Distributed estimation of the inverse of the correlation matrix for privacy preserving beamforming

In this paper, we consider a privacy preserving scenario where users in the network want to perform distributed target source estimation with a wireless acoustic sensor network (WASN), without revealing the actual source of interest to other entities in the network. This implies that users do not share the steering vector of the beamformer with any other party. For distributed multi-channel noise reduction in WASNs, distributed estimation of the inverse noise or noiseþtarget correlation matrix is an important aspect and in general a challenging problem. To make both privacy preservation and distributed multi-channel noise reduction possible, we make use of the fact that recursive estimation of the inverse correlation matrix can be structured as a consensus problem and can be realized in a distributed manner via the randomized gossip algorithm. This makes it possible to compute the MVDR in distributed manner without revealing the steering vector to any of the other entities in the network, and providing privacy about the actual source of interest. We provide theoretical analysis and numerical simulations to investigate the convergence error between the gossip-based estimated correlation matrix and the centralized estimated correlation matrix. It is shown that the convergence error accumulates across time without using a sufficient number of transmissions in the gossip-based algorithm. To eliminate this convergence error, we propose in addition a clique-based algorithm for distributed estimation of the inverse correlation matrix (CbDECM). Theoretical analysis shows that the CbDECM algorithm converges to the centralized estimate of the matrix