Training energy-based probabilistic models is confronted with apparently intractable sums, whose Monte Carlo estimation requires sampling from the estimated probability distribution in the inner loop of training. This can be approximately achieved by Markov chain Monte Carlo methods, but may still face a formidable obstacle that is the difficulty of mixing between modes with sharp concentrations of probability. Whereas an MCMC process is usually derived from a given energy function based on mathematical considerations and requires an arbitrarily long time to obtain good and varied samples, we propose to train a deep directed generative model (not a Markov chain) so that its sampling distribution approximately matches the energy function that is being trained. Inspired by generative adversarial networks, the proposed framework involves training of two models that represent dual views of the estimated probability distribution: the energy function (mapping an input configuration to a scalar energy value) and the generator (mapping a noise vector to a generated configuration), both represented by deep neural networks.
Recent work (Bengio et al., 2013) has shown howDenoising Auto-Encoders(DAE) become gener-ative models as a density estimator. However,in practice, the framework suffers from a mixingproblem in the MCMC sampling process and nodirect method to estimate the test log-likelihood.We consider a directed model with an stochas-tic identity mapping (simple corruption pro-cess) as an inference model and a DAE as agenerative model. By cascading these mod-els, we propose Cascading Denoising Auto-Encoders(CDAE) which can generate samples ofdata distribution from tractable prior distributionunder the assumpt…
Rice University（莱斯大学）；Aix-Marseille Univ（艾克斯-马赛大学）；Université de Toulon（土伦大学）；CNRS（法国国家科学研究中心）
Deep Generative Networks (DGNs) with probabilistic modeling of their output and latent space are currently trained via Variational Autoencoders (VAEs). In the absence of a known analytical form for the posterior and likelihood expectation, VAEs resort to approximations, including (Amortized) Variational Inference (AVI) and Monte-Carlo (MC) sampling. We exploit the Continuous Piecewise Affine (CPA) property of modern DGNs to derive their posterior and marginal distributions as well as the latter's first moments. These findings enable us to derive an analytical Expectation-Maximization (EM) algo…
Graph structured data are abundant in the real world. Among different graph types, directed acyclic graphs (DAGs) are of particular interest to machine learning researchers, as many machine learning models are realized as computations on DAGs, including neural networks and Bayesian networks. In this paper, we study deep generative models for DAGs, and propose a novel DAG variational autoencoder (D-VAE). To encode DAGs into the latent space, we leverage graph neural networks. We propose an asynchronous message passing scheme that allows encoding the computations on DAGs, rather than using exist…
Georgia Institute of Technology（佐治亚理工学院）；Stony Brook University（石溪大学）；Ant Financial（蚂蚁金服）
Deep generative models have been enjoying success in modeling continuous data. However it remains challenging to capture the representations for discrete structures with formal grammars and semantics, e.g., computer programs and molecular structures. How to generate both syntactically and semantically correct data still remains largely an open problem. Inspired by the theory of compiler where the syntax and semantics check is done via syntax-directed translation (SDT), we propose a novel syntax-directed variational autoencoder (SD-VAE) by introducing stochastic lazy attributes. This approach c…
Probability density estimation is a classical and well studied problem, but standard density estimation methods have historically lacked the power to model complex and high-dimensional image distributions. More recent generative models leverage the power of neural networks to implicitly learn and represent probability models over complex images. We describe methods to extract explicit probability density estimates from GANs, and explore the properties of these image density functions. We perform sanity check experiments to provide evidence that these probabilities are reasonable. However, we a…
Northeastern University（东北大学）；Khoury College of Computer Sciences（库里计算机学院）；The Australian National University（澳大利亚国立大学）；School of Computing（计算学院）
Applied category theory has recently developed libraries for computing with morphisms in interesting categories, while machine learning has developed ways of learning programs in interesting languages. Taking the analogy between categories and languages seriously, this paper defines a probabilistic generative model of morphisms in free monoidal categories over domain-specific generating objects and morphisms. The paper shows how acyclic directed wiring diagrams can model specifications for morphisms, which the model can use to generate morphisms. Amortized variational inference in the generati…
Many applications of generative models rely on the marginalization of their high-dimensional output probability distributions. Normalization functions that yield sparse probability distributions can make exact marginalization more computationally tractable. However, sparse normalization functions usually require alternative loss functions for training since the log-likelihood is undefined for sparse probability distributions. Furthermore, many sparse normalization functions often collapse the multimodality of distributions. In this work, we present $\textit{ev-softmax}$, a sparse normalization…
University of Geneva（日内瓦大学）；University of Applied Sciences Western Switzerland（瑞士西部应用科学大学）；University College London（伦敦大学学院）；Alan Turing Institute（阿兰·图灵研究所）
Despite recent advances, goal-directed generation of structured discrete data remains challenging. For problems such as program synthesis (generating source code) and materials design (generating molecules), finding examples which satisfy desired constraints or exhibit desired properties is difficult. In practice, expensive heuristic search or reinforcement learning algorithms are often employed. In this paper we investigate the use of conditional generative models which directly attack this inverse problem, by modeling the distribution of discrete structures given properties of interest. Unfo…
Carl von Ossietzky University of Oldenburg（奥尔登堡卡尔·冯·奥西茨基大学）；Frankfurt Institute for Advanced Studies (FIAS)（法兰克福高等研究所（FIAS））；Technical University of Berlin（柏林工业大学）
Classifiers for the semi-supervised setting often combine strong supervised models with additional learning objectives to make use of unlabeled data. This results in powerful though very complex models that are hard to train and that demand additional labels for optimal parameter tuning, which are often not given when labeled data is very sparse. We here study a minimalistic multi-layer generative neural network for semi-supervised learning in a form and setting as similar to standard discriminative networks as possible. Based on normalized Poisson mixtures, we derive compact and local learnin…
