90 research outputs found
Fast randomized iteration: diffusion Monte Carlo through the lens of numerical linear algebra
We review the basic outline of the highly successful diffusion Monte Carlo
technique commonly used in contexts ranging from electronic structure
calculations to rare event simulation and data assimilation, and propose a new
class of randomized iterative algorithms based on similar principles to address
a variety of common tasks in numerical linear algebra. From the point of view
of numerical linear algebra, the main novelty of the Fast Randomized Iteration
schemes described in this article is that they work in either linear or
constant cost per iteration (and in total, under appropriate conditions) and
are rather versatile: we will show how they apply to solution of linear
systems, eigenvalue problems, and matrix exponentiation, in dimensions far
beyond the present limits of numerical linear algebra. While traditional
iterative methods in numerical linear algebra were created in part to deal with
instances where a matrix (of size ) is too big to store, the
algorithms that we propose are effective even in instances where the solution
vector itself (of size ) may be too big to store or manipulate.
In fact, our work is motivated by recent DMC based quantum Monte Carlo schemes
that have been applied to matrices as large as . We
provide basic convergence results, discuss the dependence of these results on
the dimension of the system, and demonstrate dramatic cost savings on a range
of test problems.Comment: 44 pages, 7 figure
Nonnegative approximations of nonnegative tensors
We study the decomposition of a nonnegative tensor into a minimal sum of
outer product of nonnegative vectors and the associated parsimonious naive
Bayes probabilistic model. We show that the corresponding approximation
problem, which is central to nonnegative PARAFAC, will always have optimal
solutions. The result holds for any choice of norms and, under a mild
assumption, even Bregman divergences.Comment: 14 page
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