In this work, we adopt Wyner common information framework for unsupervised multi-view representation learning. Within this framework, we propose two novel formulations that enable the development of computational efficient solvers based on the alternating minimization principle. The first formulation, referred to as the {\em variational form}, enjoys a linearly growing complexity with the number of views and is based on a variational-inference tight surrogate bound coupled with a Lagrangian optimization objective function. The second formulation, i.e., the {\em representational form}, is shown to include known results as special cases. Here, we develop a tailored version from the alternating direction method of multipliers (ADMM) algorithm for solving the resulting non-convex optimization problem. In the two cases, the convergence of the proposed solvers is established in certain relevant regimes. Furthermore, our empirical results demonstrate the effectiveness of the proposed methods as compared with the state-of-the-art solvers. In a nutshell, the proposed solvers offer computational efficiency, theoretical convergence guarantees (local minima), scalable complexity with the number of views, and exceptional accuracy as compared with the state-of-the-art techniques. Our focus here is devoted to the discrete case and our results for continuous distributions are reported elsewhere.
This paper introduces a new latent variable generative model able to handle high dimensional longitudinal data and relying on variational inference. The time dependency between the observations of an input sequence is modelled using normalizing flows over the associated latent variables. The proposed method can be used to generate either fully synthetic longitudinal sequences or trajectories that are conditioned on several data in a sequence and demonstrates good robustness properties to missing data. We test the model on 6 datasets of different complexity and show that it can achieve better likelihood estimates than some competitors as well as more reliable missing data imputation. A code is made available at \url{https://github.com/clementchadebec/variational_inference_for_longitudinal_data}.
Multitemporal hyperspectral unmixing (MTHU) is a fundamental tool in the analysis of hyperspectral image sequences. It reveals the dynamical evolution of the materials (endmembers) and of their proportions (abundances) in a given scene. However, adequately accounting for the spatial and temporal variability of the endmembers in MTHU is challenging, and has not been fully addressed so far in unsupervised frameworks. In this work, we propose an unsupervised MTHU algorithm based on variational recurrent neural networks. First, a stochastic model is proposed to represent both the dynamical evolution of the endmembers and their abundances, as well as the mixing process. Moreover, a new model based on a low-dimensional parametrization is used to represent spatial and temporal endmember variability, significantly reducing the amount of variables to be estimated. We propose to formulate MTHU as a Bayesian inference problem. However, the solution to this problem does not have an analytical solution due to the nonlinearity and non-Gaussianity of the model. Thus, we propose a solution based on deep variational inference, in which the posterior distribution of the estimated abundances and endmembers is represented by using a combination of recurrent neural networks and a physically motivated model. The parameters of the model are learned using stochastic backpropagation. Experimental results show that the proposed method outperforms state of the art MTHU algorithms.
In applied Bayesian inference scenarios, users may have access to a large number of pre-existing model evaluations, for example from maximum-a-posteriori (MAP) optimization runs. However, traditional approximate inference techniques make little to no use of this available information. We propose the framework of post-process Bayesian inference as a means to obtain a quick posterior approximation from existing target density evaluations, with no further model calls. Within this framework, we introduce Variational Sparse Bayesian Quadrature (VSBQ), a method for post-process approximate inference for models with black-box and potentially noisy likelihoods. VSBQ reuses existing target density evaluations to build a sparse Gaussian process (GP) surrogate model of the log posterior density function. Subsequently, we leverage sparse-GP Bayesian quadrature combined with variational inference to achieve fast approximate posterior inference over the surrogate. We validate our method on challenging synthetic scenarios and real-world applications from computational neuroscience. The experiments show that VSBQ builds high-quality posterior approximations by post-processing existing optimization traces, with no further model evaluations.
Neyman-Scott processes (NSPs) have been applied across a range of fields to model points or temporal events with a hierarchy of clusters. Markov chain Monte Carlo (MCMC) is typically used for posterior sampling in the model. However, MCMC's mixing time can cause the resulting inference to be slow, and thereby slow down model learning and prediction. We develop the first variational inference (VI) algorithm for NSPs, and give two examples of suitable variational posterior point process distributions. Our method minimizes the inclusive Kullback-Leibler (KL) divergence for VI to obtain the variational parameters. We generate samples from the approximate posterior point processes much faster than MCMC, as we can directly estimate the approximate posterior point processes without any MCMC steps or gradient descent. We include synthetic and real-world data experiments that demonstrate our VI algorithm achieves better prediction performance than MCMC when computational time is limited.
