This paper describes an approach to simultaneously identify clusters and estimate cluster-specific regression parameters from the given data. Such an approach can be useful in learning the relationship between input and output when the regression parameters for estimating output are different in different regions of the input space. Variational Inference (VI), a machine learning approach to obtain posterior probability densities using optimization techniques, is used to identify clusters of explanatory variables and regression parameters for each cluster. From these results, one can obtain both the expected value and the full distribution of predicted output. Other advantages of the proposed approach include the elegant theoretical solution and clear interpretability of results. The proposed approach is well-suited for financial forecasting where markets have different regimes (or clusters) with different patterns and correlations of market changes in each regime. In financial applications, knowledge about such clusters can provide useful insights about portfolio performance and identify the relative importance of variables in different market regimes. An illustrative example of predicting one-day S&P change is considered to illustrate the approach and compare the performance of the proposed approach with standard regression without clusters. Due to the broad applicability of the problem, its elegant theoretical solution, and the computational efficiency of the proposed algorithm, the approach may be useful in a number of areas extending beyond the financial domain.
In this paper, we address the problem of convergence of sequential variational inference filter (VIF) through the application of a robust variational objective and Hinf-norm based correction for a linear Gaussian system. As the dimension of state or parameter space grows, performing the full Kalman update with the dense covariance matrix for a large scale system requires increased storage and computational complexity, making it impractical. The VIF approach, based on mean-field Gaussian variational inference, reduces this burden through the variational approximation to the covariance usually in the form of a diagonal covariance approximation. The challenge is to retain convergence and correct for biases introduced by the sequential VIF steps. We desire a framework that improves feasibility while still maintaining reasonable proximity to the optimal Kalman filter as data is assimilated. To accomplish this goal, a Hinf-norm based optimization perturbs the VIF covariance matrix to improve robustness. This yields a novel VIF- Hinf recursion that employs consecutive variational inference and Hinf based optimization steps. We explore the development of this method and investigate a numerical example to illustrate the effectiveness of the proposed filter.
We investigate the potential of adaptive blind equalizers based on variational inference for carrier recovery in optical communications. These equalizers are based on a low-complexity approximation of maximum likelihood channel estimation. We generalize the concept of variational autoencoder (VAE) equalizers to higher order modulation formats encompassing probabilistic constellation shaping (PCS), ubiquitous in optical communications, oversampling at the receiver, and dual-polarization transmission. Besides black-box equalizers based on convolutional neural networks, we propose a model-based equalizer based on a linear butterfly filter and train the filter coefficients using the variational inference paradigm. As a byproduct, the VAE also provides a reliable channel estimation. We analyze the VAE in terms of performance and flexibility over a classical additive white Gaussian noise (AWGN) channel with inter-symbol interference (ISI) and over a dispersive linear optical dual-polarization channel. We show that it can extend the application range of blind adaptive equalizers by outperforming the state-of-the-art constant-modulus algorithm (CMA) for PCS for both fixed but also time-varying channels. The evaluation is accompanied with a hyperparameter analysis.
Collecting paired training data is difficult in practice, but the unpaired samples broadly exist. Current approaches aim at generating synthesized training data from unpaired samples by exploring the relationship between the corrupted and clean data. This work proposes LUD-VAE, a deep generative method to learn the joint probability density function from data sampled from marginal distributions. Our approach is based on a carefully designed probabilistic graphical model in which the clean and corrupted data domains are conditionally independent. Using variational inference, we maximize the evidence lower bound (ELBO) to estimate the joint probability density function. Furthermore, we show that the ELBO is computable without paired samples under the inference invariant assumption. This property provides the mathematical rationale of our approach in the unpaired setting. Finally, we apply our method to real-world image denoising, super-resolution, and low-light image enhancement tasks and train the models using the synthetic data generated by the LUD-VAE. Experimental results validate the advantages of our method over other approaches.
We propose a novel generative saliency prediction framework that adopts an informative energy-based model as a prior distribution. The energy-based prior model is defined on the latent space of a saliency generator network that generates the saliency map based on a continuous latent variables and an observed image. Both the parameters of saliency generator and the energy-based prior are jointly trained via Markov chain Monte Carlo-based maximum likelihood estimation, in which the sampling from the intractable posterior and prior distributions of the latent variables are performed by Langevin dynamics. With the generative saliency model, we can obtain a pixel-wise uncertainty map from an image, indicating model confidence in the saliency prediction. Different from existing generative models, which define the prior distribution of the latent variables as a simple isotropic Gaussian distribution, our model uses an energy-based informative prior which can be more expressive in capturing the latent space of the data. With the informative energy-based prior, we extend the Gaussian distribution assumption of generative models to achieve a more representative distribution of the latent space, leading to more reliable uncertainty estimation. We apply the proposed frameworks to both RGB and RGB-D salient object detection tasks with both transformer and convolutional neural network backbones. We further propose an adversarial learning algorithm and a variational inference algorithm as alternatives to train the proposed generative framework. Experimental results show that our generative saliency model with an energy-based prior can achieve not only accurate saliency predictions but also reliable uncertainty maps that are consistent with human perception. Results and code are available at \url{https://github.com/JingZhang617/EBMGSOD}.
