We present a new latent-variable model employing a Gaussian mixture integrated with a feature selection procedure (the Bernoulli part of the model) which together form a "Latent Bernoulli-Gauss" distribution. The model is applied to MAP estimation, clustering, feature selection and collaborative filtering and fares favorably with the state-of-the-art latent-variable models.
Stochastic gradient descent based algorithms are typically used as the general optimization tools for most deep learning models. A Restricted Boltzmann Machine (RBM) is a probabilistic generative model that can be stacked to construct deep architectures. For RBM with Bernoulli inputs, non-Euclidean algorithm such as stochastic spectral descent (SSD) has been specifically designed to speed up the convergence with improved use of the gradient estimation by sampling methods. However, the existing algorithm and corresponding theoretical justification depend on the assumption that the possible conf…
The standard margin-based structured prediction commonly uses a maximum loss over all possible structured outputs. The large-margin formulation including latent variables not only results in a non-convex formulation but also increases the search space by a factor of the size of the latent space. Recent work has proposed the use of the maximum loss over random structured outputs sampled independently from some proposal distribution, with theoretical guarantees. We extend this work by including latent variables. We study a new family of loss functions under Gaussian perturbations and analyze the…
We explore the algebraic structure of the solution space of convex optimization problem Constrained Minimum Trace Factor Analysis (CMTFA), when the population covariance matrix $Σ_x$ has an additional latent graphical constraint, namely, a latent star topology. In particular, we have shown that CMTFA can have either a rank $ 1 $ or a rank $ n-1 $ solution and nothing in between. The special case of a rank $ 1 $ solution, corresponds to the case where just one latent variable captures all the dependencies among the observables, giving rise to a star topology. We found explicit conditions for bo…
We study the optimization landscape of the log-likelihood function and the convergence of the Expectation-Maximization (EM) algorithm in latent Gaussian tree models, i.e. tree-structured Gaussian graphical models whose leaf nodes are observable and non-leaf nodes are unobservable. We show that the unique non-trivial stationary point of the population log-likelihood is its global maximum, and establish that the expectation-maximization algorithm is guaranteed to converge to it in the single latent variable case. Our results for the landscape of the log-likelihood function in general latent tree…
Several approximate inference methods have been proposed for deep discrete latent variable models. However, non-parametric methods which have previously been successfully employed for classical sparse coding models have largely been unexplored in the context of deep models. We propose a non-parametric iterative algorithm for learning discrete latent representations in such deep models. Additionally, to learn scale invariant discrete features, we propose local data scaling variables. Lastly, to encourage sparsity in our representations, we propose a Beta-Bernoulli process prior on the latent fa…
The Gaussian-Bernoulli restricted Boltzmann machine (GB-RBM) is a useful generative model that captures meaningful features from the given $n$-dimensional continuous data. The difficulties associated with learning GB-RBM are reported extensively in earlier studies. They indicate that the training of the GB-RBM using the current standard algorithms, namely, contrastive divergence (CD) and persistent contrastive divergence (PCD), needs a carefully chosen small learning rate to avoid divergence which, in turn, results in slow learning. In this work, we alleviate such difficulties by showing that …
University of British Columbia（不列颠哥伦比亚大学）；Google Research Brain Team（谷歌研究院大脑团队）；New York University（纽约大学）；University of Toronto（多伦多大学）；Vector Institute（向量研究所）
We revisit the challenging problem of training Gaussian-Bernoulli restricted Boltzmann machines (GRBMs), introducing two innovations. We propose a novel Gibbs-Langevin sampling algorithm that outperforms existing methods like Gibbs sampling. We propose a modified contrastive divergence (CD) algorithm so that one can generate images with GRBMs starting from noise. This enables direct comparison of GRBMs with deep generative models, improving evaluation protocols in the RBM literature. Moreover, we show that modified CD and gradient clipping are enough to robustly train GRBMs with large learning…
Texas A&M University（德州农工大学）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
Semantic hashing has become a crucial component of fast similarity search in many large-scale information retrieval systems, in particular, for text data. Variational auto-encoders (VAEs) with binary latent variables as hashing codes provide state-of-the-art performance in terms of precision for document retrieval. We propose a pairwise loss function with discrete latent VAE to reward within-class similarity and between-class dissimilarity for supervised hashing. Instead of solving the optimization relying on existing biased gradient estimators, an unbiased low-variance gradient estimator is a…
Rochester Institute of Technology（罗切斯特理工学院）；Dalhousie University（达尔豪斯大学）
To improve the ability of VAE to disentangle in the latent space, existing works mostly focus on enforcing independence among the learned latent factors. However, the ability of these models to disentangle often decreases as the complexity of the generative factors increases. In this paper, we investigate the little-explored effect of the modeling capacity of a posterior density on the disentangling ability of the VAE. We note that the independence within and the complexity of the latent density are two different properties we constrain when regularizing the posterior density: while the former…
Fraunhofer ITWM（弗劳恩霍夫工业数学研究所）；Technical University of Kaiserslautern（凯泽斯劳滕工业大学）
We consider a variational autoencoder (VAE) for binary data. Our main innovations are an interpretable lower bound for its training objective, a modified initialization and architecture of such a VAE that leads to faster training, and a decision support for finding the appropriate dimension of the latent space via using a PCA. Numerical examples illustrate our theoretical result and the performance of the new architecture.
