This paper considers eigenvalue estimation for the decentralized inference problem for spectrum sensing. We propose a decentralized eigenvalue computation algorithm based on the power method, which is referred to as generalized power method GPM; it is capable of estimating the eigenvalues of a given covariance matrix under certain conditions. Furthermore, we have developed a decentralized implementation of GPM by splitting the iterative operations into local and global computation tasks. The global tasks require data exchange to be performed among the nodes. For this task, we apply an average consensus algorithm to efficiently perform the global computations. As a special case, we consider a structured graph that is a tree with clusters of nodes at its leaves. For an accelerated distributed implementation, we propose to use computation over multiple access channel (CoMAC) as a building block of the algorithm. Numerical simulations are provided to illustrate the performance of the two algorithms.
Proof of convergence of ideal GPM
The proof follows from the same token given in , since for every eigenvalue we run the conventional power iteration. In order to satisfy the assumptions in the original proof by , we need the following lemma to hold:
The matrices are all Hermitian, therefore diagonalizable.
It is straightforward if we write,
Since is Hermitian positive semi-definite, s are holding the same properties as well.
Proof of Lemma 1
For GPM algorithm with as the initial input vector, and Theorem 8.2.1 in  we have:
in which we used . We further substitute
where, the coefficients are defined as and .
Proof of Lemma 2
For GPM algorithm with is the initial input vector, and Theorem 8.2.1 in  we have:
where, and .
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