70 research outputs found
On largest volume simplices and sub-determinants
We show that the problem of finding the simplex of largest volume in the
convex hull of points in can be approximated with a factor
of in polynomial time. This improves upon the previously best
known approximation guarantee of by Khachiyan. On the other hand,
we show that there exists a constant such that this problem cannot be
approximated with a factor of , unless . % This improves over the
inapproximability that was previously known. Our hardness result holds
even if , in which case there exists a \bar c\,^{d}-approximation
algorithm that relies on recent sampling techniques, where is again a
constant. We show that similar results hold for the problem of finding the
largest absolute value of a subdeterminant of a matrix
On maximum volume submatrices and cross approximation for symmetric semidefinite and diagonally dominant matrices
The problem of finding a submatrix of maximum volume of a matrix
is of interest in a variety of applications. For example, it yields a
quasi-best low-rank approximation constructed from the rows and columns of .
We show that such a submatrix can always be chosen to be a principal submatrix
if is symmetric semidefinite or diagonally dominant. Then we analyze the
low-rank approximation error returned by a greedy method for volume
maximization, cross approximation with complete pivoting. Our bound for general
matrices extends an existing result for symmetric semidefinite matrices and
yields new error estimates for diagonally dominant matrices. In particular, for
doubly diagonally dominant matrices the error is shown to remain within a
modest factor of the best approximation error. We also illustrate how the
application of our results to cross approximation for functions leads to new
and better convergence results
Trivariate polynomial approximation on Lissajous curves
We study Lissajous curves in the 3-cube, that generate algebraic cubature
formulas on a special family of rank-1 Chebyshev lattices. These formulas are
used to construct trivariate hyperinterpolation polynomials via a single 1-d
Fast Chebyshev Transform (by the Chebfun package), and to compute discrete
extremal sets of Fekete and Leja type for trivariate polynomial interpolation.
Applications could arise in the framework of Lissajous sampling for MPI
(Magnetic Particle Imaging)
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