Normalizing flows map an independent set of latent variables to their samples using a bijective transformation. Despite the exact correspondence between samples and latent variables, their high level relationship is not well understood. In this paper we characterize the geometric structure of flows using principal manifolds and understand the relationship between latent variables and samples using contours. We introduce a novel class of normalizing flows, called principal manifold flows (PF), whose contours are its principal manifolds, and a variant for injective flows (iPF) that is more efficient to train than regular injective flows. PFs can be constructed using any flow architecture, are trained with a regularized maximum likelihood objective and can perform density estimation on all of their principal manifolds. In our experiments we show that PFs and iPFs are able to learn the principal manifolds over a variety of datasets. Additionally, we show that PFs can perform density estimation on data that lie on a manifold with variable dimensionality, which is not possible with existing normalizing flows.

In former articles, we have been looking at many different concepts related to Bayesian Networks: how to represent them in a compact manner, their reasoning patterns, and I-Map. In this article, we will take a look at D-separation and try to understand it both intuitively and mathematically.

Convolutional Neural Networks (CNNs) are artificial learning systems typically based on two operations: convolution, which implements feature extraction through filtering, and pooling, which implements dimensionality reduction. The impact of pooling in the classification performance of the CNNs has been highlighted in several previous works, and a variety of alternative pooling operators have been proposed. However, only a few of them tackle with the uncertainty that is naturally propagated from the input layer to the feature maps of the hidden layers through convolutions. In this paper we present a novel pooling operation based on (type-1) fuzzy sets to cope with the local imprecision of the feature maps, and we investigate its performance in the context of image classification. Fuzzy pooling is performed by fuzzification, aggregation and defuzzification of feature map neighborhoods. It is used for the construction of a fuzzy pooling layer that can be applied as a drop-in replacement of the current, crisp, pooling layers of CNN architectures. Several experiments using publicly available datasets show that the proposed approach can enhance the classification performance of a CNN. A comparative evaluation shows that it outperforms state-of-the-art pooling approaches.

Variational inference (VI) is a technique to approximate difficult to compute posteriors by optimization. In contrast to MCMC, VI scales to many observations. In the case of complex posteriors, however, state-of-the-art VI approaches often yield unsatisfactory posterior approximations. This paper presents Bernstein flow variational inference (BF-VI), a robust and easy-to-use method, flexible enough to approximate complex multivariate posteriors. BF-VI combines ideas from normalizing flows and Bernstein polynomial-based transformation models. In benchmark experiments, we compare BF-VI solutions with exact posteriors, MCMC solutions, and state-of-the-art VI methods including normalizing flow based VI. We show for low-dimensional models that BF-VI accurately approximates the true posterior; in higher-dimensional models, BF-VI outperforms other VI methods. Further, we develop with BF-VI a Bayesian model for the semi-structured Melanoma challenge data, combining a CNN model part for image data with an interpretable model part for tabular data, and demonstrate for the first time how the use of VI in semi-structured models.

We establish new generalisation bounds for multiclass classification by abstracting to a more general setting of discretised error types. Extending the PAC-Bayes theory, we are hence able to provide fine-grained bounds on performance for multiclass classification, as well as applications to other learning problems including discretisation of regression losses. Tractable training objectives are derived from the bounds. The bounds are uniform over all weightings of the discretised error types and thus can be used to bound weightings not foreseen at training, including the full confusion matrix in the multiclass classification case.

In this paper, we study the long-time convergence and uniform strong propagation of chaos for a class of nonlinear Markov chains for Markov chain Monte Carlo (MCMC). Our technique is quite simple, making use of recent contraction estimates for linear Markov kernels and basic techniques from Markov theory and analysis. Moreover, the same proof strategy applies to both the long-time convergence and propagation of chaos. We also show, via some experiments, that these nonlinear MCMC techniques are viable for use in real-world high-dimensional inference such as Bayesian neural networks.

The (probabilistic) graphical model consists of a collection of probability distributions that factorize according to the structure of an underlying graph [52]. The graphical model captures the complex dependencies among random variables and build large-scale multivariate statistical models, which has been used in many research areas such as hierarchical Bayesian models [27], contingency table analysis [20, 53] in categorical data analysis [1, 23, 37], constraint satisfaction [16, 15], language and speech processing [11, 31], image processing [17, 24, 28] and spatial statistics more generally [8]. In our work, we focus on the undirected graphical models, where the probability distribution factorizes according to the function defined on the cliques of the graph. The undirected graphical models have a variety of applications, including statistical physics [32], natural language processing [38], image analysis [54] and spatial statistics [43]. Specifically, we pay attention to the undirected graphical models which can be described as exponential families, a broad class of probability distributions elaborately studied in many statistical literature [4, 21, 13]. The properties of the exponential families provide some connections between the inference methods and the convex analysis [12, 29]. There are many well-known examples that are undirected graphical models viewed as exponential families, such as Ising model [32, 5], Gaussian MRF [46] and latent Dirichlet allocation [11].

We will understand how our assumptions on the data enable us to create meaningful optimization problems. In fact, we will derive commonly used criteria such as cross-entropy in classification and mean square error in regression. Finally, I am trying to answer an interview question that I encountered: What would happen if we use MSE on binary classification? To begin, let's start with a fundamental question: what is the difference between likelihood and probability? The data xxx are connected to the possible models θ\thetaθ by means of a probability P(x,θ)P(x,\theta)P(x,θ) or a probability density function (pdf) p(x,θ)p(x,\theta)p(x,θ).

Unresolved data association in ambiguous and perceptually aliased environments leads to multi-modal hypotheses on both the robot's and the environment state. To avoid catastrophic results, when operating in such ambiguous environments, it is crucial to reason about data association within Belief Space Planning (BSP). However, explicitly considering all possible data associations, the number of hypotheses grows exponentially with the planning horizon and determining the optimal action sequence quickly becomes intractable. Moreover, with hard budget constraints where some non-negligible hypotheses must be pruned, achieving performance guarantees is crucial. In this work we present a computationally efficient novel approach that utilizes only a distilled subset of hypotheses to solve BSP problems while reasoning about data association. Furthermore, to provide performance guarantees, we derive error bounds with respect to the optimal solution. We then demonstrate our approach in an extremely aliased environment, where we manage to significantly reduce computation time without compromising on the quality of the solution.

In a first part, we present a mathematical analysis of a general methodology of a probabilistic learning inference that allows for estimating a posterior probability model for a stochastic boundary value problem from a prior probability model. The given targets are statistical moments for which the underlying realizations are not available. Under these conditions, the Kullback-Leibler divergence minimum principle is used for estimating the posterior probability measure. A statistical surrogate model of the implicit mapping, which represents the constraints, is introduced. The MCMC generator and the necessary numerical elements are given to facilitate the implementation of the methodology in a parallel computing framework. In a second part, an application is presented to illustrate the proposed theory and is also, as such, a contribution to the three-dimensional stochastic homogenization of heterogeneous linear elastic media in the case of a non-separation of the microscale and macroscale. For the construction of the posterior probability measure by using the probabilistic learning inference, in addition to the constraints defined by given statistical moments of the random effective elasticity tensor, the second-order moment of the random normalized residue of the stochastic partial differential equation has been added as a constraint. This constraint guarantees that the algorithm seeks to bring the statistical moments closer to their targets while preserving a small residue.