MRC Cognition and Brain Sciences Unit（MRC认知与脑科学组）；University of Cambridge（剑桥大学）；Columbia University（哥伦比亚大学）
As deep neural networks (DNNs) are applied to increasingly challenging problems, they will need to be able to represent their own uncertainty. Modeling uncertainty is one of the key features of Bayesian methods. Using Bernoulli dropout with sampling at prediction time has recently been proposed as an efficient and well performing variational inference method for DNNs. However, sampling from other multiplicative noise based variational distributions has not been investigated in depth. We evaluated Bayesian DNNs trained with Bernoulli or Gaussian multiplicative masking of either the units (dropo…
University of Cambridge（剑桥大学）；Stanford University（斯坦福大学）
Stochastic variational inference (SVI) is emerging as the most promising candidate for scaling inference in Bayesian probabilistic models to large datasets. However, the performance of these methods has been assessed primarily in the context of Bayesian topic models, particularly latent Dirichlet allocation (LDA). Deriving several new algorithms, and using synthetic, image and genomic datasets, we investigate whether the understanding gleaned from LDA applies in the setting of sparse latent factor models, specifically beta process factor analysis (BPFA). We demonstrate that the big picture is …
Beyond Gaussian Worlds: Latent Geometry Matters for JEPAs
Université Paris-Saclay（巴黎-萨克雷大学）；CNRS（法国国家科学研究中心）；ENS Paris-Saclay（巴黎-萨克雷高等师范学校）；Centre Borelli（Borelli中心）；Advanced Track and Trace；IIE, Facultad de Ingeniería, Universidad de la República（共和国大学工程学院IIE）；Institut Universitaire de France（法国大学研究院）
Variational autoencoders (VAE) have quickly become a central tool in machine learning, applicable to a broad range of data types and latent variable models. By far the most common first step, taken by seminal papers and by core software libraries alike, is to model MNIST data using a deep network parameterizing a Bernoulli likelihood. This practice contains what appears to be and what is often set aside as a minor inconvenience: the pixel data is [0,1] valued, not {0,1} as supported by the Bernoulli likelihood. Here we show that, far from being a triviality or nuisance that is convenient to ig…
NASA Ames Research Center（美国国家航空航天局艾姆斯研究中心）；HP SCDS（惠普SCDS）；Universities Space Research Association（大学空间研究协会）；Harvard University（哈佛大学）
Deep generative learning cannot only be used for generating new data with statistical characteristics derived from input data but also for anomaly detection, by separating nominal and anomalous instances based on their reconstruction quality. In this paper, we explore the performance of three unsupervised deep generative models -- variational autoencoders (VAEs) with Gaussian, Bernoulli, and Boltzmann priors -- in detecting anomalies in flight-operations data of commercial flights consisting of multivariate time series. We devised two VAE models with discrete latent variables (DVAEs), one with…
University College London（伦敦大学学院）；University of Cambridge（剑桥大学）
In a variety of disciplines such as social sciences, psychology, medicine and economics, the recorded data are considered to be noisy measurements of latent variables connected by some causal structure. This corresponds to a family of graphical models known as the structural equation model with latent variables. While linear non-Gaussian variants have been well-studied, inference in nonparametric structural equation models is still underdeveloped. We introduce a sparse Gaussian process parameterization that defines a non-linear structure connecting latent variables, unlike common formulations …
We present the Mixed Likelihood Gaussian process latent variable model (GP-LVM), capable of modeling data with attributes of different types. The standard formulation of GP-LVM assumes that each observation is drawn from a Gaussian distribution, which makes the model unsuited for data with e.g. categorical or nominal attributes. Our model, for which we use a sampling based variational inference, instead assumes a separate likelihood for each observed dimension. This formulation results in more meaningful latent representations, and give better predictive performance for real world data with di…
University of Bristol（布里斯托大学）；University of Bath（巴斯大学）
We introduce Latent Gaussian Process Regression which is a latent variable extension allowing modelling of non-stationary multi-modal processes using GPs. The approach is built on extending the input space of a regression problem with a latent variable that is used to modulate the covariance function over the training data. We show how our approach can be used to model multi-modal and non-stationary processes. We exemplify the approach on a set of synthetic data and provide results on real data from motion capture and geostatistics.
