Probability density estimation from observed data constitutes a central task in statistics. In this brief, we focus on the problem of estimating the copula density associated with any observed data, as it fully describes the dependence between random variables. We separate univariate marginal distributions from the joint dependence structure in the data, the copula itself, and we model the latter with a neural network-based method referred to as copula density neural estimation (CODINE). Results show that the novel learning approach is capable of modeling complex distributions and can be applied for mutual information estimation and data generation.

Estimating mutual information accurately is pivotal across diverse applications, from machine learning to communications and biology, enabling us to gain insights into the inner mechanisms of complex systems. Yet, dealing with high-dimensional data presents a formidable challenge, due to its size and the presence of intricate relationships. Recently proposed neural methods employing variational lower bounds on the mutual information have gained prominence. However, these approaches suffer from either high bias or high variance, as the sample size and the structure of the loss function directly influence the training process. In this paper, we propose a novel class of discriminative mutual information estimators based on the variational representation of the $f$-divergence. We investigate the impact of the permutation function used to obtain the marginal training samples and present a novel architectural solution based on derangements. The proposed estimator is flexible since it exhibits an excellent bias/variance trade-off. The comparison with state-of-the-art neural estimators, through extensive experimentation within established reference scenarios, shows that our approach offers higher accuracy and lower complexity.

Recent advancements in generative networks have shown that it is possible to produce real-world-like data using deep neural networks. Some implicit probabilistic models that follow a stochastic procedure to directly generate data have been introduced to overcome the intractability of the posterior distribution. However, the ability to model data requires deep knowledge and understanding of its statistical dependence-which can be preserved and studied in appropriate latent spaces. In this article, we present a segmented generation process through linear and nonlinear manipulations in the same-dimensional latent space where data are projected to. Inspired by the known stochastic method to generate correlated data, we develop a segmented approach for the generation of dependent data, exploiting the concept of copula. The generation process is split into two frames: one embedding the covariance or copula information in the uniform probability space, and the other embedding the marginal distribution information in the sample domain. The proposed network structure, referred to as a segmented generative network (SGN), also provides an empirical method to sample directly from implicit copulas. To show its generality, we evaluate the presented approach in three application scenarios: a toy example, handwritten digits, and face image generation.

This article presents a novel recursive smooth trajectory (RST) generation algorithm for application in robotics and in particular for unmanned aerial vehicles (UAVs). RST builds the trajectory recursively as a smooth polynomial path, thus a closed form trajectory satisfying any arbitrary dynamic limitation that translates into kinematic constraints (e.g., position, velocity, acceleration, etc.). Uncertainties and perturbations in the constraints are also discussed, modeled, and finally included in the calculation of the polynomial trajectory coefficients. Moreover, due to its recursive formulation, RST enables an immediate extension toward the solution of a trajectory optimization problem. In particular, it is shown that the minimum-snap piecewise polynomial trajectory can be interpreted as a special case of RST. The effectiveness of the proposed algorithm is demonstrated numerically via two illustrative scenarios. Its application to a UAV structure is also discussed to highlight the advantage of a smooth path over a piecewise one. Finally, the computational complexity and memory requirements are analyzed.