61,586 research outputs found
Rigorous Polynomial Approximation using Taylor Models in Coq
International audienceOne of the most common and practical ways of representing a real function on machines is by using a polynomial approximation. It is then important to properly handle the error introduced by such an approximation. The purpose of this work is to offer guaranteed error bounds for a specific kind of rigorous polynomial approximation called Taylor model. We carry out this work in the Coq proof assistant, with a special focus on genericity and efficiency for our implementation. We give an abstract interface for rigorous polynomial approximations, parameter- ized by the type of coefficients and the implementation of polynomials, and we instantiate this interface to the case of Taylor models with inter- val coefficients, while providing all the machinery for computing them. We compare the performances of our implementation in Coq with those of the Sollya tool, which contains an implementation of Taylor models written in C. This is a milestone in our long-term goal of providing fully formally proved and efficient Taylor models
Polynomial Tensor Sketch for Element-wise Function of Low-Rank Matrix
This paper studies how to sketch element-wise functions of low-rank matrices.
Formally, given low-rank matrix A = [Aij] and scalar non-linear function f, we
aim for finding an approximated low-rank representation of the (possibly
high-rank) matrix [f(Aij)]. To this end, we propose an efficient
sketching-based algorithm whose complexity is significantly lower than the
number of entries of A, i.e., it runs without accessing all entries of [f(Aij)]
explicitly. The main idea underlying our method is to combine a polynomial
approximation of f with the existing tensor sketch scheme for approximating
monomials of entries of A. To balance the errors of the two approximation
components in an optimal manner, we propose a novel regression formula to find
polynomial coefficients given A and f. In particular, we utilize a
coreset-based regression with a rigorous approximation guarantee. Finally, we
demonstrate the applicability and superiority of the proposed scheme under
various machine learning tasks
Chebyshev Interpolation Polynomial-based Tools for Rigorous Computing
17 pagesInternational audiencePerforming numerical computations, yet being able to provide rigorous mathematical statements about the obtained result, is required in many domains like global optimization, ODE solving or integration. Taylor models, which associate to a function a pair made of a Taylor approximation polynomial and a rigorous remainder bound, are a widely used rigorous computation tool. This approach benefits from the advantages of numerical methods, but also gives the ability to make reliable statements about the approximated function. Despite the fact that approximation polynomials based on interpolation at Chebyshev nodes offer a quasi-optimal approximation to a function, together with several other useful features, an analogous to Taylor models, based on such polynomials, has not been yet well-established in the field of validated numerics. This paper presents a preliminary work for obtaining such interpolation polynomials together with validated interval bounds for approximating univariate functions. We propose two methods that make practical the use of this: one is based on a representation in Newton basis and the other uses Chebyshev polynomial basis. We compare the quality of the obtained remainders and the performance of the approaches to the ones provided by Taylor models
Energy-Efficient Flow Scheduling and Routing with Hard Deadlines in Data Center Networks
The power consumption of enormous network devices in data centers has emerged
as a big concern to data center operators. Despite many
traffic-engineering-based solutions, very little attention has been paid on
performance-guaranteed energy saving schemes. In this paper, we propose a novel
energy-saving model for data center networks by scheduling and routing
"deadline-constrained flows" where the transmission of every flow has to be
accomplished before a rigorous deadline, being the most critical requirement in
production data center networks. Based on speed scaling and power-down energy
saving strategies for network devices, we aim to explore the most energy
efficient way of scheduling and routing flows on the network, as well as
determining the transmission speed for every flow. We consider two general
versions of the problem. For the version of only flow scheduling where routes
of flows are pre-given, we show that it can be solved polynomially and we
develop an optimal combinatorial algorithm for it. For the version of joint
flow scheduling and routing, we prove that it is strongly NP-hard and cannot
have a Fully Polynomial-Time Approximation Scheme (FPTAS) unless P=NP. Based on
a relaxation and randomized rounding technique, we provide an efficient
approximation algorithm which can guarantee a provable performance ratio with
respect to a polynomial of the total number of flows.Comment: 11 pages, accepted by ICDCS'1
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