University College London（伦敦大学学院）；University of Cambridge（剑桥大学）
In a variety of disciplines such as social sciences, psychology, medicine and economics, the recorded data are considered to be noisy measurements of latent variables connected by some causal structure. This corresponds to a family of graphical models known as the structural equation model with latent variables. While linear non-Gaussian variants have been well-studied, inference in nonparametric structural equation models is still underdeveloped. We introduce a sparse Gaussian process parameterization that defines a non-linear structure connecting latent variables, unlike common formulations …
Inferentia Limited（Inferentia有限公司）；University of Sheffield（谢菲尔德大学）
Often in machine learning, data are collected as a combination of multiple conditions, e.g., the voice recordings of multiple persons, each labeled with an ID. How could we build a model that captures the latent information related to these conditions and generalize to a new one with few data? We present a new model called Latent Variable Multiple Output Gaussian Processes (LVMOGP) and that allows to jointly model multiple conditions for regression and generalize to a new condition with a few data points at test time. LVMOGP infers the posteriors of Gaussian processes together with a latent sp…
The Gaussian process latent variable model (GP-LVM) is a popular approach to non-linear probabilistic dimensionality reduction. One design choice for the model is the number of latent variables. We present a spike and slab prior for the GP-LVM and propose an efficient variational inference procedure that gives a lower bound of the log marginal likelihood. The new model provides a more principled approach for selecting latent dimensions than the standard way of thresholding the length-scale parameters. The effectiveness of our approach is demonstrated through experiments on real and simulated d…
We consider the problem of inferring a latent function in a probabilistic model of data. When dependencies of the latent function are specified by a Gaussian process and the data likelihood is complex, efficient computation often involve Markov chain Monte Carlo sampling with limited applicability to large data sets. We extend some of these techniques to scale efficiently when the problem exhibits a sequential structure. We propose an approximation that enables sequential sampling of both latent variables and associated parameters. We demonstrate strong performance in growing-data settings tha…
We introduce a Bayesian Gaussian process latent variable model that explicitly captures spatial correlations in data using a parameterized spatial kernel and leveraging structure-exploiting algebra on the model covariance matrices for computational tractability. Inference is made tractable through a collapsed variational bound with similar computational complexity to that of the traditional Bayesian GP-LVM. Inference over partially-observed test cases is achieved by optimizing a "partially-collapsed" bound. Modeling high-dimensional time series systems is enabled through use of a dynamical GP …
We introduce Latent Vector Grammars (LVeGs), a new framework that extends latent variable grammars such that each nonterminal symbol is associated with a continuous vector space representing the set of (infinitely many) subtypes of the nonterminal. We show that previous models such as latent variable grammars and compositional vector grammars can be interpreted as special cases of LVeGs. We then present Gaussian Mixture LVeGs (GM-LVeGs), a new special case of LVeGs that uses Gaussian mixtures to formulate the weights of production rules over subtypes of nonterminals. A major advantage of using…
College of Information Science and Technology（信息科学与技术学院）；The Pennsylvania State University（宾夕法尼亚州立大学）
The Dynamical Gaussian Process Latent Variable Models provide an elegant non-parametric framework for learning the low dimensional representations of the high-dimensional time-series. Real world observational studies, however, are often ill-conditioned: the observations can be noisy, not assuming the luxury of relatively complete and equally spaced like those in time series. Such conditions make it difficult to learn reasonable representations in the high dimensional longitudinal data set by way of Gaussian Process Latent Variable Model as well as other dimensionality reduction procedures. In …
University of Oxford（牛津大学）；Technical University of Denmark（丹麦技术大学）
We present a probabilistic model where the latent variable respects both the distances and the topology of the modeled data. The model leverages the Riemannian geometry of the generated manifold to endow the latent space with a well-defined stochastic distance measure, which is modeled locally as Nakagami distributions. These stochastic distances are sought to be as similar as possible to observed distances along a neighborhood graph through a censoring process. The model is inferred by variational inference based on observations of pairwise distances. We demonstrate how the new model can enco…
