相关论文: Performance of Random Lattice Algorithms

The Voronoi diagram is a certain geometric data structure which has numerous applications in various scientific and technological fields. The theory of algorithms for computing 2D Euclidean Voronoi diagrams of point sites is rich and…

In a seminal work, Micciancio & Voulgaris (2013) described a deterministic single-exponential time algorithm for the Closest Vector Problem (CVP) on lattices. It is based on the computation of the Voronoi cell of the given lattice and thus…

We give a novel algorithm for enumerating lattice points in any convex body, and give applications to several classic lattice problems, including the Shortest and Closest Vector Problems (SVP and CVP, respectively) and Integer Programming…

We analyze convergence rates of stochastic optimization procedures for non-smooth convex optimization problems. By combining randomized smoothing techniques with accelerated gradient methods, we obtain convergence rates of stochastic…

Following S\"odergren, we consider a collection of random variables on the space $X_n$ of unimodular lattices in dimension $n$: Normalizations of the angles between the $N = N(n)$ shortest vectors in a random unimodular lattice, and the…

Lattice decoders constructed with neural networks are presented. Firstly, we show how the fundamental parallelotope is used as a compact set for the approximation by a neural lattice decoder. Secondly, we introduce the notion of…

We derive large- and moderate-deviation results in random networks given as planar directed navigations on homogeneous Poisson point processes. In this non-Markovian routing scheme, starting from the origin, at each consecutive step a…

We study randomized variants of two classical algorithms: coordinate descent for systems of linear equations and iterated projections for systems of linear inequalities. Expanding on a recent randomized iterated projection algorithm of…

Fix a subset $S \subset \mathbb{R}^n$ of volume at most $c n$ that satisfies $S \cap (-S) = \emptyset$. We consider two point processes in $S$: the first is the Poisson point process of intensity one, and the second is the restriction of a…

This paper focuses on analyzing and differentiating between lattice linear problems and algorithms. It introduces a new class of algorithms called \textit{(fully) lattice linear algorithms}. A property of these algorithms is that they…

In this paper we construct polynomial lattice rules which have, in some sense, small gain coefficients using a component-by-component approach. The gain coefficients, as introduced by Owen, indicate to what degree the method improves upon…

We consider a discrete-time random walk on the nodes of an unbounded hexagonal lattice. We determine the probability generating functions, the transition probabilities and the relevant moments. The convergence of the stochastic process to a…

This work develops a methodology for analyzing large-deviation lower tails associated with geometric functionals computed on a homogeneous Poisson point process. The technique applies to characteristics expressed in terms of stabilizing…

An algorithm is developed for efficiently constructing the Lorentz covariant effective three-point vertices of the decay of a particle into two daughter particles in which all the masses and spins of the three particles can be arbitrary.…

Randomized sampling has recently been demonstrated to be an efficient technique for computing approximate low-rank factorizations of matrices for which fast methods for computing matrix vector products are available. This paper describes an…

An overview of some methods of statistical physics applied to the analysis of algorithms for optimization problems (satisfiability of Boolean constraints, vertex cover of graphs, decoding, ...) with distributions of random inputs is…

For a lattice/linear code, we define the Voronoi spherical cumulative density function (CDF) as the CDF of the $\ell_2$-norm/Hamming weight of a random vector uniformly distributed over the Voronoi cell. Using the first moment method…

The lifetime of a system of connected units under some natural assumptions can be represented as a random variable Y defined as a weighted lattice polynomial of random lifetimes of its components. As such, the concept of a random variable Y…

We characterize the complexity of the lattice decoding problem from a neural network perspective. The notion of Voronoi-reduced basis is introduced to restrict the space of solutions to a binary set. On the one hand, this problem is shown…

Improving on the Voronoi cell based techniques of Micciancio and Voulgaris (SIAM J. Comp. 13), and Sommer, Feder and Shalvi (SIAM J. Disc. Math. 09), we give a Las Vegas $\tilde{O}(2^n)$ expected time and space algorithm for CVPP (the…