Bayesian phylogenetic inference is currently done via Markov chain Monte Carlo (MCMC) with simple proposal mechanisms. This hinders exploration efficiency and often requires long runs to deliver accurate posterior estimates. In this paper, we present an alternative approach: a variational framework for Bayesian phylogenetic analysis. We propose combining subsplit Bayesian networks, an expressive graphical model for tree topology distributions, and a structured amortization of the branch lengths over tree topologies for a suitable variational family of distributions. We train the variational approximation via stochastic gradient ascent and adopt gradient estimators for continuous and discrete variational parameters separately to deal with the composite latent space of phylogenetic models. We show that our variational approach provides competitive performance to MCMC, while requiring much fewer (though more costly) iterations due to a more efficient exploration mechanism enabled by variational inference. Experiments on a benchmark of challenging real data Bayesian phylogenetic inference problems demonstrate the effectiveness and efficiency of our methods.
Most deep learning models for computational imaging regress a single reconstructed image. In practice, however, ill-posedness, nonlinearity, model mismatch, and noise often conspire to make such point estimates misleading or insufficient. The Bayesian approach models images and (noisy) measurements as jointly distributed random vectors and aims to approximate the posterior distribution of unknowns. Recent variational inference methods based on conditional normalizing flows are a promising alternative to traditional MCMC methods, but they come with drawbacks: excessive memory and compute demands for moderate to high resolution images and underwhelming performance on hard nonlinear problems. In this work, we propose C-Trumpets -- conditional injective flows specifically designed for imaging problems, which greatly diminish these challenges. Injectivity reduces memory footprint and training time while low-dimensional latent space together with architectural innovations like fixed-volume-change layers and skip-connection revnet layers, C-Trumpets outperform regular conditional flow models on a variety of imaging and image restoration tasks, including limited-view CT and nonlinear inverse scattering, with a lower compute and memory budget. C-Trumpets enable fast approximation of point estimates like MMSE or MAP as well as physically-meaningful uncertainty quantification.
Current works in the generation of personalized dialogue primarily contribute to the agent presenting a consistent personality and driving a more informative response. However, we found that the generated responses from most previous models tend to be self-centered, with little care for the user in the dialogue. Moreover, we consider that human-like conversation is essentially built based on inferring information about the persona of the other party. Motivated by this, we propose a novel personalized dialogue generator by detecting an implicit user persona. Because it is hard to collect a large number of detailed personas for each user, we attempted to model the user's potential persona and its representation from dialogue history, with no external knowledge. The perception and fader variables were conceived using conditional variational inference. The two latent variables simulate the process of people being aware of each other's persona and producing a corresponding expression in conversation. Finally, posterior-discriminated regularization was presented to enhance the training procedure. Empirical studies demonstrate that, compared to state-of-the-art methods, our approach is more concerned with the user's persona and achieves a considerable boost across the evaluations.
Missing values are common in many real-life datasets. However, most of the current machine learning methods can not handle missing values. This means that they should be imputed beforehand. Gaussian Processes (GPs) are non-parametric models with accurate uncertainty estimates that combined with sparse approximations and stochastic variational inference scale to large data sets. Sparse GPs can be used to compute a predictive distribution for missing data. Here, we present a hierarchical composition of sparse GPs that is used to predict missing values at each dimension using all the variables from the other dimensions. We call the approach missing GP (MGP). MGP can be trained simultaneously to impute all observed missing values. Specifically, it outputs a predictive distribution for each missing value that is then used in the imputation of other missing values. We evaluate MGP in one private clinical data set and four UCI datasets with a different percentage of missing values. We compare the performance of MGP with other state-of-the-art methods for imputing missing values, including variants based on sparse GPs and deep GPs. The results obtained show a significantly better performance of MGP.
