# Data Interpolants -- That's What Discriminators in Higher-order Gradient-regularized GANs Are

arXiv: https://arxiv.org/abs/2306.00785 · HTML: https://mlsanity.com/paper/2306.00785/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 48 of its 75 references that are in the map.
- First posted on arXiv 1 Jun 2023.

## Builds on (most cited first, 30 of 48)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Rethinking the Inception Architecture for Computer Vision](https://mlsanity.com/paper/1512.00567/) (2015)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Large Scale GAN Training for High Fidelity Natural Image Synthesis](https://mlsanity.com/paper/1809.11096/) (2018)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks](https://mlsanity.com/paper/1511.06434/) (2015)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [LSUN: Construction of a Large-scale Image Dataset using Deep Learning with Humans in the Loop](https://mlsanity.com/paper/1506.03365/) (2015)
- [Demystifying MMD GANs](https://mlsanity.com/paper/1801.01401/) (2018)
- [Training Generative Adversarial Networks with Limited Data](https://mlsanity.com/paper/2006.06676/) (2020)
- [Alias-Free Generative Adversarial Networks](https://mlsanity.com/paper/2106.12423/) (2021)
- [Least Squares Generative Adversarial Networks](https://mlsanity.com/paper/1611.04076/) (2016)
- [Which Training Methods for GANs do actually Converge?](https://mlsanity.com/paper/1801.04406/) (2018)
- [TensorFlow: Large-Scale Machine Learning on Heterogeneous Distributed Systems](https://mlsanity.com/paper/1603.04467/) (2016)
- [Vector-quantized Image Modeling with Improved VQGAN](https://mlsanity.com/paper/2110.04627/) (2021)
- [f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization](https://mlsanity.com/paper/1606.00709/) (2016)
- [Towards Principled Methods for Training Generative Adversarial Networks](https://mlsanity.com/paper/1701.04862/) (2017)
- [Wasserstein Auto-Encoders](https://mlsanity.com/paper/1711.01558/) (2017)
- [Generative Moment Matching Networks](https://mlsanity.com/paper/1502.02761/) (2015)
- [MMD GAN: Towards Deeper Understanding of Moment Matching Network](https://mlsanity.com/paper/1705.08584/) (2017)
- [Training GANs with Optimism](https://mlsanity.com/paper/1711.00141/) (2017)
- [The Cramer Distance as a Solution to Biased Wasserstein Gradients](https://mlsanity.com/paper/1705.10743/) (2017)
- [Stabilizing Training of Generative Adversarial Networks through Regularization](https://mlsanity.com/paper/1705.09367/) (2017)
- [How Well Generative Adversarial Networks Learn Distributions](https://mlsanity.com/paper/1811.03179/) (2018)

## Built on by (most cited first, 2 of 2)

- [Y-Diagonal Couplings: Approximating Posteriors with Conditional Wasserstein Distances](https://mlsanity.com/paper/2310.13433/) (2023)
- [Insights into Closed-form IPM-GAN Discriminator Guidance for Diffusion Modeling](https://mlsanity.com/paper/2306.01654/) (2023)

## Abstract (arXiv)

We consider the problem of optimizing the discriminator in generative adversarial networks (GANs) subject to higher-order gradient regularization. We show analytically, via the least-squares (LSGAN) and Wasserstein (WGAN) GAN variants, that the discriminator optimization problem is one of interpolation in n-dimensions. The optimal discriminator, derived using variational Calculus, turns out to be the solution to a partial differential equation involving the iterated Laplacian or the polyharmonic operator. The solution is implementable in closed-form via polyharmonic radial basis function (RBF) interpolation. In view of the polyharmonic connection, we refer to the corresponding GANs as Poly-LSGAN and Poly-WGAN. Through experimental validation on multivariate Gaussians, we show that implementing the optimal RBF discriminator in closed-form, with penalty orders m ≈lceil fracn2 rceil , results in superior performance, compared to training GAN with arbitrarily chosen discriminator architectures. We employ the Poly-WGAN discriminator to model the latent space distribution of the data with encoder-decoder-based GAN flavors such as Wasserstein autoencoders.