPR52729.2023.00351  

   Cui, J; Wu, Y N; Han, T (June 2023, Conference on Computer Vision and Pattern Recognition (CVPR 2023))

   This paper studies the fundamental problem of learning multi-layer generator models. The multi-layer generator model builds multiple layers of latent variables as a prior model on top of the generator, which benefits learning complex data distribution and hierarchical representations. However, such a prior model usually focuses on modeling inter-layer relations between latent variables by assuming non-informative (conditional) Gaussian distributions, which can be limited in model expressivity. To tackle this issue and learn more expressive prior models, we propose an energy-based model (EBM) on the joint latent space over all layers of latent variables with the multi-layer generator as its backbone. Such joint latent space EBM prior model captures the intra-layer contextual relations at each layer through layer-wise energy terms, and latent variables across different layers are jointly corrected. We develop a joint training scheme via maximum likelihood estimation (MLE), which involves Markov Chain Monte Carlo (MCMC) sampling for both prior and posterior distributions of the latent variables from different layers. To ensure efficient inference and learning, we further propose a variational training scheme where an inference model is used to amortize the costly posterior MCMC sampling. Our experiments demonstrate that the learned model can be expressive in generating high-quality images and capturing hierarchical features for better outlier detection. 

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2. A Gaussian latent variable model for incomplete mixed type data

   Ajirak, Marzieh; Djurić, Petar M (January 2023, Conference record IEEE International Conference on Acoustics Speech and Signal Processing)

   In many machine learning problems, one has to work with data of different types, including continuous, discrete, and categorical data. Further, it is often the case that many of these data are missing from the database. This paper proposes a Gaussian process framework that efficiently captures the information from mixed numerical and categorical data that effectively incorporates missing variables. First, we propose a generative model for the mixed-type data. The generative model exploits Gaussian processes with kernels constructed from the latent vectors. We also propose a method for inference of the unknowns, and in its implementation, we rely on a sparse spectrum approximation of the Gaussian processes and variational inference. We demonstrate the performance of the method for both supervised and unsupervised tasks. First, we investigate the imputation of missing variables in an unsupervised setting, and then we show the results of joint imputation and classification on IBM employee data. 

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3. Causal Inference with Latent Variables: Recent Advances and Future Prospectives

   https://doi.org/10.1145/3637528.3671450  

   Zhu, Yaochen; He, Yinhan; Ma, Jing; Hu, Mengxuan; Li, Sheng; Li, Jundong (August 2024, Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining)

   Causality lays the foundation for the trajectory of our world. Causal inference (CI), which aims to infer intrinsic causal relations among variables of interest, has emerged as a crucial research topic. Nevertheless, the lack of observation of important variables (e.g., confounders, mediators, exogenous variables, etc.) severely compromises the reliability of CI methods. The issue may arise from the inherent difficulty in measuring the variables. Additionally, in observational studies where variables are passively recorded, certain covariates might be inadvertently omitted by the experimenter. Depending on the type of unobserved variables and the specific CI task, various consequences can be incurred if these latent variables are carelessly handled, such as biased estimation of causal effects, incomplete understanding of causal mechanisms, lack of individual-level causal consideration, etc. In this survey, we provide a comprehensive review of recent developments in CI with latent variables. We start by discussing traditional CI techniques when variables of interest are assumed to be fully observed. Afterward, under the taxonomy of circumvention and inference-based methods, we provide an in-depth discussion of various CI strategies to handle latent variables, covering the tasks of causal effect estimation, mediation analysis, counterfactual reasoning, and causal discovery. Furthermore, we generalize the discussion to graph data where interference among units may exist. Finally, we offer fresh aspects for further advancement of CI with latent variables, especially new opportunities in the era of large language models (LLMs). 

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4. Algebraic properties of solutions to common information of Gaussian vectors under sparse graphical constraints

   https://doi.org/10.1109/ALLERTON.2017.8262852  

   Moharrer, Ali; Wei, Shuangqing (October 2017, 2017 55th Annual Allerton Conference on Communication, Control, and Computing (Allerton))

   We formulate Wyner's common information for random vectors x ϵ R n with joint Gaussian density. We show that finding the common information of Gaussian vectors is equivalent to maximizing a log-determinant of the additive Gaussian noise covariance matrix. We coin such optimization problem as a constrained minimum determinant factor analysis (CMDFA) problem. The convexity of such problem with necessary and sufficient conditions on CMDFA solution is shown. We study the algebraic properties of CMDFA solution space, through which we study two sparse Gaussian graphical models, namely, latent Gaussian stars, and explicit Gaussian chains. Interestingly, we show that depending on pairwise covariance values in a Gaussian graphical structure, one may not always end up with the same parameters and structure for CMDFA solution as those found via graphical learning algorithms. In addition, our results suggest that Gaussian chains have little room left for dimension reduction in terms of the number of latent variables required in their CMDFA solutions. 

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5. Thresholded graphical lasso adjusts for latent variables

   https://doi.org/10.1093/biomet/asac060  

   Wang, Minjie; Allen, Genevera I (November 2022, Biometrika)

   Summary Structural learning of Gaussian graphical models in the presence of latent variables has long been a challenging problem. Chandrasekaran et al. (2012) proposed a convex program for estimating a sparse graph plus a low-rank term that adjusts for latent variables; however, this approach poses challenges from both computational and statistical perspectives. We propose an alternative, simple solution: apply a hard-thresholding operator to existing graph selection methods. Conceptually simple and computationally attractive, the approach of thresholding the graphical lasso is shown to be graph selection consistent in the presence of latent variables under a simpler minimum edge strength condition and at an improved statistical rate. The results are extended to estimators for thresholded neighbourhood selection and constrained $$\ell_{1}$$-minimization for inverse matrix estimation as well. We show that our simple thresholded graph estimators yield stronger empirical results than existing methods for the latent variable graphical model problem, and we apply them to a neuroscience case study on estimating functional neural connections. 

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