Generating high quality texts with high diversity is important for many NLG applications, but current methods mostly focus on building deterministic models to generate higher quality texts and do not provide many options for promoting diversity. In this work, we present a novel latent structured variable model to generate high quality texts by enriching contextual representation learning of encoder-decoder models. Specifically, we introduce a stochastic function to map deterministic encoder hidden states into random context variables. The proposed stochastic function is sampled from a Gaussian process prior to (1) provide infinite number of joint Gaussian distributions of random context variables (diversity-promoting) and (2) explicitly model dependency between context variables (accurate-encoding). To address the learning challenge of Gaussian processes, we propose an efficient variational inference approach to approximate the posterior distribution of random context variables. We evaluate our method in two typical text generation tasks: paraphrase generation and text style transfer. Experimental results on benchmark datasets demonstrate that our method improves the generation quality and diversity compared with other baselines.
In this paper we propose Discretely Indexed flows (DIF) as a new tool for solving variational estimation problems. Roughly speaking, DIF are built as an extension of Normalizing Flows (NF), in which the deterministic transport becomes stochastic, and more precisely discretely indexed. Due to the discrete nature of the underlying additional latent variable, DIF inherit the good computational behavior of NF: they benefit from both a tractable density as well as a straightforward sampling scheme, and can thus be used for the dual problems of Variational Inference (VI) and of Variational density estimation (VDE). On the other hand, DIF can also be understood as an extension of mixture density models, in which the constant mixture weights are replaced by flexible functions. As a consequence, DIF are better suited for capturing distributions with discontinuities, sharp edges and fine details, which is a main advantage of this construction. Finally we propose a methodology for constructiong DIF in practice, and see that DIF can be sequentially cascaded, and cascaded with NF.
In safety-critical applications, practitioners are reluctant to trust neural networks when no interpretable explanations are available. Many attempts to provide such explanations revolve around pixel-based attributions or use previously known concepts. In this paper we aim to provide explanations by provably identifying \emph{high-level, previously unknown ground-truth concepts}. To this end, we propose a probabilistic modeling framework to derive (C)oncept (L)earning and (P)rediction (CLAP) -- a VAE-based classifier that uses visually interpretable concepts as predictors for a simple classifier. Assuming a generative model for the ground-truth concepts, we prove that CLAP is able to identify them while attaining optimal classification accuracy. Our experiments on synthetic datasets verify that CLAP identifies distinct ground-truth concepts on synthetic datasets and yields promising results on the medical Chest X-Ray dataset.
Near-term noisy intermediate-scale quantum circuits can efficiently implement implicit probabilistic models in discrete spaces, supporting distributions that are practically infeasible to sample from using classical means. One of the possible applications of such models, also known as Born machines, is probabilistic inference, which is at the core of Bayesian methods. This paper studies the use of Born machines for the problem of training binary Bayesian neural networks. In the proposed approach, a Born machine is used to model the variational distribution of the binary weights of the neural network, and data from multiple tasks is used to reduce training data requirements on new tasks. The method combines gradient-based meta-learning and variational inference via Born machines, and is shown in a prototypical regression problem to outperform conventional joint learning strategies.
Black-box variational inference (BBVI) now sees widespread use in machine learning and statistics as a fast yet flexible alternative to Markov chain Monte Carlo methods for approximate Bayesian inference. However, stochastic optimization methods for BBVI remain unreliable and require substantial expertise and hand-tuning to apply effectively. In this paper, we propose Robust and Automated Black-box VI (RABVI), a framework for improving the reliability of BBVI optimization. RABVI is based on rigorously justified automation techniques, includes just a small number of intuitive tuning parameters, and detects inaccurate estimates of the optimal variational approximation. RABVI adaptively decreases the learning rate by detecting convergence of the fixed--learning-rate iterates, then estimates the symmetrized Kullback--Leibler (KL) divergence between the current variational approximation and the optimal one. It also employs a novel optimization termination criterion that enables the user to balance desired accuracy against computational cost by comparing (i) the predicted relative decrease in the symmetrized KL divergence if a smaller learning were used and (ii) the predicted computation required to converge with the smaller learning rate. We validate the robustness and accuracy of RABVI through carefully designed simulation studies and on a diverse set of real-world model and data examples.
Structuring the latent space in probabilistic deep generative models, e.g., variational autoencoders (VAEs), is important to yield more expressive models and interpretable representations, and to avoid overfitting. One way to achieve this objective is to impose a sparsity constraint on the latent variables, e.g., via a Laplace prior. However, such approaches usually complicate the training phase, and they sacrifice the reconstruction quality to promote sparsity. In this paper, we propose a simple yet effective methodology to structure the latent space via a sparsity-promoting dictionary model, which assumes that each latent code can be written as a sparse linear combination of a dictionary's columns. In particular, we leverage a computationally efficient and tuning-free method, which relies on a zero-mean Gaussian latent prior with learnable variances. We derive a variational inference scheme to train the model. Experiments on speech generative modeling demonstrate the advantage of the proposed approach over competing techniques, since it promotes sparsity while not deteriorating the output speech quality.
