Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generative adversarial learning with optimal input dimension and its adaptive generator architecture

Zhiyao Tan, Ling Zhou, Huazhen Lin

TL;DR

The paper addresses how input dimension affects GAN generalization, identifying a minimal input dimension $d_0$ (MID) that minimizes IPM-based risk and proposing generalized GANs (G-GANs) that learn a dimension-reducing matrix $\mathbf B$ plus a compact generator via group sparsity and architecture penalties. It provides a risk decomposition and proves selection consistency for the OID and generator architecture (Theorem $t2$), with the optimal rate achieved when $m>n$ and $d=d_0$. Empirically, G-GANs outperform standard GANs on CT slices, MNIST, and FashionMNIST, identifying interpretable low-dimensional input directions in $\mathbf B z$ and substantially reducing generator complexity. These results have practical implications for stable, efficient GAN training and offer a principled approach to adaptively determine network structure aligned with the data's intrinsic dimensionality.

Abstract

We investigate the impact of the input dimension on the generalization error in generative adversarial networks (GANs). In particular, we first provide both theoretical and practical evidence to validate the existence of an optimal input dimension (OID) that minimizes the generalization error. Then, to identify the OID, we introduce a novel framework called generalized GANs (G-GANs), which includes existing GANs as a special case. By incorporating the group penalty and the architecture penalty developed in the paper, G-GANs have several intriguing features. First, our framework offers adaptive dimensionality reduction from the initial dimension to a dimension necessary for generating the target distribution. Second, this reduction in dimensionality also shrinks the required size of the generator network architecture, which is automatically identified by the proposed architecture penalty. Both reductions in dimensionality and the generator network significantly improve the stability and the accuracy of the estimation and prediction. Theoretical support for the consistent selection of the input dimension and the generator network is provided. Third, the proposed algorithm involves an end-to-end training process, and the algorithm allows for dynamic adjustments between the input dimension and the generator network during training, further enhancing the overall performance of G-GANs. Extensive experiments conducted with simulated and benchmark data demonstrate the superior performance of G-GANs. In particular, compared to that of off-the-shelf methods, G-GANs achieves an average improvement of 45.68% in the CT slice dataset, 43.22% in the MNIST dataset and 46.94% in the FashionMNIST dataset in terms of the maximum mean discrepancy or Frechet inception distance. Moreover, the features generated based on the input dimensions identified by G-GANs align with visually significant features.

Generative adversarial learning with optimal input dimension and its adaptive generator architecture

TL;DR

The paper addresses how input dimension affects GAN generalization, identifying a minimal input dimension (MID) that minimizes IPM-based risk and proposing generalized GANs (G-GANs) that learn a dimension-reducing matrix plus a compact generator via group sparsity and architecture penalties. It provides a risk decomposition and proves selection consistency for the OID and generator architecture (Theorem ), with the optimal rate achieved when and . Empirically, G-GANs outperform standard GANs on CT slices, MNIST, and FashionMNIST, identifying interpretable low-dimensional input directions in and substantially reducing generator complexity. These results have practical implications for stable, efficient GAN training and offer a principled approach to adaptively determine network structure aligned with the data's intrinsic dimensionality.

Abstract

We investigate the impact of the input dimension on the generalization error in generative adversarial networks (GANs). In particular, we first provide both theoretical and practical evidence to validate the existence of an optimal input dimension (OID) that minimizes the generalization error. Then, to identify the OID, we introduce a novel framework called generalized GANs (G-GANs), which includes existing GANs as a special case. By incorporating the group penalty and the architecture penalty developed in the paper, G-GANs have several intriguing features. First, our framework offers adaptive dimensionality reduction from the initial dimension to a dimension necessary for generating the target distribution. Second, this reduction in dimensionality also shrinks the required size of the generator network architecture, which is automatically identified by the proposed architecture penalty. Both reductions in dimensionality and the generator network significantly improve the stability and the accuracy of the estimation and prediction. Theoretical support for the consistent selection of the input dimension and the generator network is provided. Third, the proposed algorithm involves an end-to-end training process, and the algorithm allows for dynamic adjustments between the input dimension and the generator network during training, further enhancing the overall performance of G-GANs. Extensive experiments conducted with simulated and benchmark data demonstrate the superior performance of G-GANs. In particular, compared to that of off-the-shelf methods, G-GANs achieves an average improvement of 45.68% in the CT slice dataset, 43.22% in the MNIST dataset and 46.94% in the FashionMNIST dataset in terms of the maximum mean discrepancy or Frechet inception distance. Moreover, the features generated based on the input dimensions identified by G-GANs align with visually significant features.
Paper Structure (9 sections, 4 theorems, 17 equations, 4 figures, 4 tables, 2 algorithms)

This paper contains 9 sections, 4 theorems, 17 equations, 4 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Theoretical results
  3. Generalized GANs
  4. Implementation
  5. Experiments
  6. Numerical simulation
  7. The CT slice dataset
  8. MNIST and FashionMNIST
  9. Conclusion

Key Result

Proposition 2.1

Let us assume $\mathcal{F}$ is symmetric, i.e., $\phi \in \mathcal{F}$ holds if and only if $-\phi \in \mathcal{F}$. Let $g^*$ and $\hat{g}$ be the population and the empirical GAN estimators f1 and f2, respectively. Then, for any evaluation class $\mathcal{H}$, it holds that

Figures (4)

  • Figure 1: The mean (blue solid line) and standard deviation (Std, orange dotted line) of Maximum Mean Discrepancy (MMD) of SNGANs miyato2018spectral based on 10 replications with varying input dimensions and corresponding generator architectures, where $d$-$l\times w$ indicates the generator with the input dimension $d$, depth $l$ and width $w$. (M1)-(M4) refer to four numerical simulations, described in detail in Section \ref{['sec:exper']}.
  • Figure 2: Observed images (a) and generated images (b) -- (e) by WGAN-GP, G-GAN$^W$, SNGAN and G-GAN$^{SN}$, respectively for MNIST.
  • Figure 3: Observed images (a) and generated images (b)--(e) by WGAN-GP, G-GAN$^W$, SNGAN and G-GAN$^{SN}$, respectively, for FashionMNIST.
  • Figure 4: The manipulation of input variables in the MNIST and FashionMNIST datasets. Each block in the figures corresponds to the traversal of a single input variable while keeping the others fixed. Each row in the figures represents a different image. The traversal is conducted within the range of [-2, 2]. In MNIST, the selected input variables correspond to the thickness of the digit (a) and the angle of inclination of the digit (b). While in FashionMNIST, the selected input variables correspond to the fabric quantity (c) and the clothing style (d).

Theorems & Definitions (6)

  • Proposition 2.1
  • Definition 2.1
  • Definition 2.2: Minimal input dimension
  • Theorem 2.1
  • Corollary 2.1
  • Theorem 3.1