University of Copenhagen（哥本哈根大学）；Technical University of Denmark（丹麦技术大学）
Latent variable models (LVMs) learn probabilistic models of data manifolds lying in an \emph{ambient} Euclidean space. In a number of applications, a priori known spatial constraints can shrink the ambient space into a considerably smaller manifold. Additionally, in these applications the Euclidean geometry might induce a suboptimal similarity measure, which could be improved by choosing a different metric. Euclidean models ignore such information and assign probability mass to data points that can never appear as data, and vastly different likelihoods to points that are similar under the desi…
EURECOM（欧洲通信研究院）；University of California, Irvine（加州大学欧文分校）
Autoencoders and their variants are among the most widely used models in representation learning and generative modeling. However, autoencoder-based models usually assume that the learned representations are i.i.d. and fail to capture the correlations between the data samples. To address this issue, we propose a novel Sparse Gaussian Process Bayesian Autoencoder (SGPBAE) model in which we impose fully Bayesian sparse Gaussian Process priors on the latent space of a Bayesian Autoencoder. We perform posterior estimation for this model via stochastic gradient Hamiltonian Monte Carlo. We evaluate …
Universidad Tecnológica de Pereira（佩雷拉理工大学）；The University of Sheffield（谢菲尔德大学）
A latent force model is a Gaussian process with a covariance function inspired by a differential operator. Such covariance function is obtained by performing convolution integrals between Green's functions associated to the differential operators, and covariance functions associated to latent functions. In the classical formulation of latent force models, the covariance functions are obtained analytically by solving a double integral, leading to expressions that involve numerical solutions of different types of error functions. In consequence, the covariance matrix calculation is considerably …
Abstract reasoning ability is fundamental to human intelligence. It enables humans to uncover relations among abstract concepts and further deduce implicit rules from the relations. As a well-known abstract visual reasoning task, Raven's Progressive Matrices (RPM) are widely used in human IQ tests. Although extensive research has been conducted on RPM solvers with machine intelligence, few studies have considered further advancing the standard answer-selection (classification) problem to a more challenging answer-painting (generating) problem, which can verify whether the model has indeed unde…
Max Planck Institute for Biological Cybernetics（马克斯·普朗克生物控制论研究所）
In nonlinear latent variable models or dynamic models, if we consider the latent variables as confounders (common causes), the noise dependencies imply further relations between the observed variables. Such models are then closely related to causal discovery in the presence of nonlinear confounders, which is a challenging problem. However, generally in such models the observation noise is assumed to be independent across data dimensions, and consequently the noise dependencies are ignored. In this paper we focus on the Gaussian process latent variable model (GPLVM), from which we develop an ex…
Amortized Structured Stochastic Variational Inference for Gaussian Process Latent Variable Models
LMU Munich（慕尼黑大学）；Munich Center for Machine Learning (MCML)（慕尼黑机器学习中心）；Imperial College London（帝国理工学院）；University of Technology Nuremberg (UTN)（纽伦堡工业大学）；Helmholtz AI（亥姆霍兹人工智能中心）；Xyme
In deep latent Gaussian models, the latent variable is generated by a time-inhomogeneous Markov chain, where at each time step we pass the current state through a parametric nonlinear map, such as a feedforward neural net, and add a small independent Gaussian perturbation. This work considers the diffusion limit of such models, where the number of layers tends to infinity, while the step size and the noise variance tend to zero. The limiting latent object is an Itô diffusion process that solves a stochastic differential equation (SDE) whose drift and diffusion coefficient are implemented by ne…
University of Electro-Communications（电气通信大学）；Waseda University（早稻田大学）
In audio applications, one of the most important representations of audio signals is the amplitude spectrogram. It is utilized in many machine-learning-based information processing methods including the ones using the restricted Boltzmann machines (RBM). However, the ordinary Gaussian-Bernoulli RBM (the most popular RBM among its variations) cannot directly handle amplitude spectra because the Gaussian distribution is a symmetric model allowing negative values which never appear in the amplitude. In this paper, after proposing a general gamma Boltzmann machine, we propose a practical model cal…
