9,079 research outputs found
Approximation with Random Bases: Pro et Contra
In this work we discuss the problem of selecting suitable approximators from
families of parameterized elementary functions that are known to be dense in a
Hilbert space of functions. We consider and analyze published procedures, both
randomized and deterministic, for selecting elements from these families that
have been shown to ensure the rate of convergence in norm of order
, where is the number of elements. We show that both randomized and
deterministic procedures are successful if additional information about the
families of functions to be approximated is provided. In the absence of such
additional information one may observe exponential growth of the number of
terms needed to approximate the function and/or extreme sensitivity of the
outcome of the approximation to parameters. Implications of our analysis for
applications of neural networks in modeling and control are illustrated with
examples.Comment: arXiv admin note: text overlap with arXiv:0905.067
Matrix Recipes for Hard Thresholding Methods
In this paper, we present and analyze a new set of low-rank recovery
algorithms for linear inverse problems within the class of hard thresholding
methods. We provide strategies on how to set up these algorithms via basic
ingredients for different configurations to achieve complexity vs. accuracy
tradeoffs. Moreover, we study acceleration schemes via memory-based techniques
and randomized, -approximate matrix projections to decrease the
computational costs in the recovery process. For most of the configurations, we
present theoretical analysis that guarantees convergence under mild problem
conditions. Simulation results demonstrate notable performance improvements as
compared to state-of-the-art algorithms both in terms of reconstruction
accuracy and computational complexity.Comment: 26 page
Variance Reduction Techniques in Monte Carlo Methods
Monte Carlo methods are simulation algorithms to estimate a numerical quantity in a statistical model of a real system. These algorithms are executed by computer programs. Variance reduction techniques (VRT) are needed, even though computer speed has been increasing dramatically, ever since the introduction of computers. This increased computer power has stimulated simulation analysts to develop ever more realistic models, so that the net result has not been faster execution of simulation experiments; e.g., some modern simulation models need hours or days for a single ’run’ (one replication of one scenario or combination of simulation input values). Moreover there are some simulation models that represent rare events which have extremely small probabilities of occurrence), so even modern computer would take ’for ever’ (centuries) to execute a single run - were it not that special VRT can reduce theses excessively long runtimes to practical magnitudes.common random numbers;antithetic random numbers;importance sampling;control variates;conditioning;stratied sampling;splitting;quasi Monte Carlo
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