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 methods is adopted for lattice Gaussian sampling. Two MH-based algorithms are proposed, which overcome the limitation of Klein’s 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 algorithm is also proposed, which is proven to be geometrically ergodic.

[1]  László Lovász,et al.  Factoring polynomials with rational coefficients , 1982 .

[2]  Haidong Zhu,et al.  On performance of sphere decoding and Markov chain Monte Carlo detection methods , 2005, IEEE Signal Processing Letters.

[3]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[4]  G. Roberts,et al.  Polynomial convergence rates of Markov chains. , 2002 .

[5]  Richard L. Tweedie,et al.  Markov Chains and Stochastic Stability , 1993, Communications and Control Engineering Series.

[6]  Craig Gentry,et al.  Trapdoors for hard lattices and new cryptographic constructions , 2008, IACR Cryptol. ePrint Arch..

[7]  Frank R. Kschischang,et al.  Optimal nonuniform signaling for Gaussian channels , 1993, IEEE Trans. Inf. Theory.

[8]  J. Rosenthal,et al.  General state space Markov chains and MCMC algorithms , 2004, math/0404033.

[9]  Cong Ling,et al.  Achieving the AWGN channel capacity with lattice Gaussian coding , 2013, 2013 IEEE International Symposium on Information Theory.

[10]  Dana Randall,et al.  Rapidly mixing Markov chains with applications in computer science and physics , 2006, Computing in Science & Engineering.

[11]  R. Tweedie,et al.  Rates of convergence of the Hastings and Metropolis algorithms , 1996 .

[12]  J. Rosenthal.  Minorization Conditions and Convergence Rates for Markov Chain Monte Carlo , 1995 .

[13]  Cong Ling,et al.  Markov chain Monte Carlo algorithms for lattice Gaussian sampling , 2014, 2014 IEEE International Symposium on Information Theory.

[14]  N. J. A. Sloane,et al.  Sphere Packings, Lattices and Groups , 1987, Grundlehren der mathematischen Wissenschaften.

[15]  Mary Kathryn Cowles,et al.  A simulation approach to convergence rates for Markov chain Monte Carlo algorithms , 1998, Stat. Comput..

[16]  Radford M. Neal.  Optimal Proposal Distributions and Adaptive MCMC , 2008 .

[17]  Fiig Jarner,et al.  Polynomial convergence rates of Markov chainsSøren , 2000 .

[18]  Xiaodong Wang,et al.  Monte Carlo methods for signal processing , 2005 .

[19]  Craig Gentry,et al.  A fully homomorphic encryption scheme , 2009 .

[20]  Xiaodong Wang,et al.  Monte Carlo methods for signal processing: a review in the statistical signal processing context , 2005, IEEE Signal Processing Magazine.

[21]  S. F. Jarner,et al.  Geometric ergodicity of Metropolis algorithms , 2000 .

[22]  Cong Ling,et al.  Decoding by Sampling: A Randomized Lattice Algorithm for Bounded Distance Decoding , 2010, IEEE Transactions on Information Theory.

[23]  W. K. Hastings,et al.  Monte Carlo Sampling Methods Using Markov Chains and Their Applications , 1970 .

[24]  Cong Ling,et al.  Semantically Secure Lattice Codes for the Gaussian Wiretap Channel , 2012, IEEE Transactions on Information Theory.

[25]  Xiaodong Wang,et al.  Multilevel sequential monte carlo algorithms for MIMO demodulation , 2007, IEEE Transactions on Wireless Communications.

[26]  Noah Stephens-Davidowitz,et al.  Discrete Gaussian Sampling Reduces to CVP and SVP , 2015, SODA.

[27]  Jinhee Yi,et al.  Theta-function identities and the explicit formulas for theta-function and their applications , 2004 .

[28]  Brent Waters,et al.  Homomorphic Encryption from Learning with Errors: Conceptually-Simpler, Asymptotically-Faster, Attribute-Based , 2013, CRYPTO.

[29]  Cong Ling,et al.  Lattice Gaussian Sampling by Markov Chain Monte Carlo: Convergence Rate and Decoding Complexity , 2017, ArXiv.

[30]  Daniel Dadush,et al.  Solving the Shortest Vector Problem in 2n Time Using Discrete Gaussian Sampling: Extended Abstract , 2014, STOC.

[31]  R. Tweedie,et al.  Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms , 1996 .

[32]  Jean-Claude Belfiore,et al.  Lattice code design criterion for MIMO wiretap channels , 2015, 2015 IEEE Information Theory Workshop - Fall (ITW).

[33]  Haidong Zhu,et al.  Markov chain Monte Carlo algorithms for CDMA and MIMO communication systems , 2006, IEEE Transactions on Signal Processing.

[34]  B. Sundar Rajan,et al.  A Novel Monte-Carlo-Sampling-Based Receiver for Large-Scale Uplink Multiuser MIMO Systems , 2013, IEEE Transactions on Vehicular Technology.

[35]  Daniel Dadush,et al.  Solving the Closest Vector Problem in 2^n Time -- The Discrete Gaussian Strikes Again! , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[36]  László Babai,et al.  On Lovász’ lattice reduction and the nearest lattice point problem , 1986, Comb..

[37]  Alistair Sinclair,et al.  Improved Bounds for Mixing Rates of Markov Chains and Multicommodity Flow , 1992, Combinatorics, Probability and Computing.

[38]  Cong Ling,et al.  On the Proximity Factors of Lattice Reduction-Aided Decoding , 2010, IEEE Transactions on Signal Processing.

[39]  Alexandros G. Dimakis,et al.  Optimized Markov Chain Monte Carlo for Signal Detection in MIMO Systems: An Analysis of the Stationary Distribution and Mixing Time , 2013, IEEE Transactions on Signal Processing.

[40]  Rong Chen,et al.  Convergence analyses and comparisons of Markov chain Monte Carlo algorithms in digital communications , 2002, IEEE Trans. Signal Process..

[41]  Shashank Vatedka,et al.  Secure Compute-and-Forward in a Bidirectional Relay , 2012, IEEE Transactions on Information Theory.

[42]  B. Berndt.  Ramanujan’s Notebooks: Part V , 1997 .

[43]  Cong Ling,et al.  Decoding by Sampling — Part II: Derandomization and Soft-Output Decoding , 2013, IEEE Transactions on Communications.

[44]  Philip N. Klein,et al.  Finding the closest lattice vector when it's unusually close , 2000, SODA '00.

[45]  W. Banaszczyk.  New bounds in some transference theorems in the geometry of numbers , 1993 .

[46]  Frédérique E. Oggier,et al.  Lattice Codes for the Wiretap Gaussian Channel: Construction and Analysis , 2016, IEEE Trans. Inf. Theory.

[47]  Daniele Micciancio,et al.  Worst-case to average-case reductions based on Gaussian measures , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[48]  Jeffrey S. Rosenthal,et al.  Optimal Proposal Distributions and Adaptive MCMC , 2011 .

[49]  Cong Ling,et al.  Achieving AWGN Channel Capacity With Lattice Gaussian Coding , 2014, IEEE Transactions on Information Theory.