Data usually resides on a manifold, and the minimal dimension of such a manifold is called its intrinsic dimension . This fundamental data property is not considered in the generative adversarial network (GAN) model along with its its variants; such that original data and generated data often hold different intrinsic dimensions . The different intrinsic dimensions of both generated and original data may cause generated data distribution to not match original data distribution completely, and it certainly will hurt the quality of generated data. In this study, we first show that GAN is often unable to generate simulation data, holding the same intrinsic dimension as the original data with both theoretical analysis and experimental illustration. Next, we propose a new model, called Hausdorff GAN, which removes the issue of different intrinsic dimensions and introduces the Hausdorff metric into GAN training to generate higher quality data. This provides new insights into the success of Hausdorff GAN. Specifically, we utilize a mapping function to map both original and generated data into the same manifold. We then calculate the Hausdorff distance to measure the difference between the mapped original data and the mapped generated data, toward pushing generated data to the side of original data. Finally, we conduct extensive experiments (using MNIST, CIFAR10, and CelebA datasets) to demonstrate the significant performance improvement of the Hausdorff GAN in achieving the largest Inception Score and the smallest Frechet inception distance (FID) score as well as producing diverse generated data at different resolutions.

This paper researches the identification problem for the unknown parameters of the multivariate equation-error autoregressive systems. Firstly, the original identification model is decomposed into several sub-identification models according to the number of system outputs. Then, based on the characteristic that the information vector and the parameter vector are common among the sub-identification models, the coupling identification concept is used to propose a partially coupled generalized stochastic gradient algorithm. Furthermore, by expanding the scalar innovation of each subsystem model to the innovation vector, a partially coupled multi-innovation generalized stochastic gradient algorithm is proposed. Finally, the numerical simulations indicate that the proposed algorithms are effective and have good parameter estimation performances.