5,708 research outputs found
On the Mixing Time of Markov Chain Monte Carlo for Integer Least-Square Problems
In this paper, we 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 ratio (SNR) 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. We also derive the probability that there exist local minima in an integer least-square problem, which can be as high as 1/3 - 1/√5 + (2 arctan(√(5/3))/(√5Π)
Approximately Sampling Elements with Fixed Rank in Graded Posets
Graded posets frequently arise throughout combinatorics, where it is natural
to try to count the number of elements of a fixed rank. These counting problems
are often -complete, so we consider approximation algorithms for
counting and uniform sampling. We show that for certain classes of posets,
biased Markov chains that walk along edges of their Hasse diagrams allow us to
approximately generate samples with any fixed rank in expected polynomial time.
Our arguments do not rely on the typical proofs of log-concavity, which are
used to construct a stationary distribution with a specific mode in order to
give a lower bound on the probability of outputting an element of the desired
rank. Instead, we infer this directly from bounds on the mixing time of the
chains through a method we call .
A noteworthy application of our method is sampling restricted classes of
integer partitions of . We give the first provably efficient Markov chain
algorithm to uniformly sample integer partitions of from general restricted
classes. Several observations allow us to improve the efficiency of this chain
to require space, and for unrestricted integer partitions,
expected time. Related applications include sampling permutations
with a fixed number of inversions and lozenge tilings on the triangular lattice
with a fixed average height.Comment: 23 pages, 12 figure
Near-Optimal Detection in MIMO Systems using Gibbs Sampling
In this paper we study a Markov Chain Monte Carlo (MCMC) Gibbs sampler for
solving the integer least-squares problem. In digital communication the problem
is equivalent to performing Maximum Likelihood (ML) detection in Multiple-Input
Multiple-Output (MIMO) systems. While the use of MCMC methods for such problems
has already been proposed, our method is novel in that we optimize the
"temperature" parameter so that in steady state, i.e. after the Markov chain
has mixed, there is only polynomially (rather than exponentially) small
probability of encountering the optimal solution. More precisely, we obtain the
largest value of the temperature parameter for this to occur, since the higher
the temperature, the faster the mixing. This is in contrast to simulated
annealing techniques where, rather than being held fixed, the temperature
parameter is tended to zero. Simulations suggest that the resulting Gibbs
sampler provides a computationally efficient way of achieving approximative ML
detection in MIMO systems having a huge number of transmit and receive
dimensions. In fact, they further suggest that the Markov chain is rapidly
mixing. Thus, it has been observed that even in cases were ML detection using,
e.g. sphere decoding becomes infeasible, the Gibbs sampler can still offer a
near-optimal solution using much less computations.Comment: To appear in Globecom 200
Estimating the spectral gap of a trace-class Markov operator
The utility of a Markov chain Monte Carlo algorithm is, in large part,
determined by the size of the spectral gap of the corresponding Markov
operator. However, calculating (and even approximating) the spectral gaps of
practical Monte Carlo Markov chains in statistics has proven to be an extremely
difficult and often insurmountable task, especially when these chains move on
continuous state spaces. In this paper, a method for accurate estimation of the
spectral gap is developed for general state space Markov chains whose operators
are non-negative and trace-class. The method is based on the fact that the
second largest eigenvalue (and hence the spectral gap) of such operators can be
bounded above and below by simple functions of the power sums of the
eigenvalues. These power sums often have nice integral representations. A
classical Monte Carlo method is proposed to estimate these integrals, and a
simple sufficient condition for finite variance is provided. This leads to
asymptotically valid confidence intervals for the second largest eigenvalue
(and the spectral gap) of the Markov operator. In contrast with previously
existing techniques, our method is not based on a near-stationary version of
the Markov chain, which, paradoxically, cannot be obtained in a principled
manner without bounds on the spectral gap. On the other hand, it can be quite
expensive from a computational standpoint. The efficiency of the method is
studied both theoretically and empirically
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