In various practical situations, matrix factorization methods suffer from poor data quality, such as high data sparsity and low signal-to-noise ratio (SNR). Here, we consider a matrix factorization problem by utilizing auxiliary information, which is massively available in real-world applications, to overcome the challenges caused by poor data quality. Unlike existing methods that mainly rely on simple linear models to combine auxiliary information with the main data matrix, we propose to integrate gradient boosted trees in the probabilistic matrix factorization framework to effectively leverage auxiliary information (MFAI). Thus, MFAI naturally inherits several salient features of gradient boosted trees, such as the capability of flexibly modeling nonlinear relationships and robustness to irrelevant features and missing values in auxiliary information. The parameters in MFAI can be automatically determined under the empirical Bayes framework, making it adaptive to the utilization of auxiliary information and immune to overfitting. Moreover, MFAI is computationally efficient and scalable to large datasets by exploiting variational inference. We demonstrate the advantages of MFAI through comprehensive numerical results from simulation studies and real data analyses. Our approach is implemented in the R package mfair available at https://github.com/YangLabHKUST/mfair.
Click-based interactive segmentation (IS) aims to extract the target objects under user interaction. For this task, most of the current deep learning (DL)-based methods mainly follow the general pipelines of semantic segmentation. Albeit achieving promising performance, they do not fully and explicitly utilize and propagate the click information, inevitably leading to unsatisfactory segmentation results, even at clicked points. Against this issue, in this paper, we propose to formulate the IS task as a Gaussian process (GP)-based pixel-wise binary classification model on each image. To solve this model, we utilize amortized variational inference to approximate the intractable GP posterior in a data-driven manner and then decouple the approximated GP posterior into double space forms for efficient sampling with linear complexity. Then, we correspondingly construct a GP classification framework, named GPCIS, which is integrated with the deep kernel learning mechanism for more flexibility. The main specificities of the proposed GPCIS lie in: 1) Under the explicit guidance of the derived GP posterior, the information contained in clicks can be finely propagated to the entire image and then boost the segmentation; 2) The accuracy of predictions at clicks has good theoretical support. These merits of GPCIS as well as its good generality and high efficiency are substantiated by comprehensive experiments on several benchmarks, as compared with representative methods both quantitatively and qualitatively.
Stochastic models with global parameters and latent variables are common, and for which variational inference (VI) is popular. However, existing methods are often either slow or inaccurate in high dimensions. We suggest a fast and accurate VI method for this case that employs a well-defined natural gradient variational optimization that targets the joint posterior of the global parameters and latent variables. It is a hybrid method, where at each step the global parameters are updated using the natural gradient and the latent variables are generated from their conditional posterior. A fast to compute expression for the Tikhonov damped Fisher information matrix is used, along with the re-parameterization trick, to provide a stable natural gradient. We apply the approach to deep mixed models, which are an emerging class of Bayesian neural networks with random output layer coefficients to allow for heterogeneity. A range of simulations show that using the natural gradient is substantially more efficient than using the ordinary gradient, and that the approach is faster and more accurate than two cutting-edge natural gradient VI methods. In a financial application we show that accounting for industry level heterogeneity using the deep mixed model improves the accuracy of asset pricing models. MATLAB code to implement the method can be found at: https://github.com/WeibenZhang07/NG-HVI.
Variational Inference (VI) is an attractive alternative to Markov Chain Monte Carlo (MCMC) due to its computational efficiency in the case of large datasets and/or complex models with high-dimensional parameters. However, evaluating the accuracy of variational approximations remains a challenge. Existing methods characterize the quality of the whole variational distribution, which is almost always poor in realistic applications, even if specific posterior functionals such as the component-wise means or variances are accurate. Hence, these diagnostics are of practical value only in limited circumstances. To address this issue, we propose the TArgeted Diagnostic for Distribution Approximation Accuracy (TADDAA), which uses many short parallel MCMC chains to obtain lower bounds on the error of each posterior functional of interest. We also develop a reliability check for TADDAA to determine when the lower bounds should not be trusted. Numerical experiments validate the practical utility and computational efficiency of our approach on a range of synthetic distributions and real-data examples, including sparse logistic regression and Bayesian neural network models.
Previous pitch-controllable text-to-speech (TTS) models rely on directly modeling fundamental frequency, leading to low variance in synthesized speech. To address this issue, we propose PITS, an end-to-end pitch-controllable TTS model that utilizes variational inference to model pitch. Based on VITS, PITS incorporates the Yingram encoder, the Yingram decoder, and adversarial training of pitch-shifted synthesis to achieve pitch-controllability. Experiments demonstrate that PITS generates high-quality speech that is indistinguishable from ground truth speech and has high pitch-controllability without quality degradation. Code, audio samples, and demo are available at https://github.com/anonymous-pits/pits.
