On the Geometric Ergodicity of Metropolis-Hastings Algorithms for Lattice Gaussian Sampling

Sampling from the lattice Gaussian distribution is emerging as an important problem in coding and cryptography. In this paper, the classic Metropolis-Hastings (MH) algorithm from Markov chain Monte Carlo (MCMC) methods is adapted for lattice Gaussian sampling. Two MH-based algorithms are proposed, which overcome the restriction suffered by the default Klein's algorithm. The first one, referred to as the independent Metropolis-Hastings-Klein (MHK) algorithm, tries to establish a Markov chain through an independent proposal distribution. We show that the Markov chain arising from the independent MHK algorithm is uniformly ergodic, namely, it converges to the stationary distribution exponentially fast regardless of the initial state. Moreover, the rate of convergence is explicitly calculated in terms of the theta series, leading to a predictable mixing time. In order to further exploit the convergence potential, a symmetric Metropolis-Klein (SMK) algorithm is proposed. It is proven that the Markov chain induced by the SMK algorithm is geometrically ergodic, where a reasonable selection of the initial state is capable to enhance the convergence performance.Comment: Submitted to IEEE Transactions on Information Theor

On the Geometric Ergodicity of Metropolis-Hastings Algorithms for Lattice Gaussian Sampling

Sampling from the lattice Gaussian distribution has emerged as an important problem in coding, decoding and cryptography. In this paper, the classic Metropolis-Hastings (MH) algorithm in Markov chain Monte Carlo (MCMC) methods is adopted for lattice Gaussian sampling. Two MH-based algorithms are proposed, which overcome the limitation of Klein\u27s algorithm. The first one, referred to as the independent Metropolis-Hastings-Klein (MHK) algorithm, establishes a Markov chain via an independent proposal distribution. We show that the Markov chain arising from this independent MHK algorithm is uniformly ergodic, namely, it converges to the stationary distribution exponentially fast regardless of the initial state. Moreover, the rate of convergence is analyzed in terms of the theta series, leading to predictable mixing time. A symmetric Metropolis-Klein (SMK) algorithm is also proposed, which is proven to be geometrically ergodic

Lattice Gaussian Sampling by Markov Chain Monte Carlo: Bounded Distance Decoding and Trapdoor Sampling

Sampling from the lattice Gaussian distribution plays an important role in various research fields. In this paper, the Markov chain Monte Carlo (MCMC)-based sampling technique is advanced in several fronts. Firstly, the spectral gap for the independent Metropolis-Hastings-Klein (MHK) algorithm is derived, which is then extended to Peikert's algorithm and rejection sampling; we show that independent MHK exhibits faster convergence. Then, the performance of bounded distance decoding using MCMC is analyzed, revealing a flexible trade-off between the decoding radius and complexity. MCMC is further applied to trapdoor sampling, again offering a trade-off between security and complexity. Finally, the independent multiple-try Metropolis-Klein (MTMK) algorithm is proposed to enhance the convergence rate. The proposed algorithms allow parallel implementation, which is beneficial for practical applications.Comment: submitted to Transaction on Information Theor

Further results on independent Metropolis-Hastings-Klein sampling

Sampling from a lattice Gaussian distribution is emerging as an important problem in coding and cryptography. This paper gives a further analysis of the independent Metropolis-Hastings-Klein (MHK) algorithm we presented at ISIT 2015. We derive the exact spectral gap of the induced Markov chain, which dictates the convergence rate of the independent MHK algorithm. Then, we apply the independent MHK algorithm to lattice decoding and obtained the decoding complexity for solving the CVP as Õ(e∥Bx-c∥2 / mini ∥b̂i∥2). Finally, the tradeoff between decoding radius and complexity is also established

