用因子结构提升扩散模型在小样本高维金融数据中的生成能力
Diffusion Factor Models: Generating High-Dimensional Returns with Factor Structure
- 将资产收益的低维因子结构融入扩散模型得分函数设计
- 理论证明误差随因子数k增长，而非资产数d，支持数千资产场景
- 适合风险管理和投资组合优化中的高维金融模拟需求
金融情景模拟对风险管理和投资组合优化至关重要，但在高维和小样本情形下仍具挑战性。我们提出一种扩散因子模型，将潜在因子结构融入生成式扩散过程，融合计量经济学与现代生成式AI，以应对维度灾难与数据稀缺问题。通过利用资产收益中固有的低维因子结构，我们采用时变正交投影分解扩散模型的关键得分函数，并将其嵌入神经网络架构设计。我们建立了严格的统计保证，证明得分估计误差为O(d^{5/2} n^{-2/(k+5)})，生成分布误差为O(d^{5/4} n^{-1/2(k+5)})，主要依赖内在因子数k而非资产数d，突破经典非参数统计学中依赖维度的限制，使该框架适用于含数千资产的市场。数值实验表明，在小样本条件下对隐含子空间的恢复性能优越。实证分析显示该框架在构建均值-方差最优投资组合与因子投资组合方面具有经济意义。本工作首次实现因子结构与扩散模型的理论整合，为有限数据下的高维金融模拟提供了一种严谨方法。代码已公开于https://github.com/xymmmm00/diffusion_factor_model。
原文摘要 · Abstract (English)
Financial scenario simulation is essential for risk management and portfolio optimization, yet it remains challenging especially in high-dimensional and small data settings common in finance. We propose a diffusion factor model that integrates latent factor structure into generative diffusion processes, bridging econometrics with modern generative AI to address the challenges of the curse of dimensionality and data scarcity in financial simulation. By exploiting the low-dimensional factor structure inherent in asset returns, we decompose the score function--a key component in diffusion models--using time-varying orthogonal projections, and this decomposition is incorporated into the design of neural network architectures. We derive rigorous statistical guarantees, establishing nonasymptotic error bounds for both score estimation at O(d^{5/2} n^{-2/(k+5)}) and generated distribution at O(d^{5/4} n^{-1/2(k+5)}), primarily driven by the intrinsic factor dimension k rather than the number of assets d, surpassing the dimension-dependent limits in the classical nonparametric statistics literature and making the framework viable for markets with thousands of assets. Numerical studies confirm superior performance in latent subspace recovery under small data regimes. Empirical analysis demonstrates the economic significance of our framework in constructing mean-variance optimal portfolios and factor portfolios. This work presents the first theoretical integration of factor structure with diffusion models, offering a principled approach for high-dimensional financial simulation with limited data. Our code is available at https://github.com/xymmmm00/diffusion_factor_model.
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