Robust training of implicit generative models for multivariate and heavy-tailed distributions with an invariant statistical loss

Implicit generative models are often trained adversarially, which can yield unstable dynamics and mode collapse. The invariant statistical loss (ISL) offers a fully sample-based alternative by comparing empirical ranks of real and generated samples. In this work, we formally characterize ISL as a proper divergence over continuous distributions and establish key regularity properties, showing that it is continuous and differentiable, thereby enabling stable gradient-based optimization without adversarial games. We further enhance ISL along two practical axes. First, to better model heavy-tailed data, where Gaussian latent priors can limit tail expressivity, we introduce Pareto-ISL, which replaces Gaussian noise with a generalized Pareto latent distribution to improve the representation of both typical and extreme events. Second, to handle multivariate data at scale, we propose ISL-slicing: a computationally efficient procedure that projects samples onto random one-dimensional subspaces, computes rank-based losses per projection, and averages them to capture high-dimensional structure. Experiments demonstrate improved tail fidelity with Pareto-ISL and show that ISL-slicing scales effectively to high dimensions. Specifically, in high dimensional settings we show that ISL can be used either as a standalone criterion or as a strong pretraining objective for subsequent adversarial fine-tuning.