1,291 research outputs found
New Algorithms for Solving Tropical Linear Systems
The problem of solving tropical linear systems, a natural problem of tropical
mathematics, has already proven to be very interesting from the algorithmic
point of view: it is known to be in but no polynomial time
algorithm is known, although counterexamples for existing pseudopolynomial
algorithms are (and have to be) very complex.
In this work, we continue the study of algorithms for solving tropical linear
systems. First, we present a new reformulation of Grigoriev's algorithm that
brings it closer to the algorithm of Akian, Gaubert, and Guterman; this lets us
formulate a whole family of new algorithms, and we present algorithms from this
family for which no known superpolynomial counterexamples work. Second, we
present a family of algorithms for solving overdetermined tropical systems. We
show that for weakly overdetermined systems, there are polynomial algorithms in
this family. We also present a concrete algorithm from this family that can
solve a tropical linear system defined by an matrix with maximal
element in time , and this time matches the complexity of the best of
previously known algorithms for feasibility testing.Comment: 17 page
Optimal Uniform Convergence Rates for Sieve Nonparametric Instrumental Variables Regression
We study the problem of nonparametric regression when the regressor is
endogenous, which is an important nonparametric instrumental variables (NPIV)
regression in econometrics and a difficult ill-posed inverse problem with
unknown operator in statistics. We first establish a general upper bound on the
sup-norm (uniform) convergence rate of a sieve estimator, allowing for
endogenous regressors and weakly dependent data. This result leads to the
optimal sup-norm convergence rates for spline and wavelet least squares
regression estimators under weakly dependent data and heavy-tailed error terms.
This upper bound also yields the sup-norm convergence rates for sieve NPIV
estimators under i.i.d. data: the rates coincide with the known optimal
-norm rates for severely ill-posed problems, and are power of
slower than the optimal -norm rates for mildly ill-posed problems. We then
establish the minimax risk lower bound in sup-norm loss, which coincides with
our upper bounds on sup-norm rates for the spline and wavelet sieve NPIV
estimators. This sup-norm rate optimality provides another justification for
the wide application of sieve NPIV estimators. Useful results on
weakly-dependent random matrices are also provided
Strong regularity of matrices in a discrete bounded bottleneck algebra
AbstractThe results concerning strong regularity of matrices over bottleneck algebras are reviewed. We extend the known conditions to the discrete bounded case and modify the known algorithms for testing strong regularity
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