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We propose a novel GAN framework for non-parametric density estimation with high-dimensional data. This framework is based on a novel density estimator, called the hyperbolic cross density estimator, which enjoys nice convergence properties in the mixed Sobolev spaces. As modifications of the usual Sobolev spaces, the mixed Sobolev spaces are more suitable for describing high-dimensional density functions. We prove that, unlike other existing approaches, the proposed GAN framework does not suffer the curse of dimensionality and can achieve the optimal convergence rate of $O_p(n^{-1/2})$, with $n$ data points in an arbitrary fixed dimension. We also study the universality of GANs in terms of the existence of ReLU networks which can approximate the density functions in the mixed Sobolev spaces up to any accuracy level.
Adversarial training is a popular method to give neural nets robustness against adversarial perturbations. In practice adversarial training leads to low robust training loss. However, a rigorous explanation for why this happens under natural conditio
We undertake a precise study of the non-asymptotic properties of vanilla generative adversarial networks (GANs) and derive theoretical guarantees in the problem of estimating an unknown $d$-dimensional density $p^*$ under a proper choice of the class
We study minimax density estimation on the product space $mathbb{R}^{d_1}timesmathbb{R}^{d_2}$. We consider $L^p$-risk for probability density functions defined over regularity spaces that allow for different level of smoothness in each of the variab
In this paper, we construct neural networks with ReLU, sine and $2^x$ as activation functions. For general continuous $f$ defined on $[0,1]^d$ with continuity modulus $omega_f(cdot)$, we construct ReLU-sine-$2^x$ networks that enjoy an approximation
Density ratio estimation serves as an important technique in the unsupervised machine learning toolbox. However, such ratios are difficult to estimate for complex, high-dimensional data, particularly when the densities of interest are sufficiently di