Coarse Grained Exponential Variational Autoencoders

Variational autoencoders (VAE) often use Gaussian or category distribution to model the inference process. This puts a limit on variational learning because this simplified assumption does not match the true posterior distribution, which is usually much more sophisticated. To break this limitation and apply arbitrary parametric distribution during inference, this paper derives a \emph{semi-continuous} latent representation, which approximates a continuous density up to a prescribed precision, and is much easier to analyze than its continuous counterpart because it is fundamentally discrete. We showcase the proposition by applying polynomial exponential family distributions as the posterior, which are universal probability density function generators. Our experimental results show consistent improvements over commonly used VAE models

An Optimization Framework for Semi-Supervised and Transfer Learning using Multiple Classifiers and Clusterers

Unsupervised models can provide supplementary soft constraints to help classify new, "target" data since similar instances in the target set are more likely to share the same class label. Such models can also help detect possible differences between training and target distributions, which is useful in applications where concept drift may take place, as in transfer learning settings. This paper describes a general optimization framework that takes as input class membership estimates from existing classifiers learnt on previously encountered "source" data, as well as a similarity matrix from a cluster ensemble operating solely on the target data to be classified, and yields a consensus labeling of the target data. This framework admits a wide range of loss functions and classification/clustering methods. It exploits properties of Bregman divergences in conjunction with Legendre duality to yield a principled and scalable approach. A variety of experiments show that the proposed framework can yield results substantially superior to those provided by popular transductive learning techniques or by naively applying classifiers learnt on the original task to the target data

Geometric Losses for Distributional Learning

Building upon recent advances in entropy-regularized optimal transport, and upon Fenchel duality between measures and continuous functions , we propose a generalization of the logistic loss that incorporates a metric or cost between classes. Unlike previous attempts to use optimal transport distances for learning, our loss results in unconstrained convex objective functions, supports infinite (or very large) class spaces, and naturally defines a geometric generalization of the softmax operator. The geometric properties of this loss make it suitable for predicting sparse and singular distributions, for instance supported on curves or hyper-surfaces. We study the theoretical properties of our loss and show-case its effectiveness on two applications: ordinal regression and drawing generation

Constructing the Matrix Multilayer Perceptron and its Application to the VAE

Like most learning algorithms, the multilayer perceptrons (MLP) is designed to learn a vector of parameters from data. However, in certain scenarios we are interested in learning structured parameters (predictions) in the form of symmetric positive definite matrices. Here, we introduce a variant of the MLP, referred to as the matrix MLP, that is specialized at learning symmetric positive definite matrices. We also present an application of the model within the context of the variational autoencoder (VAE). Our formulation of the VAE extends the vanilla formulation to the cases where the recognition and the generative networks can be from the parametric family of distributions with dense covariance matrices. Two specific examples are discussed in more detail: the dense covariance Gaussian and its generalization, the power exponential distribution. Our new developments are illustrated using both synthetic and real data

Relaxed Wasserstein with Applications to GANs

Wasserstein Generative Adversarial Networks (WGANs) provide a versatile class of models, which have attracted great attention in various applications. However, this framework has two main drawbacks: (i) Wasserstein-1 (or Earth-Mover) distance is restrictive such that WGANs cannot always fit data geometry well; (ii) It is difficult to achieve fast training of WGANs. In this paper, we propose a new class of \textit{Relaxed Wasserstein} (RW) distances by generalizing Wasserstein-1 distance with Bregman cost functions. We show that RW distances achieve nice statistical properties while not sacrificing the computational tractability. Combined with the GANs framework, we develop Relaxed WGANs (RWGANs) which are not only statistically flexible but can be approximated efficiently using heuristic approaches. Experiments on real images demonstrate that the RWGAN with Kullback-Leibler (KL) cost function outperforms other competing approaches, e.g., WGANs, even with gradient penalty.Comment: Accepted by ICASSP 2021; add the reference

On The Chain Rule Optimal Transport Distance

We define a novel class of distances between statistical multivariate distributions by solving an optimal transportation problem on their marginal densities with respect to a ground distance defined on their conditional densities. By using the chain rule factorization of probabilities, we show how to perform optimal transport on a ground space being an information-geometric manifold of conditional probabilities. We prove that this new distance is a metric whenever the chosen ground distance is a metric. Our distance generalizes both the Wasserstein distances between point sets and a recently introduced metric distance between statistical mixtures. As a first application of this Chain Rule Optimal Transport (CROT) distance, we show that the ground distance between statistical mixtures is upper bounded by this optimal transport distance and its fast relaxed Sinkhorn distance, whenever the ground distance is joint convex. We report on our experiments which quantify the tightness of the CROT distance for the total variation distance, the square root generalization of the Jensen-Shannon divergence, the Wasserstein $W_p$ metric and the R\'enyi divergence between mixtures.Comment: 23 page