Normalizing flows model probability distributions through an expressive tractable density. They transform a simple base distribution, such as a Gaussian, through a sequence of invertible functions, which are referred to as layers. These layers typically use neural networks to become very expressive. Flows are ubiquitous in machine learning and have been applied to image generation, text modeling, variational inference, approximating Boltzmann distributions, and many other problems. Here, we present normflows, a Python package for normalizing flows. It allows to build normalizing flow models from a suite of base distributions, flow layers, and neural networks. The package is implemented in the popular deep learning framework PyTorch, which simplifies the integration of flows in larger machine learning models or pipelines. It supports most of the common normalizing flow architectures, such as Real NVP, Glow, Masked Autoregressive Flows, Neural Spline Flows, Residual Flows, and many more. The package can be easily installed via pip and the code is publicly available on GitHub.
In this paper, we propose energy-based sample adaptation at test time for domain generalization. Where previous works adapt their models to target domains, we adapt the unseen target samples to source-trained models. To this end, we design a discriminative energy-based model, which is trained on source domains to jointly model the conditional distribution for classification and data distribution for sample adaptation. The model is optimized to simultaneously learn a classifier and an energy function. To adapt target samples to source distributions, we iteratively update the samples by energy minimization with stochastic gradient Langevin dynamics. Moreover, to preserve the categorical information in the sample during adaptation, we introduce a categorical latent variable into the energy-based model. The latent variable is learned from the original sample before adaptation by variational inference and fixed as a condition to guide the sample update. Experiments on six benchmarks for classification of images and microblog threads demonstrate the effectiveness of our proposal.
Sampling a probability distribution with an unknown normalization constant is a fundamental problem in computational science and engineering. This task may be cast as an optimization problem over all probability measures, and an initial distribution can be evolved to the desired minimizer dynamically via gradient flows. Mean-field models, whose law is governed by the gradient flow in the space of probability measures, may also be identified; particle approximations of these mean-field models form the basis of algorithms. The gradient flow approach is also the basis of algorithms for variational inference, in which the optimization is performed over a parameterized family of probability distributions such as Gaussians, and the underlying gradient flow is restricted to the parameterized family. By choosing different energy functionals and metrics for the gradient flow, different algorithms with different convergence properties arise. In this paper, we concentrate on the Kullback-Leibler divergence after showing that, up to scaling, it has the unique property that the gradient flows resulting from this choice of energy do not depend on the normalization constant. For the metrics, we focus on variants of the Fisher-Rao, Wasserstein, and Stein metrics; we introduce the affine invariance property for gradient flows, and their corresponding mean-field models, determine whether a given metric leads to affine invariance, and modify it to make it affine invariant if it does not. We study the resulting gradient flows in both probability density space and Gaussian space. The flow in the Gaussian space may be understood as a Gaussian approximation of the flow. We demonstrate that the Gaussian approximation based on the metric and through moment closure coincide, establish connections between them, and study their long-time convergence properties showing the advantages of affine invariance.
Uncertainty quantification is an important task in machine learning - a task in which standardneural networks (NNs) have traditionally not excelled. This can be a limitation for safety-critical applications, where uncertainty-aware methods like Gaussian processes or Bayesian linear regression are often preferred. Bayesian neural networks are an approach to address this limitation. They assume probability distributions for all parameters and yield distributed predictions. However, training and inference are typically intractable and approximations must be employed. A promising approximation is NNs with Bayesian last layer (BLL). They assume distributed weights only in the linear output layer and yield a normally distributed prediction. To approximate the intractable Bayesian neural network, point estimates of the distributed weights in all but the last layer should be obtained by maximizing the marginal likelihood. This has previously been challenging, as the marginal likelihood is expensive to evaluate in this setting. We present a reformulation of the log-marginal likelihood of a NN with BLL which allows for efficient training using backpropagation. Furthermore, we address the challenge of uncertainty quantification for extrapolation points. We provide a metric to quantify the degree of extrapolation and derive a method to improve the uncertainty quantification for these points. Our methods are derived for the multivariate case and demonstrated in a simulation study. In comparison to Bayesian linear regression with fixed features, and a Bayesian neural network trained with variational inference, our proposed method achieves the highest log-predictive density on test data.
When factorized approximations are used for variational inference (VI), they tend to underestimate the uncertainty -- as measured in various ways -- of the distributions they are meant to approximate. We consider two popular ways to measure the uncertainty deficit of VI: (i) the degree to which it underestimates the componentwise variance, and (ii) the degree to which it underestimates the entropy. To better understand these effects, and the relationship between them, we examine an informative setting where they can be explicitly (and elegantly) analyzed: the approximation of a Gaussian,~$p$, with a dense covariance matrix, by a Gaussian,~$q$, with a diagonal covariance matrix. We prove that $q$ always underestimates both the componentwise variance and the entropy of $p$, \textit{though not necessarily to the same degree}. Moreover we demonstrate that the entropy of $q$ is determined by the trade-off of two competing forces: it is decreased by the shrinkage of its componentwise variances (our first measure of uncertainty) but it is increased by the factorized approximation which delinks the nodes in the graphical model of $p$. We study various manifestations of this trade-off, notably one where, as the dimension of the problem grows, the per-component entropy gap between $p$ and $q$ becomes vanishingly small even though $q$ underestimates every componentwise variance by a constant multiplicative factor. We also use the shrinkage-delinkage trade-off to bound the entropy gap in terms of the problem dimension and the condition number of the correlation matrix of $p$. Finally we present empirical results on both Gaussian and non-Gaussian targets, the former to validate our analysis and the latter to explore its limitations.
