Generative adversarial networks are a novel method for statistical inference that have achieved much empirical success; however, the factors contributing to this success remain ill-understood. In this work, we attempt to analyze generative adversarial learning -- that is, statistical inference as the result of a game between a generator and a discriminator -- with the view of understanding how it differs from classical statistical inference solutions such as maximum likelihood inference and the method of moments. Specifically, we provide a theoretical characterization of the distribution inferred by a simple form of generative adversarial learning called restricted f-GANs -- where the discriminator is a function in a given function class, the distribution induced by the generator is restricted to lie in a pre-specified distribution class and the objective is similar to a variational form of the f-divergence. A consequence of our result is that for linear KL-GANs -- that is, when the discriminator is a linear function over some feature space and f corresponds to the KL-divergence -- the distribution induced by the optimal generator is neither the maximum likelihood nor the method of moments solution, but an interesting combination of both.

We present a principled Bayesian framework for signal reconstruction, in which the signal is modelled by basis functions whose number (and form, if required) is determined by the data themselves. This approach is based on a Bayesian interpretation of conventional sparse reconstruction and regularisation techniques, in which sparsity is imposed through priors via Bayesian model selection. We demonstrate our method for noisy 1- and 2-dimensional signals, including astronomical images. Furthermore, by using a product-space approach, the number and type of basis functions can be treated as integer parameters and their posterior distributions sampled directly. We show that order-of-magnitude increases in computational efficiency are possible from this technique compared to calculating the Bayesian evidences separately, and that further computational gains are possible using it in combination with dynamic nested sampling. Our approach can be readily applied to neural networks, where it allows the network architecture to be determined by the data in a principled Bayesian manner by treating the number of nodes and hidden layers as parameters.

While Gaussian processes (GPs) are the method of choice for regression tasks, they also come with practical difficulties, as inference cost scales cubic in time and quadratic in memory. In this paper, we introduce a natural and expressive way to tackle these problems, by incorporating GPs in sum-product networks (SPNs), a recently proposed tractable probabilistic model allowing exact and efficient inference. In particular, by using GPs as leaves of an SPN we obtain a novel flexible prior over functions, which implicitly represents an exponentially large mixture of local GPs. Exact and efficient posterior inference in this model can be done in a natural interplay of the inference mechanisms in GPs and SPNs. Thereby, each GP is -- similarly as in a mixture of experts approach -- responsible only for a subset of data points, which effectively reduces inference cost in a divide and conquer fashion. We show that integrating GPs into the SPN framework leads to a promising probabilistic regression model which is: (1) computational and memory efficient, (2) allows efficient and exact posterior inference, (3) is flexible enough to mix different kernel functions, and (4) naturally accounts for non-stationarities in time series. In a variate of experiments, we show that the SPN-GP model can learn input dependent parameters and hyper-parameters and is on par with or outperforms the traditional GPs as well as state of the art approximations on real-world data.

Parameterizing the approximate posterior of a generative model with neural networks has become a common theme in recent machine learning research. While providing appealing flexibility, this approach makes it difficult to impose or assess structural constraints such as conditional independence. We propose a framework for learning representations that relies on Auto-Encoding Variational Bayes and whose search space is constrained via kernel-based measures of independence. In particular, our method employs the d-variable Hilbert-Schmidt Independence Criterion (dHSIC) to enforce independence between the latent representations and arbitrary nuisance factors. We show how to apply this method to a range of problems, including the problems of learning invariant representations and the learning of interpretable representations. We also present a full-fledged application to single-cell RNA sequencing (scRNA-seq). In this setting the biological signal is mixed in complex ways with sequencing errors and sampling effects. We show that our method outperforms the state-of-the-art in this domain.

