FGM-HD: Boosting Generation Diversity of Fractal Generative Models through Hausdorff Dimension Induction

2025-11-12

AI Summary

Fractal generation models (FGMs) struggle to simultaneously achieve high diversity and visual fidelity. Method: This paper introduces the Hausdorff dimension (HD)—a geometric measure of structural complexity—as a learnable prior into FGMs for the first time. We propose a learnable HD estimator and an HD-aware end-to-end training and inference framework: (i) during training, we design an HD-augmented hybrid loss and a momentum-driven hyperparameter scheduling strategy; (ii) during inference, we incorporate HD-guided rejection sampling. The approach requires no additional annotations or architectural modifications. Contribution/Results: Our method significantly enhances controllability over structural complexity and improves diversity representation. On ImageNet, it achieves a 39% increase in generation diversity while maintaining competitive Fréchet Inception Distance (FID). These results validate HD as an effective geometric prior for fractal generation modeling and establish a new paradigm for geometry-aware generative learning.

Technology Category

• Computer Vision: Generative Adversarial Networks (GANs) for Vision
• Machine Learning: Deep Generative Models & Autoencoders
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization
• Web Mining and Content Analysis: Web data generation and simulation

Abstract

Improving the diversity of generated results while maintaining high visual quality remains a significant challenge in image generation tasks. Fractal Generative Models (FGMs) are efficient in generating high-quality images, but their inherent self-similarity limits the diversity of output images. To address this issue, we propose a novel approach based on the Hausdorff Dimension (HD), a widely recognized concept in fractal geometry used to quantify structural complexity, which aids in enhancing the diversity of generated outputs. To incorporate HD into FGM, we propose a learnable HD estimation method that predicts HD directly from image embeddings, addressing computational cost concerns. However, simply introducing HD into a hybrid loss is insufficient to enhance diversity in FGMs due to: 1) degradation of image quality, and 2) limited improvement in generation diversity. To this end, during training, we adopt an HD-based loss with a monotonic momentum-driven scheduling strategy to progressively optimize the hyperparameters, obtaining optimal diversity without sacrificing visual quality. Moreover, during inference, we employ HD-guided rejection sampling to select geometrically richer outputs. Extensive experiments on the ImageNet dataset demonstrate that our FGM-HD framework yields a 39% improvement in output diversity compared to vanilla FGMs, while preserving comparable image quality. To our knowledge, this is the very first work introducing HD into FGM. Our method effectively enhances the diversity of generated outputs while offering a principled theoretical contribution to FGM development.

Problem

• Enhancing fractal generative model diversity while maintaining visual quality
• Addressing self-similarity limitations in fractal-based image generation
• Incorporating Hausdorff dimension to quantify and improve structural complexity

Innovation

• Hausdorff Dimension estimation from image embeddings
• Monotonic momentum scheduling for hyperparameter optimization
• HD-guided rejection sampling during inference phase