Xiongming Dai

Xiongming Dai, Gerald Baumgartner

We propose a generalized minimum discrepancy, which derives from Legendre's ODE and spherical harmonic theoretics to provide a new criterion of equidistributed pointsets on the sphere. A continuous and derivative kernel in terms of elementary functions is established to simplify the computation of the generalized minimum discrepancy. We consider the deterministic point generated from Pycke's statistics to integrate a Franke function for the sphere and investigate the discrepancies of points systems embedding with different kernels. Quantitive experiments are conducted and the results are analyzed. Our deduced model can explore latent point systems, that have the minimum discrepancy without the involvement of pseudodifferential operators and Beltrami operators, by the use of derivatives. Compared to the random point generated from the Monte Carlo method, only a few points generated by our method are required to approximate the target in arbitrary dimensions.

Xiongming Dai, Gerald Baumgartner

A resampling scheme provides a way to switch low-weight particles for sequential Monte Carlo with higher-weight particles representing the objective distribution. The less the variance of the weight distribution is, the more concentrated the effective particles are, and the quicker and more accurate it is to approximate the hidden Markov model, especially for the nonlinear case. We propose a repetitive deterministic domain with median ergodicity for resampling and have achieved the lowest variances compared to the other resampling methods. As the size of the deterministic domain $M\ll N$ (the size of population), given a feasible size of particles, our algorithm is faster than the state of the art, which is verified by theoretical deduction and experiments of a hidden Markov model in both the linear and non-linear cases.

Xiongming Dai, Gerald Baumgartner

Markov chain Monte Carlo (MCMC) provides a feasible method for inferring Hidden Markov models, however, it is often computationally prohibitive, especially constrained by the curse of dimensionality, as the Monte Carlo sampler traverses randomly taking small steps within uncertain regions in the parameter space. We are the first to consider the posterior distribution of the objective as a mapping of samples in an infinite-dimensional Euclidean space where deterministic submanifolds are embedded and propose a new criterion by maximizing the weighted Riesz polarization quantity, to discretize rectifiable submanifolds via pairwise interaction. We study the characteristics of Chebyshev particles and embed them into sequential MCMC, a novel sampler with a high acceptance ratio that proposes only a few evaluations. We have achieved high performance from the experiments for parameter inference in a linear Gaussian state-space model with synthetic data and a non-linear stochastic volatility model with real-world data.