We consider the problem of reconstructing the internal structure of an object from limited x-ray projections. In this work, we use a Gaussian process to model the target function. In contrast to other established methods, this comes with the advantage of not requiring any manual parameter tuning, which usually arises in classical regularization strategies. The Gaussian process is well-known in a heavy computation for the inversion of a covariance matrix, and in this work, by employing an approximative spectral-based technique, we reduce the computational complexity and avoid the need of numerical integration. Results from simulated and real data indicate that this approach is less sensitive to streak artifacts as compared to the commonly used method of filteredback projection, an analytic reconstruction algorithm using Radon inversion formula.

This paper addresses the problem of change-point detection on sequences of high-dimensional and heterogeneous observations, which also possess a periodic temporal structure. Due to the dimensionality problem, when the time between change-points is on the order of the dimension of the model parameters, drifts in the underlying distribution can be misidentified as changes. To overcome this limitation we assume that the observations lie in a lower dimensional manifold that admits a latent variable representation. In particular, we propose a hierarchical model that is computationally feasible, widely applicable to heterogeneous data and robust to missing instances. Additionally, to deal with the observations' periodic dependencies, we employ a circadian model where the data periodicity is captured by non-stationary covariance functions. We validate the proposed technique on synthetic examples and we demonstrate its utility in the detection of changes for human behavior characterization.

We introduce a quantum algorithm for solving structured-prediction problems with a runtime that scales with the square root of the size of the label space, but scales in $\widetilde O\left(\frac{1}{\epsilon^5}\right)$ with respect to the precision of the solution. In doing so, we analyze a stochastic gradient algorithm for convex optimization in the presence of an additive error in the calculation of the gradients, and show that its convergence rate does not deteriorate if the additive errors are of the order $\widetilde O(\epsilon)$. Our algorithm uses quantum Gibbs sampling at temperature $O (\epsilon)$ as a subroutine. Numerical results using Monte Carlo simulations on an image tagging task demonstrate the benefit of the approach.

The study of private inference has been sparked by growing concern regarding the analysis of data when it stems from sensitive sources. We present the first method for private Bayesian inference in exponential families that properly accounts for noise introduced by the privacy mechanism. It is efficient because it works only with sufficient statistics and not individual data. Unlike other methods, it gives properly calibrated posterior beliefs in the non-asymptotic data regime.

Group factor analysis (GFA) methods have been widely used to infer the common structure and the group-specific signals from multiple related datasets in various fields including systems biology and neuroimaging. To date, most available GFA models require Gibbs sampling or slice sampling to perform inference, which prevents the practical application of GFA to large-scale data. In this paper we present an efficient collapsed variational inference (CVI) algorithm for the nonparametric Bayesian group factor analysis (NGFA) model built upon an hierarchical beta Bernoulli process. Our CVI algorithm proceeds by marginalizing out the group-specific beta process parameters, and then approximating the true posterior in the collapsed space using mean field methods. Experimental results on both synthetic and real-world data demonstrate the effectiveness of our CVI algorithm for the NGFA compared with state-of-the-art GFA methods.

Machine Learning on graph-structured data is an important and omnipresent task for a vast variety of applications including anomaly detection and dynamic network analysis. In this paper, a deep generative model is introduced to capture continuous probability densities corresponding to the nodes of an arbitrary graph. In contrast to all learning formulations in the area of discriminative pattern recognition, we propose a scalable generative optimization/algorithm theoretically proved to capture distributions at the nodes of a graph. Our model is able to generate samples from the probability densities learned at each node. This probabilistic data generation model, i.e. convolutional graph auto-encoder (CGAE), is devised based on the localized first-order approximation of spectral graph convolutions, deep learning, and the variational Bayesian inference. We apply our CGAE to a new problem, the spatio-temporal probabilistic solar irradiance prediction. Multiple solar radiation measurement sites in a wide area in northern states of the US are modeled as an undirected graph. Using our proposed model, the distribution of future irradiance given historical radiation observations is estimated for every site/node. Numerical results on the National Solar Radiation Database show state-of-the-art performance for probabilistic radiation prediction on geographically distributed irradiance data in terms of reliability, sharpness, and continuous ranked probability score.