Lattice sampling algorithms for communications

In this thesis, we investigate the problem of decoding for wireless communications from the perspective of lattice sampling. In particular, computationally efficient lattice sampling algorithms are exploited to enhance the system performance, which enjoys the system tradeoff between performance and complexity through the sample size. Based on this idea, several novel lattice sampling algorithms are presented in this thesis. First of all, in order to address the inherent issues in the random sampling, derandomized sampling algorithm is proposed. Specifically, by setting a probability threshold to sample candidates, the whole sampling procedure becomes deterministic, leading to considerable performance improvement and complexity reduction over to the randomized sampling. According to the analysis and optimization, the correct decoding radius is given with the optimized parameter setting. Moreover, the upper bound on the sample size, which corresponds to near-maximum likelihood (ML) performance, is also derived. After that, the proposed derandomized sampling algorithm is introduced into the soft-output decoding of MIMO bit-interleaved coded modulation (BICM) systems to further improve the decoding performance. According to the demonstration, we show that the derandomized sampling algorithm is able to achieve the near-maximum a posteriori (MAP) performance in the soft-output decoding. We then extend the well-known Markov Chain Monte Carlo methods into the samplings from lattice Gaussian distribution, which has emerged as a common theme in lattice coding and decoding, cryptography, mathematics. We firstly show that the statistical Gibbs sampling is capable to perform the lattice Gaussian sampling. Then, a more efficient algorithm referred to as Gibbs-Klein sampling is proposed, which samples multiple variables block by block using Klein’s algorithm. After that, for the sake of convergence rate, we introduce the conventional statistical Metropolis-Hastings (MH) sampling into lattice Gaussian distributions and three MH-based sampling algorithms are then proposed. The first one, named as MH multivariate sampling algorithm, is demonstrated to have a faster convergence rate than Gibbs-Klein sampling. Next, the symmetrical distribution generated by Klein’s algorithm is taken as the proposal distribution, which offers an efficient way to perform the Metropolis sampling over high-dimensional models. Finally, the independent Metropolis-Hastings-Klein (MHK) algorithm is proposed, where the Markov chain arising from it is proved to converge to the stationary distribution exponentially fast. Furthermore, its convergence rate can be explicitly calculated in terms of the theta series, making it possible to predict the exact mixing time of the underlying Markov chain.Open Acces

Sliced lattice Gaussian sampling: convergence improvement and decoding optimization

Sampling from the lattice Gaussian distribution has emerged as a key problem in coding and decoding while Markov chain Monte Carlo (MCMC) methods from statistics offer an effective way to solve it. In this paper, the sliced lattice Gaussian sampling algorithm is proposed to further improve the convergence performance of the Markov chain targeting at lattice Gaussian sampling. We demonstrate that the Markov chain arising from it is uniformly ergodic, namely, it converges exponentially fast to the stationary distribution. Meanwhile, the convergence rate of the underlying Markov chain is also investigated, and we show the proposed sliced sampling algorithm entails a better convergence performance than the independent Metropolis-Hastings-Klein (IMHK) sampling algorithm. On the other hand, the decoding performance based on the proposed sampling algorithm is analyzed, where the optimization with respect to the standard deviation σ>0 of the target lattice Gaussian distribution is given. After that, a judicious mechanism based on distance judgement and dynamic updating for choosing σ is proposed for a better decoding performance. Finally, simulation results based on multiple-input multiple-output (MIMO) detection are presented to confirm the performance gain by the convergence enhancement and the parameter optimization

Markov chain Monte Carlo Methods For Lattice Gaussian Sampling:Convergence Analysis and Enhancement

Sampling from lattice Gaussian distribution has emerged as an important problem in coding, decoding and cryptography. In this paper, the classic Gibbs algorithm from Markov chain Monte Carlo (MCMC) methods is demonstrated to be geometrically ergodic for lattice Gaussian sampling, which means the Markov chain arising from it converges exponentially fast to the stationary distribution. Meanwhile, the exponential convergence rate of Markov chain is also derived through the spectral radius of forward operator. Then, a comprehensive analysis regarding to the convergence rate is carried out and two sampling schemes are proposed to further enhance the convergence performance. The first one, referred to as Metropolis-within-Gibbs (MWG) algorithm, improves the convergence by refining the state space of the univariate sampling. On the other hand, the blocked strategy of Gibbs algorithm, which performs the sampling over multivariate at each Markov move, is also shown to yield a better convergence rate than the traditional univariate sampling. In order to perform blocked sampling efficiently, Gibbs-Klein (GK) algorithm is proposed, which samples block by block using Klein's algorithm. Furthermore, the validity of GK algorithm is demonstrated by showing its ergodicity. Simulation results based on MIMO detections are presented to confirm the convergence gain brought by the proposed Gibbs sampling schemes.Comment: Submitted to IEEE Transaction on Communication

Symmetric mettropolis-within-Gibbs algorithm for lattice Gaussian sampling

As a key sampling scheme in Markov chain Monte Carlo (MCMC) methods, Gibbs sampling is widely used in various research fields due to its elegant univariate conditional sampling, especially in tacking with multidimensional sampling systems. In this paper, a Gibbs-based sampler named as symmet- ric Metropolis-within-Gibbs (SMWG) algorithm is proposed for lattice Gaussian sampling. By adopting a symmetric Metropolis- Hastings (MH) step into the Gibbs update, we show the Markov chain arising from it is geometrically ergodic, which converges exponentially fast to the stationary distribution. Moreover, by optimizing its symmetric proposal distribution, the convergence efficiency can be further enhanced