Multiple pass streaming algorithms for learning mixtures of distributions in Rd

We present a multiple pass streaming algorithm for learning the density function of a mixture of $k$ uniform distributions over rectangles (cells) in $\reals^d$, for any $d>0$. Our learning model is: samples drawn according to the mixture are placed in {\it arbitrary order} in a data stream that may only be accessed sequentially by an algorithm with a very limited random access memory space. Our algorithm makes $2\ell+1$ passes, for any $\ell>0$, and requires memory at most $\tilde O(\epsilon^{-2/\ell}k^2d^4+(2k)^d)$. This exhibits a strong memory-space tradeoff: a few more passes significantly lowers its memory requirements, thus trading one of the two most important resources in streaming computation for the other. Chang and Kannan \cite{chang06} first considered this problem for $d=1, 2$. Our learning algorithm is especially appropriate for situations where massive data sets of samples are available, but practical computation with such large inputs requires very restricted models of computation