Chebyshev multivariate polynomial approximation and point reduction procedure
- Authors: Sukhorukova, Nadezda , Ugon, Julien , Yost, David
- Date: 2021
- Type: Text , Journal article
- Relation: Constructive Approximation Vol. 53, no. 3 (2021), p. 529-544
- Relation: http://purl.org/au-research/grants/arc/DP180100602
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- Description: We apply the methods of nonsmooth and convex analysis to extend the study of Chebyshev (uniform) approximation for univariate polynomial functions to the case of general multivariate functions (not just polynomials). First of all, we give new necessary and sufficient optimality conditions for multivariate approximation, and a geometrical interpretation of them which reduces to the classical alternating sequence condition in the univariate case. Then, we present a procedure for verification of necessary and sufficient optimality conditions that is based on our generalization of the notion of alternating sequence to the case of multivariate polynomials. Finally, we develop an algorithm for fast verification of necessary optimality conditions in the multivariate polynomial case. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.
Generalised rational approximation and its application to improve deep learning classifiers
- Authors: Peiris, V , Sharon, Nir , Sukhorukova, Nadezda , Ugon, Julien
- Date: 2021
- Type: Text , Journal article
- Relation: Applied Mathematics and Computation Vol. 389, no. (2021), p.
- Relation: https://purl.org/au-research/grants/arc/DP180100602
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- Description: A rational approximation (that is, approximation by a ratio of two polynomials) is a flexible alternative to polynomial approximation. In particular, rational functions exhibit accurate estimations to nonsmooth and non-Lipschitz functions, where polynomial approximations are not efficient. We prove that the optimisation problems appearing in the best uniform rational approximation and its generalisation to a ratio of linear combinations of basis functions are quasiconvex even when the basis functions are not restricted to monomials. Then we show how this fact can be used in the development of computational methods. This paper presents a theoretical study of the arising optimisation problems and provides results of several numerical experiments. We apply our approximation as a preprocessing step to deep learning classifiers and demonstrate that the classification accuracy is significantly improved compared to the classification of the raw signals. © 2020
- Description: This research was supported by the Australian Research Council (ARC), Solving hard Chebyshev approximation problems through nonsmooth analysis (Discovery Project DP180100602 ). This research was partially sponsored by Tel Aviv-Swinburne Research Collaboration Grant (2019).
Two curve Chebyshev approximation and its application to signal clustering
- Authors: Sukhorukova, Nadezda
- Date: 2019
- Type: Text , Journal article
- Relation: Applied Mathematics and Computation Vol. 356, no. (2019), p. 42-49
- Relation: http://purl.org/au-research/grants/arc/DP180100602
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- Description: In this paper, we extend a number of important results of the classical Chebyshev approximation theory to the case of simultaneous approximation of two or more functions. The need for this extension is application driven, since such kind of problems appears in the area of curve (signal) clustering. In this paper, we propose a new efficient algorithm for signal clustering and develop a procedure that allows one to reuse the results obtained at the previous iteration without recomputing the cluster centres from scratch. This approach is based on the extension of the classical de la Vallee-Poussin procedure originally developed for polynomial approximation. We also develop necessary and sufficient optimality conditions for two curve Chebyshev approximation, which is our core tool for curve clustering. These results are based on application of nonsmooth convex analysis. (C) 2019 Elsevier Inc. All rights reserved. In this paper, we extend a number of important results of the classical Chebyshev approximation theory to the case of simultaneous approximation of two or more functions. The need for this extension is application driven, since such kind of problems appears in the area of curve (signal) clustering. In this paper, we propose a new efficient algorithm for signal clustering and develop a procedure that allows one to reuse the results obtained at the previous iteration without recomputing the cluster centres from scratch. This approach is based on the extension of the classical de la Vallee-Poussin procedure originally developed for polynomial approximation. We also develop necessary and sufficient optimality conditions for two curve Chebyshev approximation, which is our core tool for curve clustering. These results are based on application of nonsmooth convex analysis. (C) 2019 Elsevier Inc. All rights reserved.
