Linear-time nearest point algorithms for Coxeter lattices

The Coxeter lattices, which we denote $A_{n/m}$, are a family of lattices containing many of the important lattices in low dimensions. This includes $A_n$, $E_7$, $E_8$ and their duals $A_n^*$, $E_7^*$ and $E_8^*$. We consider the problem of finding a nearest point in a Coxeter lattice. We describe two new algorithms, one with worst case arithmetic complexity $O(n\log{n})$ and the other with worst case complexity O(n) where $n$ is the dimension of the lattice. We show that for the particular lattices $A_n$ and $A_n^*$ the algorithms reduce to simple nearest point algorithms that already exist in the literature.Comment: submitted to IEEE Transactions on Information Theor

Finding a closest point in a lattice of Voronoi's first kind

We show that for those lattices of Voronoi's first kind with known obtuse superbasis, a closest lattice point can be computed in $O(n^4)$ operations where $n$ is the dimension of the lattice. To achieve this a series of relevant lattice vectors that converges to a closest lattice point is found. We show that the series converges after at most $n$ terms. Each vector in the series can be efficiently computed in $O(n^3)$ operations using an algorithm to compute a minimum cut in an undirected flow network

Maximum-likelihood period estimation from sparse, noisy timing data

The problem of estimating the period of a periodic point process is considered when the observations are sparse and noisy. There is a class of estimators that operate by maximizing an objective function over an interval of possible periods, notably the periodogram estimator of Fogel & Gavish and the line-search algorithms of Sidiropoulos et al. and Clarkson. For numerical calculation, the interval is sampled. However, it is not known how fine the sampling must be in order to ensure statistically accurate results. In this paper, a new estimator is proposed which eliminates the need for sampling. For the proposed statistical model, it calculates a maximum- likelihood estimate. It is shown that the expected arithmetic complexity of the algorithm is O(n3 log n) where n is the number of observations. Numerical simulations demonstrate the superior statistical performance of the new estimator

Approximate maximum-likelihood period estimation from sparse, noisy timing data

The problem of estimating the period of a series of periodic events is considered under the condition where the measurements of the times of occurrence are noisy and sparse. The problem is common to bit synchronisation in telecommunications and pulse-train parameter estimation in electronic support, among other applications. Two new algorithms are presented which represent different compromises between computational and statistical efficiency. The first extends the separable least squares line search (SLS2) algorithms of Sidiropoulos et al., having very low computational complexity while attaining good statistical accuracy. The second is an approximate maximum-likelihood algorithm, based on a low complexity lattice search, and is found to achieve excellent accuracy