Pavlo Tkachenko

A linear functional strategy for regularized ranking

Regularization schemes are frequently used for performing ranking tasks. This topic has been intensively studied in recent years. However, to be effective a regularization scheme should be equipped with a suitable strategy for choosing a regularization parameter. In the present study we discuss an approach, which is based on the idea of a linear combination of regularized rankers corresponding to different values of the regularization parameter. The coefficients of the linear combination are estimated by means of the so-called linear functional strategy. We provide a theoretical justification of the proposed approach and illustrate them by numerical experiments. Some of them are related with ranking the risk of nocturnal hypoglycemia of diabetes patients.

On the convergence rate and some applications of regularized ranking algorithms

This paper studies the ranking problem in the context of the regularization theory that allows a simultaneous analysis of a wide class of ranking algorithms. Some of them were previously studied separately. For such ones, our analysis gives a better convergence rate compared to the reported in the literature. We also supplement our theoretical results with numerical illustrations and discuss the application of ranking to the problem of estimating the risk from errors in blood glucose measurements of diabetic patients.

Nyström type subsampling analyzed as a regularized projection

In the statistical learning theory the Nyström type subsampling methods are considered as tools for dealing with big data. In this paper we consider Nyström subsampling as a special form of the projected Lavrentiev regularization, and study it using the approaches developed in the regularization theory. As a result, we prove that the same capacity independent learning rates that are quaranteed for standard algorithms running with quadratic computational complexity can be obtained with subquadratic complexity by the Nyström subsampling approach, provided that the subsampling size is chosen properly. We propose a priori rule for choosing the subsampling size and a posteriori strategy for dealing with uncertainty in the choice of it. The theoretical results are illustrated by numerical experiments.

Glycemic Control Indices and Their Aggregation in the Prediction of Nocturnal Hypoglycemia From Intermittent Blood Glucose Measurements

Background: Despite the risk associated with nocturnal hypoglycemia (NH) there are only a few methods aiming at the prediction of such events based on intermittent blood glucose monitoring data. One of the first methods that potentially can be used for NH prediction is based on the low blood glucose index (LBGI) and suggested, for example, in Accu-Chek® Connect as a hypoglycemia risk indicator. On the other hand, nowadays there are other glucose control indices (GCI), which could be used for NH prediction in the same spirit as LBGI. In the present study we propose a general approach of combining NH predictors constructed from different GCI. Methods: The approach is based on a recently developed strategy for aggregating ranking algorithms in machine learning. NH predictors have been calibrated and tested on data extracted from clinical trials, performed in EU FP7-funded project DIAdvisor. Then, to show a portability of the method we have tested it on another dataset that was received from EU Horizon 2020-funded project AMMODIT. Results: We exemplify the proposed approach by aggregating NH predictors that have been constructed based on 4 GCI associated with hypoglycemia. Even though these predictors have been preliminary optimized to exhibit better performance on the considered dataset, our aggregation approach allows a further performance improvement. On the dataset, where a portability of the proposed approach has been demonstrated, the aggregating predictor has exhibited the following performance: sensitivity 77%, specificity 83.4%, positive predictive value 80.2%, negative predictive value 80.6%, which is higher than conventionally considered as acceptable. Conclusion: The proposed approach shows potential to be used in telemedicine systems for NH prediction.

Prediction of nocturnal hypoglycemia by an aggregation of previously known prediction approaches

BACKGROUND AND OBJECTIVE Nocturnal hypoglycemia (NH) is common in patients with insulin-treated diabetes. Despite the risk associated with NH, there are only a few methods aiming at the prediction of such events based on intermittent blood glucose monitoring data and none has been validated for clinical use. Here we propose a method of combining several predictors into a new one that will perform at the level of the best involved one, or even outperform all individual candidates. METHODS The idea of the method is to use a recently developed strategy for aggregating ranking algorithms. The method has been calibrated and tested on data extracted from clinical trials, performed in the European FP7-funded project DIAdvisor. Then we have tested the proposed approach on other datasets to show the portability of the method. This feature of the method allows its simple implementation in the form of a diabetic smartphone app. RESULTS On the considered datasets the proposed approach exhibits good performance in terms of sensitivity, specificity and predictive values. Moreover, the resulting predictor automatically performs at the level of the best involved method or even outperforms it. CONCLUSION We propose a strategy for a combination of NH predictors that leads to a method exhibiting a reliable performance and the potential for everyday use by any patient who performs self-monitoring of blood glucose.

Two-parameter regularization of ill-posed spherical pseudo-differential equations in the space of continuous functions

In this paper, a two-step regularization method is used to solve an ill-posed spherical pseudo-differential equation in the presence of noisy data. For the first step of regularization we approximate the data by means of a spherical polynomial that minimizes a functional with a penalty term consisting of the squared norm in a Sobolev space. The second step is a regularized collocation method. An error bound is obtained in the uniform norm, which is potentially smaller than that for either the noise reduction alone or the regularized collocation alone. We discuss an a posteriori parameter choice, and present some numerical experiments, which support the claimed superiority of the two-step method.

Parameter Choice Strategies for Least-squares Approximation of Noisy Smooth Functions on the Sphere

We consider a polynomial reconstruction of smooth functions from their noisy values at discrete nodes on the unit sphere by a variant of the regularized least-squares method of An et al. [SIAM J. Numer. Anal., 50 (2012), pp. 1513--1534]. As nodes we use the points of a positive-weight cubature formula that is exact for all spherical polynomials of degree up to 2M, where M is the degree of the reconstructing polynomial. We first obtain a reconstruction error bound in terms of the regularization parameter and the penalization parameters in the regularization operator. Then we discuss a priori and a posteriori strategies for choosing these parameters. Finally, we give numerical examples illustrating the theoretical results.

Regularization by the Linear Functional Strategy with Multiple Kernels

The choice of the kernel is known to be a challenging and central problem of kernel based supervised learning. Recent applications and significant amount of literature have shown that using multiple kernels (the so-called Multiple Kernel Learning (MKL)) instead of a single one can enhance the interpretability of the learned function and improve performances. However, a comparison of existing MKL-algorithms shows that though there may not be large differences in terms of accuracy, there is difference between MKL-algorithms in complexity as given by the training time, for example. In this paper we present a promising approach for training the MKL-machine by the linear functional strategy, which is either faster or more accurate than previously known ones.

Pointwise Computation in an Ill-Posed Spherical Pseudo-Differential Equation

Abstract In the present paper, we consider the approximation of the solution of an ill-posed spherical pseudo-differential equation at a given point. While the methods for approximating the whole solution are well-studied in Hilbert spaces, such as the space of square-summable functions, the computation of values of the solution at given points is much less studied. This can be explained, in particular, by the fact that for square-summable functions the functional of pointwise evaluation is, in general, not well defined. To overcome this limitation we adjust the regularized least-squares method of An, Chen, Sloan and Womersley [Siam J. Numer. Anal. 50 (2012), no. 3, 1513–1534] by using a special a posteriori parameter choice rule. We also illustrate our theoretical findings by numerical results for the reconstruction of the solution at a given point.

Joint Inversion of Multiple Observations

Joint inversion becomes increasingly important with the availability of various types of measurements related to the same quantity. Questions arising in this context are how to combine the different data sets in the first place and, secondly, how to choose the multiple parameters that naturally occur in such a combination. This chapter discusses some recently proposed techniques addressing these issues. Additionally, we distinguish the two cases when all underlying problems are ill posed (e.g., satellite data only) and when some of them are not ill posed (e.g., satellite data is complemented by data at the Earth surface). Theoretical discussions of the topics above are presented as well as numerical experiments with different settings of simulated data.