4,412 research outputs found
Accurate and Efficient Expression Evaluation and Linear Algebra
We survey and unify recent results on the existence of accurate algorithms
for evaluating multivariate polynomials, and more generally for accurate
numerical linear algebra with structured matrices. By "accurate" we mean that
the computed answer has relative error less than 1, i.e., has some correct
leading digits. We also address efficiency, by which we mean algorithms that
run in polynomial time in the size of the input. Our results will depend
strongly on the model of arithmetic: Most of our results will use the so-called
Traditional Model (TM). We give a set of necessary and sufficient conditions to
decide whether a high accuracy algorithm exists in the TM, and describe
progress toward a decision procedure that will take any problem and provide
either a high accuracy algorithm or a proof that none exists. When no accurate
algorithm exists in the TM, it is natural to extend the set of available
accurate operations by a library of additional operations, such as , dot
products, or indeed any enumerable set which could then be used to build
further accurate algorithms. We show how our accurate algorithms and decision
procedure for finding them extend to this case. Finally, we address other
models of arithmetic, and the relationship between (im)possibility in the TM
and (in)efficient algorithms operating on numbers represented as bit strings.Comment: 49 pages, 6 figures, 1 tabl
Distance to the Nearest Stable Metzler Matrix
This paper considers the non-convex problem of finding the nearest Metzler
matrix to a given possibly unstable matrix. Linear systems whose state vector
evolves according to a Metzler matrix have many desirable properties in
analysis and control with regard to scalability. This motivates the question,
how close (in the Frobenius norm of coefficients) to the nearest Metzler matrix
are we? Dropping the Metzler constraint, this problem has recently been studied
using the theory of dissipative Hamiltonian (DH) systems, which provide a
helpful characterization of the feasible set of stable matrices. This work uses
the DH theory to provide a block coordinate descent algorithm consisting of a
quadratic program with favourable structural properties and a semidefinite
program for which recent diagonal dominance results can be used to improve
tractability.Comment: To Appear in Proc. of 56th IEEE CD
Kronecker Sum Decompositions of Space-Time Data
In this paper we consider the use of the space vs. time Kronecker product
decomposition in the estimation of covariance matrices for spatio-temporal
data. This decomposition imposes lower dimensional structure on the estimated
covariance matrix, thus reducing the number of samples required for estimation.
To allow a smooth tradeoff between the reduction in the number of parameters
(to reduce estimation variance) and the accuracy of the covariance
approximation (affecting estimation bias), we introduce a diagonally loaded
modification of the sum of kronecker products representation [1]. We derive a
Cramer-Rao bound (CRB) on the minimum attainable mean squared predictor
coefficient estimation error for unbiased estimators of Kronecker structured
covariance matrices. We illustrate the accuracy of the diagonally loaded
Kronecker sum decomposition by applying it to video data of human activity.Comment: 5 pages, 8 figures, accepted to CAMSAP 201
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