In this work, we develop a scalable approach for a flexible latent factor model for high-dimensional dynamical systems. Each latent factor process has its own correlation and variance parameters, and the orthogonal factor loading matrix can be either fixed or estimated. We utilize an orthogonal factor loading matrix that avoids computing the inversion of the posterior covariance matrix at each time of the Kalman filter, and derive closed-form expressions in an expectation-maximization algorithm for parameter estimation, which substantially reduces the computational complexity without approximation. Our study is motivated by inversely estimating slow slip events from geodetic data, such as continuous GPS measurements. Extensive simulated studies illustrate higher accuracy and scalability of our approach compared to alternatives. By applying our method to geodetic measurements in the Cascadia region, our estimated slip better agrees with independently measured seismic data of tremor events. The substantial acceleration from our method enables the use of massive noisy data for geological hazard quantification and other applications.

The demand of modeling and forecasting high-dimensional matrix time series brings the opportunities with challenges. Let Y t " p y i,j,tq be a p ˆ q matrix recorded at time t, where y i,j,trepresents the value of, for example, the j -th variable on the i -th individual at time t . A popular approach to model Y t in the existing literature is via the so-called Tucker decomposition, namely the matrix Tucker-factor model. See, for example, Wang et al. (2019), Chen et al. (2020) and Chen and Chen (2022). It represents a high-dimensional matrix time series as a linear combination of a lower-dimensional matrix process. The Tucker decomposition can be viewed as a natural extension of the factor model for vector time series considered in Lam and Yao (2012) and Chang et al. (2015). Similarly we can only identify the factor loading spaces (the linear spaces spanned by the columns of the factor loading matrices) in the matrix Tucker-factor model while the factor loading matrices themselves are not uniquely defined. Parallel to the Tucker decomposition for matrix time series Y t, Chang et al. (2023) and Han et al. (2024) consider to model Y t via the so-called canonical polyadic (CP) decomposition, namely the matrix CP-factor model. It provides a more comprehensive dimensionality reduction as the dynamic structure of a matrix time series is driven by a vector process rather than a matrix process. Furthermore the factor loading matrices in the matrix CP-factor model can be identified uniquely upto the column reflection and permutation indeterminacy under some regularity conditions. The CP-factor model for matrix time series specified in Chang et al. (2023) admits the form Y t " AX tB J ` ε t, t ě 1, (1) where X t " diag p x tq with x t " p x t, 1, .. .

Adverse posttraumatic neuropsychiatric sequelae (APNS) are common among veterans and millions of Americans after traumatic exposures, resulting in substantial burdens for trauma survivors and society. Despite numerous studies conducted on APNS over the past decades, there has been limited progress in understanding the underlying neurobiological mechanisms due to several unique challenges. One of these challenges is the reliance on subjective self-report measures to assess APNS, which can easily result in measurement errors and biases (e.g., recall bias). To mitigate this issue, in this paper, we investigate the potential of leveraging the objective longitudinal mobile device data to identify homogeneous APNS states and study the dynamic transitions and potential risk factors of APNS after trauma exposure. To handle specific challenges posed by longitudinal mobile device data, we developed exploratory hidden Markov factor models and designed a Stabilized Expectation-Maximization algorithm for parameter estimation. Simulation studies were conducted to evaluate the performance of parameter estimation and model selection. Finally, to demonstrate the practical utility of the method, we applied it to mobile device data collected from the Advancing Understanding of RecOvery afteR traumA (AURORA) study.

Mining and learning the shape variability of underlying population has benefited the applications including parametric shape modeling, 3D animation, and image segmentation. The current statistical shape modeling method works well on learning unstructured shape variations without obvious pose changes (relative rotations of the body parts). Studying the pose variations within a shape population involves segmenting the shapes into different articulated parts and learning the transformations of the segmented parts. This paper formulates the pose learning problem as mixtures of factor analyzers. The segmentation is obtained by components posterior probabilities and the rotations in pose variations are learned by the factor loading matrices. To guarantee that the factor loading matrices are composed by rotation matrices, constraints are imposed and the corresponding closed form optimal solution is derived. Based on the proposed method, the pose variations are automatically learned from the given shape populations. The method is applied in motion animation where new poses are generated by interpolating the existing poses in the training set. The obtained results are smooth and realistic.

