Sliced inverse regression (SIR), which includes linear discriminant analysis (LDA) as a special case, is a popular and powerful dimension reduction tool. In this article, we extend SIR to address the challenges of decentralized data, prioritizing privacy and communication efficiency. Our approach, named as federated sliced inverse regression (FSIR), facilitates collaborative estimation of the sufficient dimension reduction subspace among multiple clients, solely sharing local estimates to protect sensitive datasets from exposure. To guard against potential adversary attacks, FSIR further employs diverse perturbation strategies, including a novel vectorized Gaussian mechanism that guarantees differential privacy at a low cost of statistical accuracy. Additionally, FSIR naturally incorporates a collaborative variable screening step, enabling effective handling of high-dimensional client data. Theoretical properties of FSIR are established for both low-dimensional and high-dimensional settings, supported by extensive numerical experiments and real data analysis.

翻译：切片逆回归（Sliced Inverse Regression, SIR）以线性判别分析（Linear Discriminant Analysis, LDA）为特例，是一种广泛应用的强大降维工具。本文扩展SIR以应对去中心化数据带来的挑战，优先考虑隐私保护与通信效率。我们提出的方法称为联邦切片逆回归（Federated Sliced Inverse Regression, FSIR），通过多个客户端仅共享局部估计值以保护敏感数据集不被泄露，促进对充分降维子空间的协同估计。为抵御潜在对手攻击，FSIR进一步采用多样化的扰动策略，包括一种新型向量化高斯机制，该机制以较低的统计精度代价保证差分隐私。此外，FSIR自然嵌入协作变量筛选步骤，可有效处理高维客户端数据。针对低维和高维场景，我们建立了FSIR的理论性质，并通过大量数值实验与真实数据分析加以验证。