Decentralized Computation of Tyler's $M$-Estimator

We study decentralized computation of Tyler's $M$-estimator over a peer-to-peer network. A central difficulty is that the estimator may exist for the pooled data while being undefined at individual nodes, whose local weighted-scatter matrices can be singular. We propose a decentralized majorization–minimization method (D-MM) that combines local weighted-scatter updates with finitely many rounds of neighbor averaging and requires the Tyler existence condition only for the pooled observations. Relative Loewner-order bounds preserve positive definiteness after finite consensus, while Hilbert's projective metric controls both inter-node disagreement and the error caused by heterogeneous local iterates. Under summable consensus errors, all node estimates converge to the pooled Tyler estimator, with explicit finite-step and communication guarantees for polynomial schedules. We further show that, on the determinant-one manifold, the strict occupancy condition induces positive data-dependent geodesic curvature on the invariant region. This yields an instance-dependent linear convergence rate for the unregularized centralized Tyler iteration, sharpening the recent deterministic $O(1/k)$ objective-gap guarantee without regularization. Under geometrically decaying consensus errors, D-MM inherits this contraction and converges linearly. Experiments confirm the predicted effects of communication, network connectivity, and data geometry, and demonstrate agreement with pooled Tyler estimation in a rolling portfolio application.