Current top-notch deep learning (DL) based vision models are primarily based on exploring and exploiting the inherent correlations between training data samples and their associated labels. However, a known practical challenge is their degraded performance against "noisy" data, induced by different circumstances such as spurious correlations, irrelevant contexts, domain shift, and adversarial attacks. In this work, we incorporate this binary information of "existence of noise" as treatment into image classification tasks to improve prediction accuracy by jointly estimating their treatment effects. Motivated from causal variational inference, we propose a transformer-based architecture, Treatment Learning Causal Transformer (TLT), that uses a latent generative model to estimate robust feature representations from current observational input for noise image classification. Depending on the estimated noise level (modeled as a binary treatment factor), TLT assigns the corresponding inference network trained by the designed causal loss for prediction. We also create new noisy image datasets incorporating a wide range of noise factors (e.g., object masking, style transfer, and adversarial perturbation) for performance benchmarking. The superior performance of TLT in noisy image classification is further validated by several refutation evaluation metrics. As a by-product, TLT also improves visual salience methods for perceiving noisy images.
Locally Linear Embedding (LLE) is a nonlinear spectral dimensionality reduction and manifold learning method. It has two main steps which are linear reconstruction and linear embedding of points in the input space and embedding space, respectively. In this work, we look at the linear reconstruction step from a stochastic perspective where it is assumed that every data point is conditioned on its linear reconstruction weights as latent factors. The stochastic linear reconstruction of LLE is solved using expectation maximization. We show that there is a theoretical connection between three fundamental dimensionality reduction methods, i.e., LLE, factor analysis, and probabilistic Principal Component Analysis (PCA). The stochastic linear reconstruction of LLE is formulated similar to the factor analysis and probabilistic PCA. It is also explained why factor analysis and probabilistic PCA are linear and LLE is a nonlinear method. This work combines and makes a bridge between two broad approaches of dimensionality reduction, i.e., the spectral and probabilistic algorithms.
Stochastic planning can be reduced to probabilistic inference in large discrete graphical models, but hardness of inference requires approximation schemes to be used. In this paper we argue that such applications can be disentangled along two dimensions. The first is the direction of information flow in the idealized exact optimization objective, i.e., forward vs backward inference. The second is the type of approximation used to compute this objective, e.g., Belief Propagation (BP) vs mean field variational inference (MFVI). This new categorization allows us to unify a large amount of isolated efforts in prior work explaining their connections and differences as well as potential improvements. An extensive experimental evaluation over large stochastic planning problems shows the advantage of forward BP over several algorithms based on MFVI. An analysis of practical limitations of MFVI motivates a novel algorithm, collapsed state variational inference (CSVI), which provides a tighter approximation and achieves comparable planning performance with forward BP.
In this study, a novel self-supervised learning (SSL) method is proposed, which considers SSL in terms of variational inference to learn not only representation but also representation uncertainties. SSL is a method of learning representations without labels by maximizing the similarity between image representations of different augmented views of an image. Meanwhile, variational autoencoder (VAE) is an unsupervised representation learning method that trains a probabilistic generative model with variational inference. Both VAE and SSL can learn representations without labels, but their relationship has not been investigated in the past. Herein, the theoretical relationship between SSL and variational inference has been clarified. Furthermore, a novel method, namely variational inference SimSiam (VI-SimSiam), has been proposed. VI-SimSiam can predict the representation uncertainty by interpreting SimSiam with variational inference and defining the latent space distribution. The present experiments qualitatively show that VI- SimSiam could learn uncertainty by comparing input images and predicted uncertainties. Additionally, we described a relationship between estimated uncertainty and classification accuracy.
Recently, there is a revival of interest in low-rank matrix completion-based unsupervised learning through the lens of dual-graph regularization, which has significantly improved the performance of multidisciplinary machine learning tasks such as recommendation systems, genotype imputation and image inpainting. While the dual-graph regularization contributes a major part of the success, computational costly hyper-parameter tunning is usually involved. To circumvent such a drawback and improve the completion performance, we propose a novel Bayesian learning algorithm that automatically learns the hyper-parameters associated with dual-graph regularization, and at the same time, guarantees the low-rankness of matrix completion. Notably, a novel prior is devised to promote the low-rankness of the matrix and encode the dual-graph information simultaneously, which is more challenging than the single-graph counterpart. A nontrivial conditional conjugacy between the proposed priors and likelihood function is then explored such that an efficient algorithm is derived under variational inference framework. Extensive experiments using synthetic and real-world datasets demonstrate the state-of-the-art performance of the proposed learning algorithm for various data analysis tasks.