证明稀疏贝叶斯神经网络在低维结构函数上可达到最优后验收缩率。
Posterior Contraction for Sparse Neural Networks in Besov Spaces with Intrinsic Dimensionality
- 基于稀疏或连续压缩先验，实现对低维结构函数的自适应建模。
- 后验收缩率依赖于函数的内在维度，可突破高维诅咒限制。
- 适用于加性、乘性等复杂结构函数，适合高维稀疏建模任务。
本文证明，稀疏贝叶斯神经网络在各向异性Besov空间及其层次组合上可实现最优后验收缩率。这些结构反映了潜在函数的内在维度，从而缓解维度灾难问题。分析表明，采用稀疏或连续压缩先验的贝叶斯神经网络能达到依赖于真实结构内在维度的最优收缩率。此外，该先验支持速率自适应，即使真实函数光滑度未知，后验仍能以最优速率收缩。所提框架涵盖广泛函数类，包括加性和乘性Besov函数作为特例。研究深化了贝叶斯神经网络的理论基础，为高维结构性估计中的实际有效性提供了严格依据。
原文摘要 · Abstract (English)
This work establishes that sparse Bayesian neural networks achieve optimal posterior contraction rates over anisotropic Besov spaces and their hierarchical compositions. These structures reflect the intrinsic dimensionality of the underlying function, thereby mitigating the curse of dimensionality. Our analysis shows that Bayesian neural networks equipped with either sparse or continuous shrinkage priors attain the optimal rates which are dependent on the intrinsic dimension of the true structures. Moreover, we show that these priors enable rate adaptation, allowing the posterior to contract at the optimal rate even when the smoothness level of the true function is unknown. The proposed framework accommodates a broad class of functions, including additive and multiplicative Besov functions as special cases. These results advance the theoretical foundations of Bayesian neural networks and provide rigorous justification for their practical effectiveness in high-dimensional, structured estimation problems.
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