In unsupervised learning, dimensionality reduction is an important tool for data exploration and visualization. Because these aims are typically open-ended, it can be useful to frame the problem as looking for patterns that are enriched in one dataset relative to another. These pairs of datasets occur commonly, for instance a population of interest vs. control or signal vs. signal free recordings.However, there are few methods that work on sets of data as opposed to data points or sequences. Here, we present a probabilistic model for dimensionality reduction to discover signal that is enriched in the target dataset relative to the background dataset. The data in these sets do not need to be paired or grouped beyond set membership. By using a probabilistic model where some structure is shared amongst the two datasets and some is unique to the target dataset, we are able to recover interesting structure in the latent space of the target dataset. The method also has the advantages of a probabilistic model, namely that it allows for the incorporation of prior information, handles missing data, and can be generalized to different distributional assumptions. We describe several possible variations of the model and demonstrate the application of the technique to de-noising, feature selection, and subgroup discovery settings.

Group factor analysis (GFA) methods have been widely used to infer the common structure and the group-specific signals from multiple related datasets in various fields including systems biology and neuroimaging. To date, most available GFA models require Gibbs sampling or slice sampling to perform inference, which prevents the practical application of GFA to large-scale data. In this paper we present an efficient collapsed variational inference (CVI) algorithm for the nonparametric Bayesian group factor analysis (NGFA) model built upon an hierarchical beta Bernoulli process. Our CVI algorithm proceeds by marginalizing out the group-specific beta process parameters, and then approximating the true posterior in the collapsed space using mean field methods. Experimental results on both synthetic and real-world data demonstrate the effectiveness of our CVI algorithm for the NGFA compared with state-of-the-art GFA methods.

We introduce three novel semi-parametric extensions of probabilistic canonical correlation analysis with identifiability guarantees. We consider moment matching techniques for estimation in these models. For that, by drawing explicit links between the new models and a discrete version of independent component analysis (DICA), we first extend the DICA cumulant tensors to the new discrete version of CCA. By further using a close connection with independent component analysis, we introduce generalized covariance matrices, which can replace the cumulant tensors in the moment matching framework, and, therefore, improve sample complexity and simplify derivations and algorithms significantly. As the tensor power method or orthogonal joint diagonalization are not applicable in the new setting, we use non-orthogonal joint diagonalization techniques for matching the cumu-lants. We demonstrate performance of the proposed models and estimation techniques on experiments with both synthetic and real datasets.

We consider the problem of discriminative factor analysis for data that are in general non-Gaussian. A Bayesian model based on the ranks of the data is proposed. We first introduce a new {\em max-margin} version of the rank-likelihood. A discriminative factor model is then developed, integrating the max-margin rank-likelihood and (linear) Bayesian support vector machines, which are also built on the max-margin principle. The discriminative factor model is further extended to the {\em nonlinear} case through mixtures of local linear classifiers, via Dirichlet processes. Fully local conjugacy of the model yields efficient inference with both Markov Chain Monte Carlo and variational Bayes approaches. Extensive experiments on benchmark and real data demonstrate superior performance of the proposed model and its potential for applications in computational biology.

We propose a new variational EM algorithm for fitting factor analysis models with mixed continuous and categorical observations. The algorithm is based on a simple quadratic bound to the log-sum-exp function. In the special case of fully observed binary data, the bound we propose is significantly faster than previous variational methods. We show that EM is significantly more robust in the presence of missing data compared to treating the latent factors as parameters, which is the approach used by exponential family PCA and other related matrix-factorization methods. A further benefit of the variational approach is that it can easily be extended to the case of mixtures of factor analyzers, as we show. We present results on synthetic and real data sets demonstrating several desirable properties of our proposed method.

We propose a nonparametric Bayesian factor regression model that accounts for uncertainty in the number of factors, and the relationship between factors. To accomplish this, we propose a sparse variant of the Indian Buffet Process and couple this with a hierarchical model over factors, based on Kingman's coalescent. We apply this model to two problems (factor analysis and factor regression) in gene-expression data analysis.