Gaussian process state-space models (GPSSMs) provide a principled and flexible approach to modeling the dynamics of a latent state, which is observed at discrete-time points via a likelihood model. However, inference in GPSSMs is computationally and statistically challenging due to the large number of latent variables in the model and the strong temporal dependencies between them. In this paper, we propose a new method for inference in Bayesian GPSSMs, which overcomes the drawbacks of previous approaches, namely over-simplified assumptions, and high computational requirements. Our method is based on free-form variational inference via stochastic gradient Hamiltonian Monte Carlo within the inducing-variable formalism. Furthermore, by exploiting our proposed variational distribution, we provide a collapsed extension of our method where the inducing variables are marginalized analytically. We also showcase results when combining our framework with particle MCMC methods. We show that, on six real-world datasets, our approach can learn transition dynamics and latent states more accurately than competing methods.
Inference on modern Bayesian Neural Networks (BNNs) often relies on a variational inference treatment, imposing violated assumptions of independence and the form of the posterior. Traditional MCMC approaches avoid these assumptions at the cost of increased computation due to its incompatibility to subsampling of the likelihood. New Piecewise Deterministic Markov Process (PDMP) samplers permit subsampling, though introduce a model specific inhomogenous Poisson Process (IPPs) which is difficult to sample from. This work introduces a new generic and adaptive thinning scheme for sampling from these IPPs, and demonstrates how this approach can accelerate the application of PDMPs for inference in BNNs. Experimentation illustrates how inference with these methods is computationally feasible, can improve predictive accuracy, MCMC mixing performance, and provide informative uncertainty measurements when compared against other approximate inference schemes.
Electronic health records (EHR) often contain sensitive medical information about individual patients, posing significant limitations to sharing or releasing EHR data for downstream learning and inferential tasks. We use normalizing flows (NF), a family of deep generative models, to estimate the probability density of a dataset with differential privacy (DP) guarantees, from which privacy-preserving synthetic data are generated. We apply the technique to an EHR dataset containing patients with pulmonary hypertension. We assess the learning and inferential utility of the synthetic data by comparing the accuracy in the prediction of the hypertension status and variational posterior distribution of the parameters of a physics-based model. In addition, we use a simulated dataset from a nonlinear model to compare the results from variational inference (VI) based on privacy-preserving synthetic data, and privacy-preserving VI obtained from directly privatizing NFs for VI with DP guarantees given the original non-private dataset. The results suggest that synthetic data generated through differentially private density estimation with NF can yield good utility at a reasonable privacy cost. We also show that VI obtained from differentially private NF based on the free energy bound loss may produce variational approximations with significantly altered correlation structure, and loss formulations based on alternative dissimilarity metrics between two distributions might provide improved results.
Latent variable models (LVMs) with discrete compositional latents are an important but challenging setting due to a combinatorially large number of possible configurations of the latents. A key tradeoff in modeling the posteriors over latents is between expressivity and tractable optimization. For algorithms based on expectation-maximization (EM), the E-step is often intractable without restrictive approximations to the posterior. We propose the use of GFlowNets, algorithms for sampling from an unnormalized density by learning a stochastic policy for sequential construction of samples, for this intractable E-step. By training GFlowNets to sample from the posterior over latents, we take advantage of their strengths as amortized variational inference algorithms for complex distributions over discrete structures. Our approach, GFlowNet-EM, enables the training of expressive LVMs with discrete compositional latents, as shown by experiments on non-context-free grammar induction and on images using discrete variational autoencoders (VAEs) without conditional independence enforced in the encoder.
In this work, we advocate for the importance of singular learning theory (SLT) as it pertains to the theory and practice of variational inference in Bayesian neural networks (BNNs). To begin, using SLT, we lay to rest some of the confusion surrounding discrepancies between downstream predictive performance measured via e.g., the test log predictive density, and the variational objective. Next, we use the SLT-corrected asymptotic form for singular posterior distributions to inform the design of the variational family itself. Specifically, we build upon the idealized variational family introduced in \citet{bhattacharya_evidence_2020} which is theoretically appealing but practically intractable. Our proposal takes shape as a normalizing flow where the base distribution is a carefully-initialized generalized gamma. We conduct experiments comparing this to the canonical Gaussian base distribution and show improvements in terms of variational free energy and variational generalization error.