# A Characteristic Function Approach to Deep Implicit Generative Modeling

arXiv: https://arxiv.org/abs/1909.07425 · HTML: https://mlsanity.com/paper/1909.07425/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Sep 2019.

## Built on by (most cited first, 26 of 26)

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- [Learning Task-relevant Representations for Generalization via Characteristic Functions of Reward Sequence Distributions](https://mlsanity.com/paper/2205.10218/) (2022)
- [PCF-GAN: generating sequential data via the characteristic function of measures on the path space](https://mlsanity.com/paper/2305.12511/) (2023)
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- [The Fourier Discrepancy Function](https://mlsanity.com/paper/2102.02979/) (2021)
- [Rethinking Multidimensional Discriminator Output for Generative Adversarial Networks](https://mlsanity.com/paper/2109.03378/) (2021)
- [CF-GO-Net: A Universal Distribution Learner via Characteristic Function Networks with Graph Optimizers](https://mlsanity.com/paper/2409.12610/) (2024)
- [ReLU integral probability metric and its applications](https://mlsanity.com/paper/2504.18897/) (2025)
- [Learning Adaptive Distribution Alignment with Neural Characteristic Function for Graph Domain Adaptation](https://mlsanity.com/paper/2602.10489/) (2026)
- [Simple and Effective Specialized Representations for Fair Classifiers](https://mlsanity.com/paper/2505.11740/) (2025)
- [CFL: On the Use of Characteristic Function Loss for Domain Alignment in Machine Learning](https://mlsanity.com/paper/2511.02148/) (2025)
- [Towards Principled Dataset Distillation: A Spectral Distribution Perspective](https://mlsanity.com/paper/2603.01698/) (2026)
- [Drift, Then Spread: Learning One-Step Generative Policies for Online Reinforcement Learning](https://mlsanity.com/paper/2610.15662/) (2026)

## Abstract (arXiv)

Implicit Generative Models (IGMs) such as GANs have emerged as effective data-driven models for generating samples, particularly images. In this paper, we formulate the problem of learning an IGM as minimizing the expected distance between characteristic functions. Specifically, we minimize the distance between characteristic functions of the real and generated data distributions under a suitably-chosen weighting distribution. This distance metric, which we term as the characteristic function distance (CFD), can be (approximately) computed with linear time-complexity in the number of samples, in contrast with the quadratic-time Maximum Mean Discrepancy (MMD). By replacing the discrepancy measure in the critic of a GAN with the CFD, we obtain a model that is simple to implement and stable to train. The proposed metric enjoys desirable theoretical properties including continuity and differentiability with respect to generator parameters, and continuity in the weak topology. We further propose a variation of the CFD in which the weighting distribution parameters are also optimized during training; this obviates the need for manual tuning, and leads to an improvement in test power relative to CFD. We demonstrate experimentally that our proposed method outperforms WGAN and MMD-GAN variants on a variety of unsupervised image generation benchmarks.