稀疏典型相关分析的最小最大与自适应迁移学习
Minimax and Adaptive Transfer Learning for Sparse Canonical Correlation Analysis
- School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出稀疏CCA的迁移学习框架，通过异质源数据提升目标估计，建立最小最大最优速率并设计自适应算法，无需稀疏度先验，同时提出源检测程序并推广sin-theta定理。
AI中文摘要：
我们针对具有多个异质源数据集的高维稀疏典型相关分析（CCA）开发了一个迁移学习框架。我们的目标是通过借用相关源的信息来改进目标典型系数矩阵的估计。我们通过联合协方差结构中的差异来刻画源-目标相似性，并在联合预测损失和单独预测损失下建立最小最大最优速率，明确量化了有效样本量增大带来的收益以及异质性造成的代价。为达到这些速率，我们首先基于去偏和截断归一化构建了一个预言机迁移估计器。随后，我们开发了一种计算高效的程序，该程序无需预先知道稀疏度水平即可自适应地达到最小最大速率，利用聚合来控制迁移偏差，并采用截断归一化方案在尺度敏感预测损失下保持迁移收益。进一步提出了一种数据驱动的源检测程序，以识别可能提供信息的辅助数据集。我们的分析还产生了一个广义的sin-theta扰动定理，该定理去除了稀疏CCA问题中通常施加的关于前导典型相关性的可比性条件。
英文摘要：
We develop a transfer learning framework for high-dimensional sparse canonical correlation analysis (CCA) with multiple heterogeneous source datasets. Our goal is to improve estimation of the target canonical coefficient matrices by borrowing information from related sources. We characterize source-target similarity through discrepancies in the joint covariance structure and establish minimax optimal rates under both joint and separate prediction losses, explicitly quantifying the gain from enlarged effective sample size and the cost of heterogeneity. To attain these rates, we first construct an oracle transfer estimator based on debiasing and truncated normalization. We then develop a computationally efficient procedure that adaptively achieves the minimax rate without prior knowledge of the sparsity levels, using aggregation to control transfer bias and a truncated normalization scheme to preserve transfer gains under the scale-sensitive prediction loss. A data-driven source detection procedure is further proposed to identify potentially informative auxiliary datasets. Our analysis also yields a generalized sin-theta perturbation theorem that removes a commonly imposed comparability condition on leading canonical correlations in the sparse CCA problem.
