Mixture-of-expert (MOE) models are popular methods in machine learning, since they can model heterogeneous behaviour across the space of the data using an ensemble collection of learners. These models are especially useful for modelling dynamic data as time-dependent data often exhibit non-stationarity and heavy-tailed errors, which may be inappropriate to model with a typical single expert model. We propose a mixture of Student-$t$ processes with an adaptive structure for the covariance and noise behaviour for each mixture. Moreover, we use a sequential Monte Carlo (SMC) sampler to perform online inference as data arrive in real time. We demonstrate the superiority of our proposed approach over other models on synthetic and real-world datasets to prove the necessity of the novel method.

The multi-view Gaussian process latent variable model (MV-GPLVM) aims to learn a unified representation from multi-view data but is hindered by challenges such as limited kernel expressiveness and low computational efficiency. To overcome these issues, we first introduce a new duality between the spectral density and the kernel function. By modeling the spectral density with a bivariate Gaussian mixture, we then derive a generic and expressive kernel termed Next-Gen Spectral Mixture (NG-SM) for MV-GPLVMs. To address the inherent computational inefficiency of the NG-SM kernel, we propose a random Fourier feature approximation. Combined with a tailored reparameterization trick, this approximation enables scalable variational inference for both the model and the unified latent representations. Numerical evaluations across a diverse range of multi-view datasets demonstrate that our proposed method consistently outperforms state-of-the-art models in learning meaningful latent representations.

Random feature latent variable models (RFLVMs) are state-of-the-art tools for uncovering structure in high-dimensional, non-Gaussian data. However, their reliance on Monte Carlo sampling significantly limits scalability, posing challenges for large-scale applications. To overcome these limitations, we develop a scalable RFLVM framework based on variational Bayesian inference (VBI), a deterministic and optimization-based alternative to sampling methods. Applying VBI to RFLVMs is nontrivial due to two key challenges: (i) the lack of an explicit probability density function (PDF) for Dirichlet process (DP) mixing weights, and (ii) the inefficiency of existing VBI approaches when handling the high-dimensional variational parameters of RFLVMs. To address these issues, we adopt the stick-breaking construction for the DP, which provides an explicit and tractable PDF over mixing weights, and propose a novel inference algorithm, block coordinate descent variational inference (BCD-VI), which partitions variational parameters into blocks and applies tailored solvers to optimize them efficiently. The resulting scalable model, referred to as SRFLVM, supports various likelihoods; we demonstrate its effectiveness under Gaussian and logistic settings. Extensive experiments on diverse benchmark datasets show that SRFLVM achieves superior scalability, computational efficiency, and performance in latent representation learning and missing data imputation, consistently outperforming state-of-the-art latent variable models, including deep generative approaches.

Gaussian process latent variable models (GPLVMs) are a versatile family of unsupervised learning models commonly used for dimensionality reduction. However, common challenges in modeling data with GPLVMs include inadequate kernel flexibility and improper selection of the projection noise, leading to a type of model collapse characterized by vague latent representations that do not reflect the underlying data structure. This paper addresses these issues by, first, theoretically examining the impact of projection variance on model collapse through the lens of a linear GPLVM. Second, we tackle model collapse due to inadequate kernel flexibility by integrating the spectral mixture (SM) kernel and a differentiable random Fourier feature (RFF) kernel approximation, which ensures computational scalability and efficiency through off-the-shelf automatic differentiation tools for learning the kernel hyperparameters, projection variance, and latent representations within the variational inference framework. The proposed GPLVM, named advisedRFLVM, is evaluated across diverse datasets and consistently outperforms various salient competing models, including state-of-the-art variational autoencoders (VAEs) and other GPLVM variants, in terms of informative latent representations and missing data imputation.

Variational autoencoders (VAE) are one of the most prominent deep generative models for learning the underlying statistical distribution of high-dimensional data. However, training VAEs suffers from a severe issue called posterior collapse; that is, the learned posterior distribution collapses to the assumed/pre-selected prior distribution. This issue limits the capacity of the learned posterior distribution to convey data information. Previous work has proposed a heuristic training scheme to mitigate this issue, in which the core idea is to train the encoder and the decoder in an alternating fashion. However, there is still no theoretical interpretation of this scheme, and this paper, for the first time, fills in this gap by inspecting the previous scheme under the lens of the expectation maximization (EM) framework. Under this framework, we propose a novel EM-type training algorithm that gives a controllable optimization process and it allows for further extensions, e.g., employing implicit distribution models. Experimental results have corroborated the superior performance of the proposed EM-type VAE training algorithm in terms of various metrics.

