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.

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