Ensuring convergence of the MMSE iteration for interference avoidance to the global optimum

Viswanath and Anantharam (1999) characterize the sum capacity of multiaccess vector channels. For a given number of users, received powers, spreading gain, and noise covariance matrix in a code-division multiple-access (CDMA) system, Viswanath and Anantharam present a combinatorial algorithm to generate a set of signature sequences that achieves the maximum sum capacity. These sets also minimize a performance measure called generalized total square correlation (TSC_g). Ulukus and Yates (2001) propose an iterative algorithm suitable for distributed implementation: at each step, one signature sequence is replaced by its linear minimum mean-square error (MMSE) filter. This algorithm results in a decrease of TSC_g at each step. The MMSE iteration has fixed points not only at the optimal configurations which attain the global minimum TSC_g but also at other configurations which are suboptimal. The authors of claim that simulations show that when starting with random sequences, the algorithm converges to optimum sets of sequences, but they give no formal proof. We show that the TSC_g function has no local minima, in the sense that given any suboptimal set of sequences, there exist arbitrarily close sets with lower TSC_g. Therefore, only the optimal sets are stable fixed points of the MMSE iteration. We define a noisy version of the MMSE iteration as follows: after replacing all the signature sequences, one at a time, by their linear MMSE filter, we add a bounded random noise to all the sequences. Using our observation about the TSC_g function, we can prove that if we choose the bound on the noise adequately, making it decrease to zero, the noisy MMSE iteration converges to the set of optimal configurations with probability one for any initial set of sequences.