Invenia Labs（因维尼亚实验室）；University College London（伦敦大学学院）；University of Cambridge（剑桥大学）
A simple and widely adopted approach to extend Gaussian processes (GPs) to multiple outputs is to model each output as a linear combination of a collection of shared, unobserved latent GPs. An issue with this approach is choosing the number of latent processes and their kernels. These choices are typically done manually, which can be time consuming and prone to human biases. We propose Gaussian Process Automatic Latent Process Selection (GP-ALPS), which automatically chooses the latent processes by turning off those that do not meaningfully contribute to explaining the data. We develop a varia…
School of Engineering, University of Warwick, UK（华威大学工程学院，英国）；Warwick Center for Predictive Modelling, University of Warwick, UK（华威大学预测建模中心，英国）；Department of Statistics, University of Warwick, UK（华威大学统计系，英国）；Division of Translational Medicine, Warwick Medical School, UK（华威医学院转化医学部，英国）；Dept of Clinical Biochemistry, Division of Pathology, University Hospital Coventry and Warwickshire, Coventry, UK（考文垂和华威郡大学医院病理学部临床生物化学系，考文垂，英国）
We introduce a Bayesian framework for inference with a supervised version of the Gaussian process latent variable model. The framework overcomes the high correlations between latent variables and hyperparameters by using an unbiased pseudo estimate for the marginal likelihood that approximately integrates over the latent variables. This is used to construct a Markov Chain to explore the posterior of the hyperparameters. We demonstrate the procedure on simulated and real examples, showing its ability to capture uncertainty and multimodality of the hyperparameters and improved uncertainty quanti…
We introduce a methodology for nonlinear inverse problems using a variational Bayesian approach where the unknown quantity is a spatial field. A structured Bayesian Gaussian process latent variable model is used both to construct a low-dimensional generative model of the sample-based stochastic prior as well as a surrogate for the forward evaluation. Its Bayesian formulation captures epistemic uncertainty introduced by the limited number of input and output examples, automatically selects an appropriate dimensionality for the learned latent representation of the data, and rigorously propagates…
Bayesian inference on non-Gaussian data is often non-analytic and requires computationally expensive approximations such as sampling or variational inference. We propose an approximate inference framework primarily designed to be computationally cheap while still achieving high approximation quality. The concept, which we call Laplace Matching, involves closed-form, approximate, bi-directional transformations between the parameter spaces of exponential families. These are constructed from Laplace approximations under custom-designed basis transformations. The mappings can then be leveraged to …
Thanks to the reparameterization trick, deep latent Gaussian models have shown tremendous success recently in learning latent representations. The ability to couple them however with nonparamet-ric priors such as the Dirichlet Process (DP) hasn't seen similar success due to its non parameteriz-able nature. In this paper, we present an alternative treatment of the variational posterior of the Dirichlet Process Deep Latent Gaussian Mixture Model (DP-DLGMM), where we show that the prior cluster parameters and the variational posteriors of the beta distributions and cluster hidden variables can be…
Center for Mathematical Modeling (CMM), Universidad de Chile（智利大学数学建模中心）；Universidad de Chile（智利大学）；Advanced Mining Technology Center, Universidad de Chile（智利大学先进采矿技术中心）
In sensing applications, sensors cannot always measure the latent quantity of interest at the required resolution, sometimes they can only acquire a blurred version of it due the sensor's transfer function. To recover latent signals when only noisy mixed measurements of the signal are available, we propose the Gaussian process mixture of measurements (GPMM), which models the latent signal as a Gaussian process (GP) and allows us to perform Bayesian inference on such signal conditional to a set of noisy mixture of measurements. We describe how to train GPMM, that is, to find the hyperparameters…
Lucerne University of Applied Sciences and Arts（卢塞恩应用科学与艺术大学）
