Distributed Inference for Manifold M-Estimators

We develop one-round confidence regions for manifold $M$-estimators from distributed data. An independent initial estimate places all workers in a common chart, where reported scores, Hessians and score second moments determine an estimate and its covariance. Under fixed-dimensional regularity, we separate the nonlinear error from the additional error caused by using only the master worker's Hessian. This decomposition yields explicit accuracy and machine-growth conditions for centralized first-order inference, with confidence regions invariant to orthonormal frame choice. For principal component analysis, the same summaries recover the exact centralized subspace, reducing the requirement to consistency. Spherical and subspace experiments connect coverage to accuracy and Hessian estimation, and quantify the communication trade-off between one-round recentring and two-round covariance estimation. A Dry Bean application illustrates uncertainty for feature contributions to a principal subspace.