Geometric data perturbation enables one-shot representation sharing for privacy-preserving collaborative learning: each participant applies a secret distance-preserving transformation to its private data and uploads the resulting representation to a central analyst. We study analyst-participant collusion, in which a colluding participant discloses its data and transformation to help the analyst reconstruct another participant's data. Independent participant-specific transformations block direct inversion through a disclosed common transformation but leave uploads in incompatible coordinate systems, degrading pooled learning. Data Collaboration analysis restores compatibility by aligning transformed copies of a common anchor matrix withheld from the analyst. We show that, when the centered anchor matrix has full column rank, a colluder who discloses it enables exact recovery of every participant's transformation and inversion of noiseless private representations. Adding noise to private-data representations leaves this transformation-recovery channel intact and reduces leakage at a substantial utility cost. Instead, we perturb the anchor representations: each participant perturbs only its transformed anchor representation, preserving the geometry of its private-data upload while turning known-anchor transformation recovery into a noisy estimation problem. The analyst estimates the alignment using a spectral estimator for a generalized orthogonal Procrustes problem. We analyze recovery attacks against this protocol and compare both noise placements on the CelebA and VGGFace2 facial image datasets. Under the evaluated collusion attacks, noisy-anchor alignment retains higher downstream accuracy at low identity-linkage levels. Participant-count experiments examine the utility gains and limitations of larger collaborations at comparable measured linkage.

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