Stratification and optimal resampling for sequential Monte Carlo

Summary Sequential Monte Carlo algorithms are widely accepted as powerful computational tools for making inference with dynamical systems. A key step in sequential is resampling, which plays the role of steering algorithm towards future dynamics. Several strategies have been used practice, including multinomial residual optimal stratified resampling and transport resampling. In one-dimensional cases, we show that equivalent to on sorted particles, both minimize variance well expected squared energy distance between original resampled empirical distributions. For general $d$-dimensional if particles first using Hilbert curve, $O(m^{-(1+2/d)})$, an improvement over best previously known rate $O(m^{-(1+1/d)})$, where $m$ number particles. We this improved ordered schemes, conjectured Gerber et al. (2019). also present almost-sure bound Wasserstein Hilbert-curve-resampled light these results, dimension $d>1$ mean square error quasi-Monte $n$ can be $O(n^{-1-4/\{d(d+4)\}})$ curve a specific low-discrepancy set chosen. To our knowledge, convergence lower than $o(n^{-1})$.