In this paper, we provide a method to learn the directed structure of a Bayesian network using data. The data is accessed by making conditional probability queries to a black-box model. We introduce a notion of simplicity of representation of conditional probability tables for the nodes in the Bayesian network, that we call "low rankness". We connect this notion to the Fourier transformation of real valued set functions and propose a method which learns the exact directed structure of a `low rank` Bayesian network using very few queries. We formally prove that our method correctly recovers the…
We show that the only parameter prior for complete Gaussian DAG models that satisfies global parameter independence, complete model equivalence, and some weak regularity assumptions, is the normal-Wishart distribution. Our analysis is based on the following new characterization of the Wishart distribution: let W be an n x n, n >= 3, positive-definite symmetric matrix of random variables and f(W) be a pdf of W. Then, f(W) is a Wishart distribution if and only if W_{11}-W_{12}W_{22}^{-1}W_{12}' is independent of {W_{12}, W_{22}} for every block partitioning W_{11}, W_{12}, W_{12}', W_{22} of W. …
Efficient unsupervised training and inference in deep generative models remains a challenging problem. One basic approach, called Helmholtz machine, involves training a top-down directed generative model together with a bottom-up auxiliary model used for approximate inference. Recent results indicate that better generative models can be obtained with better approximate inference procedures. Instead of improving the inference procedure, we here propose a new model which guarantees that the top-down and bottom-up distributions can efficiently invert each other. We achieve this by interpreting bo…
Indian Institute of Technology Delhi（德里印度理工学院）；Université de Montréal（蒙特利尔大学）
For discrete data, the likelihood $P(x)$ can be rewritten exactly and parametrized into $P(X = x) = P(X = x | H = f(x)) P(H = f(x))$ if $P(X | H)$ has enough capacity to put no probability mass on any $x'$ for which $f(x')\neq f(x)$, where $f(\cdot)$ is a deterministic discrete function. The log of the first factor gives rise to the log-likelihood reconstruction error of an autoencoder with $f(\cdot)$ as the encoder and $P(X|H)$ as the (probabilistic) decoder. The log of the second term can be seen as a regularizer on the encoded activations $h=f(x)$, e.g., as in sparse autoencoders. Both enco…
The pattern theory of Grenander is a mathematical framework where patterns are represented by probability models on random variables of algebraic structures. In this paper, we review three families of probability models, namely, the discriminative models, the descriptive models, and the generative models. A discriminative model is in the form of a classifier. It specifies the conditional probability of the class label given the input signal. A descriptive model specifies the probability distribution of the signal, based on an energy function defined on the signal. A generative model assumes th…
We develop simple methods for constructing parameter priors for model choice among Directed Acyclic Graphical (DAG) models. In particular, we introduce several assumptions that permit the construction of parameter priors for a large number of DAG models from a small set of assessments. We then present a method for directly computing the marginal likelihood of every DAG model given a random sample with no missing observations. We apply this methodology to Gaussian DAG models which consist of a recursive set of linear regression models. We show that the only parameter prior for complete Gaussian…
Deep directed generative models have attracted much attention recently due to their generative modeling nature and powerful data representation ability. In this paper, we review different structures of deep directed generative models and the learning and inference algorithms associated with the structures. We focus on a specific structure that consists of layers of Bayesian Networks due to the property of capturing inherent and rich dependencies among latent variables. The major difficulty of learning and inference with deep directed models with many latent variables is the intractable inferen…
Flow-based generative models as iterative algorithms in probability space
H. Milton Stewart School of Industrial and Systems Engineering (ISyE) at the Georgia Institute of Technology（佐治亚理工学院H.米尔顿·斯图尔特工业与系统工程学院）；Mathematics Department at Duke University（杜克大学数学系）
Learning and inference movement is a very challenging problem due to its high dimensionality and dependency to varied environments or tasks. In this paper, we propose an effective probabilistic method for learning and inference of basic movements. The motion planning problem is formulated as learning on a directed graphic model and deep generative model is used to perform learning and inference from demonstrations. An important characteristic of this method is that it flexibly incorporates the task descriptors and context information for long-term planning and it can be combined with dynamic s…
A central tenet of probabilistic programming is that a model is specified exactly once in a canonical representation which is usable by inference algorithms. We describe JointDistributions, a family of declarative representations of directed graphical models in TensorFlow Probability.
We present an architecture which lets us train deep, directed generative models with many layers of latent variables. We include deterministic paths between all latent variables and the generated output, and provide a richer set of connections between computations for inference and generation, which enables more effective communication of information throughout the model during training. To improve performance on natural images, we incorporate a lightweight autoregressive model in the reconstruction distribution. These techniques permit end-to-end training of models with 10+ layers of latent v…
We marry ideas from deep neural networks and approximate Bayesian inference to derive a generalised class of deep, directed generative models, endowed with a new algorithm for scalable inference and learning. Our algorithm introduces a recognition model to represent approximate posterior distributions, and that acts as a stochastic encoder of the data. We develop stochastic back-propagation -- rules for back-propagation through stochastic variables -- and use this to develop an algorithm that allows for joint optimisation of the parameters of both the generative and recognition model. We demon…