Latent Gaussian models and boosting are widely used techniques in statistics and machine learning. Tree-boosting shows excellent prediction accuracy on many data sets, but potential drawbacks are that it assumes conditional independence of samples, produces discontinuous predictions for, e.g., spatial data, and it can have difficulty with high-cardinality categorical variables. Latent Gaussian models, such as Gaussian process and grouped random effects models, are flexible prior models which explicitly model dependence among samples and which allow for efficient learning of predictor functions…
Gaussian process latent variable models (GPLVM) are a flexible and non-linear approach to dimensionality reduction, extending classical Gaussian processes to an unsupervised learning context. The Bayesian incarnation of the GPLVM Titsias and Lawrence, 2010] uses a variational framework, where the posterior over latent variables is approximated by a well-behaved variational family, a factorized Gaussian yielding a tractable lower bound. However, the non-factories ability of the lower bound prevents truly scalable inference. In this work, we study the doubly stochastic formulation of the Bayesia…
Context-aware recommender systems (CARS) have gained increasing attention due to their ability to utilize contextual information. Compared to traditional recommender systems, CARS are, in general, able to generate more accurate recommendations. Latent factors approach accounts for a large proportion of CARS. Recently, a non-linear Gaussian Process (GP) based factorization method was proven to outperform the state-of-the-art methods in CARS. Despite its effectiveness, GP model-based methods can suffer from over-fitting and may not be able to determine the impact of each context automatically. I…
Variational autoencoders often assume isotropic Gaussian priors and mean-field posteriors, hence do not exploit structure in scenarios where we may expect similarity or consistency across latent variables. Gaussian process variational autoencoders alleviate this problem through the use of a latent Gaussian process, but lead to a cubic inference time complexity. We propose a more scalable extension of these models by leveraging the independence of the auxiliary features, which is present in many datasets. Our model factorizes the latent kernel across these features in different dimensions, lead…
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Technical University of Munich（慕尼黑工业大学）
Planning is a powerful approach to control problems with known environment dynamics. In unknown environments the agent needs to learn a model of the system dynamics to make planning applicable. This is particularly challenging when the underlying states are only indirectly observable through images. We propose to learn a deep latent Gaussian process dynamics (DLGPD) model that learns low-dimensional system dynamics from environment interactions with visual observations. The method infers latent state representations from observations using neural networks and models the system dynamics in the …
University of Michigan（密歇根大学）；Technicolor Research Center（特艺集团研究中心）
Gaussian graphical models (GGM) have been widely used in many high-dimensional applications ranging from biological and financial data to recommender systems. Sparsity in GGM plays a central role both statistically and computationally. Unfortunately, real-world data often does not fit well to sparse graphical models. In this paper, we focus on a family of latent variable Gaussian graphical models (LVGGM), where the model is conditionally sparse given latent variables, but marginally non-sparse. In LVGGM, the inverse covariance matrix has a low-rank plus sparse structure, and can be learned in …
University of Michigan-Dearborn（密歇根大学迪尔伯恩分校）；University of Central Florida（中佛罗里达大学）
A latent function decomposition method is proposed for forecasting the capacity of lithium-ion battery cells. The method uses the Multi-Output Gaussian Process, a generative machine learning framework for multi-task and transfer learning. The MCGP decomposes the available capacity trends from multiple battery cells into latent functions. The latent functions are then convolved over kernel smoothers to reconstruct and/or forecast capacity trends of the battery cells. Besides the high prediction accuracy the proposed method possesses, it provides uncertainty information for the predictions and c…
University of Toronto（多伦多大学）；Cornell University（康奈尔大学）
This paper considers binary classification of high-dimensional features under a postulated model with a low-dimensional latent Gaussian mixture structure and non-vanishing noise. A generalized least squares estimator is used to estimate the direction of the optimal separating hyperplane. The estimated hyperplane is shown to interpolate on the training data. While the direction vector can be consistently estimated as could be expected from recent results in linear regression, a naive plug-in estimate fails to consistently estimate the intercept. A simple correction, that requires an independent…
