On the Mixing Time of Markov Chain Monte Carlo for Integer Least-Square Problems
In this paper, the authors study the mixing time of Markov Chain Monte Carlo (MCMC) for integer Least-Square (LS) optimization problems. It is found that the mixing time of MCMC for integer LS problems depends on the structure of the underlying lattice. More specifically, the mixing time of MCMC is closely related to whether there is a local minimum in the lattice structure. For some lattices, the mixing time of the Markov chain is independent of the Signal-to-Noise (SNR) Ratio and grows polynomially in the problem dimension; while for some lattices, the mixing time grows unboundedly as SNR grows. Both theoretical and empirical results suggest that to ensure fast mixing, the temperature for MCMC should often grow positively as the SNR increases.