Gaussian process-based latent variable models are flexible and theoretically grounded tools for nonlinear dimension reduction, but generalizing to non-Gaussian data likelihoods within this nonlinear framework is statistically challenging. Here, we use random features to develop a family of nonlinear dimension reduction models that are easily extensible to non-Gaussian data likelihoods; we call these random feature latent variable models (RFLVMs). By approximating a nonlinear relationship between the latent space and the observations with a function that is linear with respect to random features, we induce closed-form gradients of the posterior distribution with respect to the latent variable. This allows the RFLVM framework to support computationally tractable nonlinear latent variable models for a variety of data likelihoods in the exponential family without specialized derivations. Our generalized RFLVMs produce results comparable with other state-of-the-art dimension reduction methods on diverse types of data, including neural spike train recordings, images, and text data.

A multi-output Gaussian process (GP) is a flexible Bayesian nonparametric framework that has proven useful in jointly modeling the physiological states of patients in medical time series data. However, capturing the short-term effects of drugs and therapeutic interventions on patient physiological state remains challenging. We propose a novel approach that models the effect of interventions as a hybrid Gaussian process composed of a GP capturing patient baseline physiology convolved with a latent force model capturing effects of treatments on specific physiological features. The combination of a multi-output GP with a time-marked kernel GP leads to a well-characterized model of patients’ physiological state across a hospital stay, including response to interventions. Our model leads to analytically tractable cross-covariance functions that allow for scalable inference. Our hierarchical model includes estimates of patient-specific effects but allows sharing of support across patients. Our approach achieves competitive predictive performance on challenging hospital data, where we recover patient-specific response to the administration of three common drugs: one antihypertensive drug and two anticoagulants.

Bayesian nonparametric (BNP) models provide elegant methods for discovering underlying latent features within a data set, but inference in such models can be slow. We exploit the fact that completely random measures, which commonly-used models like the Dirichlet process and the beta-Bernoulli process can be expressed using, are decomposable into independent sub-measures. We use this decomposition to partition the latent measure into a finite measure containing only instantiated components, and an infinite measure containing all other components. We then select different inference algorithms for the two components: uncollapsed samplers mix well on the finite measure, while collapsed samplers mix well on the infinite, sparsely occupied tail. The resulting hybrid algorithm can be applied to a wide class of models, and can be easily distributed to allow scalable inference without sacrificing asymptotic convergence guarantees.

In many applications, observed data are influenced by some combination of latent causes. For example, suppose sensors are placed inside a building to record responses such as temperature, humidity, power consumption and noise levels. These random, observed responses are typically affected by many unobserved, latent factors (or features) within the building such as the number of individuals, the turning on and off of electrical devices, power surges, etc. These latent factors are usually present for a contiguous period of time before disappearing; further, multiple factors could be present at a time. This paper develops new probabilistic methodology and inference methods for random object generation influenced by latent features exhibiting temporal persistence. Every datum is associated with subsets of a potentially infinite number of hidden, persistent features that account for temporal dynamics in an observation. The ensuing class of dynamic models constructed by adapting the Indian Buffet Process — a probability measure on the space of random, unbounded binary matrices — finds use in a variety of applications arising in operations, signal processing, biomedicine, marketing, image analysis, etc. Illustrations using synthetic and real data are provided.

Training Gaussian process-based models typically involves an $O(N^3)$ computational bottleneck due to inverting the covariance matrix. Popular methods for overcoming this matrix inversion problem cannot adequately model all types of latent functions, and are often not parallelizable. However, judicious choice of model structure can ameliorate this problem. A mixture-of-experts model that uses a mixture of $K$ Gaussian processes offers modeling flexibility and opportunities for scalable inference. Our embarrassingly parallel algorithm combines low-dimensional matrix inversions with importance sampling to yield a flexible, scalable mixture-of-experts model that offers comparable performance to Gaussian process regression at a much lower computational cost.

In this letter, we take a new approach for time of arrival geo-localization. We show that the main sources of error in metropolitan areas are due to environmental imperfections that bias our solutions, and that we can rely on a probabilistic model to learn and compensate for them. The resulting localization error is validated using measurements from a live LTE cellular network to be less than 10 meters, representing an order-of-magnitude improvement.

The last decade has witnessed an explosion in the development of models, theory and computational algorithms for ``big data'' analysis. In particular, distributed inference has served as a natural and dominating paradigm for statistical inference. However, the existing literature on parallel inference almost exclusively focuses on Euclidean data and parameters. While this assumption is valid for many applications, it is increasingly more common to encounter problems where the data or the parameters lie on a non-Euclidean space, like a manifold for example. Our work aims to fill a critical gap in the literature by generalizing parallel inference algorithms to optimization on manifolds. We show that our proposed algorithm is both communication efficient and carries theoretical convergence guarantees. In addition, we demonstrate the performance of our algorithm to the estimation of Fr\'echet means on simulated spherical data and the low-rank matrix completion problem over Grassmann manifolds applied to the Netflix prize data set.