AI 中文总结
本文从统计视角利用广义斯坦因公式，提出基于SVD的CNN滤波器学习方法，经模拟与真实数据验证，其性能优于Adam等算法，还可拓展至非线性降维。
AI 中文摘要
卷积神经网络（Convolutional Neural Networks, CNNs）无疑已革新了图像数据分析与计算机视觉领域。作为CNNs的基石，卷积操作使网络能够提取抽象特征并揭示图像数据中的隐藏关系。本文从统计视角利用经典工具——斯坦因公式，研究卷积滤波器的估计问题。我们首先将CNNs构建为具有矩阵值输入的通用指标模型，其中卷积滤波器可视为指标向量。此外，我们提出一种基于奇异值分解（Singular Value Decomposition, SVD）的新方法，利用一阶斯坦因公式的广义版本准确学习卷积滤波器。理论分析表明，我们的估计达到最优收敛速率，与链接函数已知的广义线性模型的收敛速率相当。大量模拟研究与真实数据分析表明，我们的方法优于Adam等流行深度学习算法。值得注意的是，该方法不仅适用于滤波器估计，还可应用于非线性降维，为表征学习提供可行途径。
英文摘要
Convolutional Neural Networks (CNNs) have undoubtedly revolutionized image data analysis and the field of computer vision. As the cornerstone of CNNs, the convolution operation enables the networks to extract abstract features and uncover hidden relationships in the image data. This paper considers the problem of estimating convolution filters from a statistical perspective using a classical tool --- Stein's formula. We first formulate CNNs into a general index model with matrix-valued input, where convolution filters can be viewed as index vectors. Furthermore, we propose a novel singular value decomposition (SVD) based approach to accurately learn the convolution filters based on a generalized version of the first-order Stein's formula. Theoretical analysis suggests that our estimation achieves an optimal convergence rate, comparable to that of generalized linear models where the link function is known. Extensive simulation studies and real data analyses demonstrate that our approach outperforms popular deep learning algorithms, such as Adam. Notably, our method extends beyond filter estimation and can be applied to nonlinear dimension reduction, providing a viable pathway for representation learning.