Copula Variational Bayes inference via information geometry

Variational Bayes (VB), also known as independent mean-field approximation, has become a popular method for Bayesian network inference in recent years. Its application is vast, e.g. in neural network, compressed sensing, clustering, etc. to name just a few. In this paper, the independence constraint in VB will be relaxed to a conditional constraint class, called copula in statistics. Since a joint probability distribution always belongs to a copula class, the novel copula VB (CVB) approximation is a generalized form of VB. Via information geometry, we will see that CVB algorithm iteratively projects the original joint distribution to a copula constraint space until it reaches a local minimum Kullback-Leibler (KL) divergence. By this way, all mean-field approximations, e.g. iterative VB, Expectation-Maximization (EM), Iterated Conditional Mode (ICM) and k-means algorithms, are special cases of CVB approximation. For a generic Bayesian network, an augmented hierarchy form of CVB will also be designed. While mean-field algorithms can only return a locally optimal approximation for a correlated network, the augmented CVB network, which is an optimally weighted average of a mixture of simpler network structures, can potentially achieve the globally optimal approximation for the first time. Via simulations of Gaussian mixture clustering, the classification's accuracy of CVB will be shown to be far superior to that of state-of-the-art VB, EM and k-means algorithms.Comment: IEEE Transactions on Information Theor

Learning the Information Divergence

Information divergence that measures the difference between two nonnegative matrices or tensors has found its use in a variety of machine learning problems. Examples are Nonnegative Matrix/Tensor Factorization, Stochastic Neighbor Embedding, topic models, and Bayesian network optimization. The success of such a learning task depends heavily on a suitable divergence. A large variety of divergences have been suggested and analyzed, but very few results are available for an objective choice of the optimal divergence for a given task. Here we present a framework that facilitates automatic selection of the best divergence among a given family, based on standard maximum likelihood estimation. We first propose an approximated Tweedie distribution for the beta-divergence family. Selecting the best beta then becomes a machine learning problem solved by maximum likelihood. Next, we reformulate alpha-divergence in terms of beta-divergence, which enables automatic selection of alpha by maximum likelihood with reuse of the learning principle for beta-divergence. Furthermore, we show the connections between gamma and beta-divergences as well as R\'enyi and alpha-divergences, such that our automatic selection framework is extended to non-separable divergences. Experiments on both synthetic and real-world data demonstrate that our method can quite accurately select information divergence across different learning problems and various divergence families.Comment: 12 pages, 7 figure

Faster Stochastic Variational Inference using Proximal-Gradient Methods with General Divergence Functions

Several recent works have explored stochastic gradient methods for variational inference that exploit the geometry of the variational-parameter space. However, the theoretical properties of these methods are not well-understood and these methods typically only apply to conditionally-conjugate models. We present a new stochastic method for variational inference which exploits the geometry of the variational-parameter space and also yields simple closed-form updates even for non-conjugate models. We also give a convergence-rate analysis of our method and many other previous methods which exploit the geometry of the space. Our analysis generalizes existing convergence results for stochastic mirror-descent on non-convex objectives by using a more general class of divergence functions. Beyond giving a theoretical justification for a variety of recent methods, our experiments show that new algorithms derived in this framework lead to state of the art results on a variety of problems. Further, due to its generality, we expect that our theoretical analysis could also apply to other applications.Comment: Published in UAI 2016. We have made the following change in this revision: instead of expressing convergence rate results in terms of the iterate difference, we state them in terms of the iterate distance divided by the step-size (a measure of first-order optimality). We also removed some claims about the performance with a fixed step siz

Unbiased Estimation Equation under ff-Separable Bregman Distortion Measures

We discuss unbiased estimation equations in a class of objective function using a monotonically increasing function $f$ and Bregman divergence. The choice of the function $f$ gives desirable properties such as robustness against outliers. In order to obtain unbiased estimation equations, analytically intractable integrals are generally required as bias correction terms. In this study, we clarify the combination of Bregman divergence, statistical model, and function $f$ in which the bias correction term vanishes. Focusing on Mahalanobis and Itakura-Saito distances, we provide a generalization of fundamental existing results and characterize a class of distributions of positive reals with a scale parameter, which includes the gamma distribution as a special case. We discuss the possibility of latent bias minimization when the proportion of outliers is large, which is induced by the extinction of the bias correction term