## resampling and [GPU] parallelism

**I**n a recent note posted on arXiv, Lawrence Murray compares the implementation of resampling schemes for parallel systems like GPUs. Given a system of weighted particles, *(x _{i},ω_{i})*, there are several ways of drawing a sample according to those weights:

- regular
*multinomial resampling*, where each point in the (new) sample is one of the*(x*, with probability_{i},ω_{i})*(x*, meaning there is a uniform generated for each point;_{i},ω_{i}) *stratified resampling*, where the weights are added, divided into equal pieces and a uniform is sampled on each piece, which means that points with large weights are sampled at least once and those with small weights at most once;*systematic resampling*, which is the same as the above except that*the same*uniform is used for each piece,*Metropolis resampling*, where a Markov chain converges to the distribution (*ω*,…,_{1}*ω*on {1,…,P},_{P})

**T**he three first resamplers are common in the particle system literature (incl. Nicolas Chopin’s PhD thesis), but difficult to adapt to GPUs (and I always feel uncomfortable with the fact that systematic uses *a single uniform*!), while the last one is more unusual, but actually well-fitted for a parallel implementation. While Lawrence Murray suggests using Raftery and Lewis’ (1992) assessment of the required number of Metropolis iterations to “achieve convergence”, I would instead suggest taking advantage of the toric nature of the space (as represented above) to run a random walk and wait for the equivalent of a complete cycle. In any case, this is a cool illustration of the new challenges posed by parallel implementations (like the development of proper random generators).

March 22, 2012 at 4:43 am

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