Characterization theorem for best polynomial spline approximation with free knots, variable degree and fixed tails
- Authors: Crouzeix, Jean-Pierre , Sukhorukova, Nadezda , Ugon, Julien
- Date: 2017
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 172, no. 3 (2017), p. 950-964
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- Description: In this paper, we derive a necessary condition for a best approximation by piecewise polynomial functions of varying degree from one interval to another. Based on these results, we obtain a characterization theorem for the polynomial splines with fixed tails, that is the value of the spline is fixed in one or more knots (external or internal). We apply nonsmooth nonconvex analysis to obtain this result, which is also a necessary and sufficient condition for inf-stationarity in the sense of Demyanov-Rubinov. This paper is an extension of a paper where similar conditions were obtained for free tails splines. The main results of this paper are essential for the development of a Remez-type algorithm for free knot spline approximation.
Chebyshev approximation by linear combinations of fixed knot polynomial splines with weighting functions
- Authors: Sukhorukova, Nadezda , Ugon, Julien
- Date: 2016
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 171, no. 2 (2016), p. 536-549
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- Description: In this paper, we derive conditions for best uniform approximation by fixed knots polynomial splines with weighting functions. The theory of Chebyshev approximation for fixed knots polynomial functions is very elegant and complete. Necessary and sufficient optimality conditions have been developed leading to efficient algorithms for constructing optimal spline approximations. The optimality conditions are based on the notion of alternance (maximal deviation points with alternating deviation signs). In this paper, we extend these results to the case when the model function is a product of fixed knots polynomial splines (whose parameters are subject to optimization) and other functions (whose parameters are predefined). This problem is nonsmooth, and therefore, we make use of convex and nonsmooth analysis to solve it.
A necessary optimality condition for free knots linear splines: Special cases
- Authors: Sukhorukova, Nadezda
- Date: 2010
- Type: Text , Journal article
- Relation: Pacific Journal of Optimization Vol. 6, no. 2, Suppl. 1 (2010), p. 305-317
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- Description: In this paper, we study the problem of best Chebyshev approximation by linear splines. We construct linear splines as a max - min of linear functions. Then we apply nonsmooth optimisation techniques to analyse and solve the corresponding optimisation problems. This approach allows us to identify and introduce a new important property of linear spline knots (regular and irregular). Using this property, we derive a necessary optimality condition for the case of regular knots. This condition is stronger than those existing in the literature. We also present a numerical example which demonstrates the difference between the old and the new optimality conditions.
Uniform approximation by the highest defect continuous polynomial splines : Necessary and sufficient optimality conditions and their generalisations
- Authors: Sukhorukova, Nadezda
- Date: 2010
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 147, no. 2 (2010), p. 378-394
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- Description: In this paper necessary and sufficient optimality conditions for uniform approximation of continuous functions by polynomial splines with fixed knots are derived. The obtained results are generalisations of the existing results obtained for polynomial approximation and polynomial spline approximation. The main result is two-fold. First, the generalisation of the existing results to the case when the degree of the polynomials, which compose polynomial splines, can vary from one subinterval to another. Second, the construction of necessary and sufficient optimality conditions for polynomial spline approximation with fixed values of the splines at one or both borders of the corresponding approximation interval. © 2010 Springer Science+Business Media, LLC.
Vallee poussin theorem and remez algorithm in the case of generalised degree polynomial spline approximation
- Authors: Sukhorukova, Nadezda
- Date: 2010
- Type: Text , Journal article
- Relation: Pacific Journal of Optimization Vol. 6, no. 1 (2010), p. 103-114
- Full Text: false
- Description: The classical Remez algorithm was developed for constructing the best polynomial approximations for continuous and discrete functions in an interval. In this paper the classical Remez algorithm is generalised to the problem of polynomial spline (piece-wise polynomial) approximation with the spline defect equal to the spline degree. Also, the values of the splines in the end points of the approximation interval may be fixed Polynomial splines combine simplicity of polynomials and flexibility, which allows one to significantly decrease the degree of the corresponding polynomials and oscillations of deviation functions. Therefore polynomial splines are a powerful tool for function and data approximation. The generalisation of the Remez algorithm developed in this research has been tested on several approximation problems. The results of the numerical experiments are presented.