  Sequential Monte Carlo (SMC) is a class of algorithms that approximate
high-dimensional expectations of a Markov chain. SMC algorithms typically
include a resampling step. There are many possible ways to resample, but the
relative advantages of different resampling schemes remains poorly understood.
Here, a theoretical framework for comparing resampling schemes is presented.
The framework uses resampling matrices to provide a simple description for the
SMC resampling step. The framework identifies the matrix resampling scheme that
gives the lowest possible error. The framework leads to new asymptotic error
formulas that can be used to compare different resampling schemes.
