Geometric Data Perturbation (GDP) enables one-shot, privacy-preserving collaborative learning: each participant applies a distance-preserving transformation to its private data and uploads only the resulting representation to a central analyst. We study GDP under analyst-participant collusion, in which the analyst combines all uploaded representations with the private data and transformations disclosed by colluding participants to recover a non-colluding participant's private data. Participant-specific independent transformations resist this attack but map participants' data into incompatible representation spaces, degrading downstream model performance. Shared-anchor alignment from Data Collaboration (DC) analysis restores compatibility and improves utility, but we show that disclosing the DC anchor matrix enables exact recovery of non-colluding participants' private data even in the presence of collusion. Adding noise directly to the private-data representations mitigates this vulnerability but substantially reduces utility. We propose adding noise to the anchor representations instead. Each participant independently transforms its private data and the shared anchor matrix, perturbs only the resulting anchor representation, and uploads both representations in a single round. Using the noisy anchor representations, the analyst aligns the private-data representations by solving a Generalized Orthogonal Procrustes Problem. We characterize alignment and recovery errors, specialize a conservative sufficient condition for convergence of the alignment to our setting, and analyze three recovery attacks. Experiments on MNIST and CelebA show that, across the evaluated attacks and deployment settings, anchor noise achieves higher learning accuracy than private-data noise at comparable measured leakage, yielding a more favorable privacy-utility trade-off under the specified collusion model.
Preserving data privacy is an important topic in structural data management and data mining. However, the issue of privacy leakage in distributed causal structure learning is a persistent challenge, especially in cases where data transmission and computation are required. In this paper, we propose a method based on fully homomorphic encryption (FHE) that performs calculations on ciphertexts, keeping data encrypted in transition and computation. Nevertheless, adopting FHE to causal structure learning is challenging due to the high computation cost and limited support on division as well as logarithm operations in FHE. To tackle this challenge, we propose a series of novel techniques including (i) circuit simplification for better efficiency, (ii) approximation of division and logarithm through Newton-Raphson Reciprocal and Taylor expansion, and (iii) a batching technique with SIMD-acceleration to enhance the whole learning process. Additionally, our method can be easily extended beyond FHE by demonstration of its portability to support differential privacy. Empirical results show that our method achieves high consistency and comparable causal structure with the plaintext version in the datasets tested. Last, our method is efficient and practical to complete learning causal structures in tens of minutes even under the privacy protection of FHE.