MCMC methods for integer least-squares problems

Abstract

We consider the problem of finding the least-squares solution to a system of linear equations where the unknown vector has integer entries (or, more precisely, has entries belonging to a subset of the integers), yet where the coefficient matrix and given vector are comprised of real numbers. Geometrically, this problem is equivalent to finding the closest lattice point to a given point and is known to be NP hard. In communication applications, however, the given vector is not arbitrary, but is a lattice point perturbed by some noise vector. Therefore it is of interest to study the computational complexity of various algorithms as a function of the noise variance or, often more appropriately, the SNR. In this paper, we apply a particular version of the Monte Carlo Markov chain (MCMC) approach to solving this problem, which is called a "heat bath". We show that there is a trade-off between the mixing time of the Markov chain (how long it takes until the chain reaches its stationary distribution) and how long it takes for the algorithm to find the optimal solution once the chain has mixed. The complexity of the algorithm is essentially the sum of these two times. More specifically, the higher the temperature, the faster the mixing, yet the slower the discovery of the optimal solution in steady state. Conversely, the lower the temperature, the slower the mixing, yet the faster the discovery of the optimal solution once the chain is mixed. We first show that for the probability of error of the maximum-likelihood (ML) solution to go to zero the SNR must scale at least as 2 ln N + α(N), where N is the ambient problem dimension and α(N) is any sequence that tends to positive infinity. We further obtain the optimal value of the temperature such that the average time required to encounter the optimal solution in steady state is polynomial. Simulations show that, with this choice of the temperature parameter, the optimal solution can be found in reasonable time. This suggests that the Markov chain mixes in polynomial-time, though we have not been able to prove this. It seems reasonable to conjecture that for SNR scaling as O((ln(N)) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1+∈</sup> ), and for appropriate choice of the temperature parameter, the heat bath algorithm finds the optimal solution in polynomial-time.