Nadezda Sukhorukova

Convex optimisation-based methods for K-complex detection

We develop three convex optimisation-based models for automatic detection of K-complexes.They extract key features of an EEG signal (a biological application).They significantly reduce the dimension of the problem and the computational time.They enhance the classification accuracy of an EEG signal in presence of K-complex.K-complexes are successfully detected in an EEG background. K-complex is a special type of electroencephalogram (EEG, brain activity) waveform that is used in sleep stage scoring. An automated detection of K-complexes is a desirable component of sleep stage monitoring. This automation is difficult due to the ambiguity of the scoring rules, complexity and extreme size of data. We develop three convex optimisation models that extract key features of EEG signals. These features are essential for detecting K-complexes. Our models are based on approximation of the original signals by sine functions with piecewise polynomial amplitudes. Then, the parameters of the corresponding approximations (rather than raw data) are used to detect the presence of K-complexes. The proposed approach significantly reduces the dimension of the classification problem (by extracting essential features) and the computational time while the classification accuracy is improved in most cases. Numerical results show that these models are efficient for detecting K-complexes.

Linear least squares problems involving fixed knots polynomial splines and their singularity study

In this paper, we study a class of approximation problems appearing in data approximation and signal processing. The approximations are constructed as combinations of polynomial splines (piecewise polynomials) whose parameters are subject to optimisation and so called prototype functions whose choice is based on the application rather than optimisation. We investigate two types of models, namely Model 1 and Model 2 in order to analyse the singularity of their system matrices. The main difference between these two models is that in Model 2, the signal is shifted vertically (signal biasing) by a polynomial spline function. The corresponding optimisation problems can be formulated as Linear Least Squares Problems (LLSPs). If the system matrix is non-singular, then, the corresponding problem can be solved inexpensively and efficiently, while for singular cases, slower (but more robust) methods have to be used. To choose a better suited method for solving the corresponding LLSPs we have developed a singularity verification rule. In this paper, we develop necessary and sufficient conditions for non-singularity of Model 1 and sufficient conditions for non-singularity of Model 2. These conditions can be verified much faster than the direct singularity verification of the system matrices. Therefore, the algorithm efficiency can be improved by choosing a suitable method for solving the corresponding LLSPs.

Chebyshev Multivariate Polynomial Approximation: Alternance Interpretation

In this paper, we derive optimality conditions for Chebyshev approximation of multivariate functions. The theory of Chebyshev (uniform) approximation for univariate functions was developed in the late nineteenth and twentieth century. The optimality conditions are based on the notion of alternance (maximal deviation points with alternating deviation signs). It is not clear, however, how to extend the notion of alternance to the case of multivariate functions. There have been several attempts to extend the theory of Chebyshev approximation to the case of multivariate functions. We propose an alternative approach, which is based on the notion of convexity and nonsmooth analysis.

Characterization Theorem for Best Polynomial Spline Approximation with Free Knots, Variable Degree and Fixed Tails

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

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.