Explicit Density Approximation for Neural Implicit Samplers Using a Bernstein-Based Convex Divergence

• URL: http://arxiv.org/abs/2506.04700v2
• Date: Wed, 05 Nov 2025 20:10:41 GMT
• Title: Explicit Density Approximation for Neural Implicit Samplers Using a Bernstein-Based Convex Divergence
• Authors: José Manuel de Frutos, Manuel A. Vázquez, Pablo M. Olmos, Joaquín Míguez,
• Abstract summary: We introduce dual-ISL, a novel likelihood-free objective for training implicit generative models.<n>We show that these theoretical advantages translate into practical ones.
• Score: 6.110760886913874
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Rank-based statistical metrics, such as the invariant statistical loss (ISL), have recently emerged as robust and practically effective tools for training implicit generative models. In this work, we introduce dual-ISL, a novel likelihood-free objective for training implicit generative models that interchanges the roles of the target and model distributions in the ISL framework, yielding a convex optimization problem in the space of model densities. We prove that the resulting rank-based discrepancy $d_K$ is i) continuous under weak convergence and with respect to the $L^1$ norm, and ii) convex in its first argument-properties not shared by classical divergences such as KL or Wasserstein distances. Building on this, we develop a theoretical framework that interprets $d_K$ as an $L^2$-projection of the density ratio $q = p/\tilde p$ onto a Bernstein polynomial basis, from which we derive exact bounds on the truncation error, precise convergence rates, and a closed-form expression for the truncated density approximation. We further extend our analysis to the multivariate setting via random one-dimensional projections, defining a sliced dual-ISL divergence that retains both convexity and continuity. We empirically show that these theoretical advantages translate into practical ones. Specifically, across several benchmarks dual-ISL converges more rapidly, delivers markedly smoother and more stable training, and more effectively prevents mode collapse than classical ISL and other leading implicit generative methods-while also providing an explicit density approximation.

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