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Regularization Of Inverse Problems

This paper presents a methodological review of models for predicting blood glucose (BG) concentration, risks and BG events. The surveyed models are classified into three categories, and they are presented in summary tables containing the most relevant data regarding the experimental setup for fitting and testing each model as well as the input signals and the performance metrics. Each category exhibits trends that are presented and discussed. This document aims to be a compact guide to determine the modeling options that are currently being exploited for personalized BG prediction.

A review of personalized blood glucose prediction strategies for T1DM patients

Within the framework of statistical learning theory we analyze in detail the so-called elastic-net regularization scheme proposed by Zou and Hastie [H. Zou, T. Hastie, Regularization and variable selection via the elastic net, J. R. Stat. Soc. Ser. B, 67(2) (2005) 301-320] for the selection of groups of correlated variables. To investigate the statistical properties of this scheme and in particular its consistency properties, we set up a suitable mathematical framework. Our setting is random-design regression where we allow the response variable to be vector-valued and we consider prediction functions which are linear combinations of elements (features) in an infinite-dimensional dictionary. Under the assumption that the regression function admits a sparse representation on the dictionary, we prove that there exists a particular ''elastic-net representation'' of the regression function such that, if the number of data increases, the elastic-net estimator is consistent not only for prediction but also for variable/feature selection. Our results include finite-sample bounds and an adaptive scheme to select the regularization parameter. Moreover, using convex analysis tools, we derive an iterative thresholding algorithm for computing the elastic-net solution which is different from the optimization procedure originally proposed in the above-cited work.

Elastic-net regularization in learning theory

Background: Estimation of future glucose concentrations is a crucial task for diabetes management. Predicted glucose values can be used for early hypoglycemic/hyperglycemic alarms or for adjustment of insulin injections or insulin infusion rates of manual or automated pumps. Continuous glucose monitoring (CGM) technologies provide glucose readings at a high frequency and consequently detailed insight into the subject's glucose variations. The objective of this research is to develop reliable subject-specific glucose prediction models using CGM data. Methods: Two separate patient databases collected under hospitalized (disturbance-free) and normal daily life conditions are used for validation of the proposed glucose prediction algorithm. Both databases consist of glucose concentration data collected at 5-min intervals using a CGM device. Using time-series analysis, low-order linear models are developed from patients' own CGM data. The time-series models are integrated with recursive identification...

Estimation of future glucose concentrations with subject-specific recursive linear models.

The strategy of early stopping is a regularization technique based on choosing a stopping time for an iterative algorithm. Focusing on non-parametric regression in a reproducing kernel Hilbert space, we analyze the early stopping strategy for a form of gradient-descent applied to the least-squares loss function. We propose a data-dependent stopping rule that does not involve hold-out or cross-validation data, and we prove upper bounds on the squared error of the resulting function estimate, measured in either the $L^2(P)$ and $L^2(P_n)$ norm. These upper bounds lead to minimax-optimal rates for various kernel classes, including Sobolev smoothness classes and other forms of reproducing kernel Hilbert spaces. We show through simulation that our stopping rule compares favorably to two other stopping rules, one based on hold-out data and the other based on Stein's unbiased risk estimate. We also establish a tight connection between our early stopping strategy and the solution path of a kernel ridge regression estimator.

Early stopping and non-parametric regression: An optimal data-dependent stopping rule

We consider the question of 30-minute prediction of blood glucose levels measured by continuous glucose monitoring devices, using clinical data. While most studies of this nature deal with one patient at a time, we take a certain percentage of patients in the data set as training data, and test on the remainder of the patients; i.e., the machine need not re-calibrate on the new patients in the data set. We demonstrate how deep learning can outperform shallow networks in this example. One novelty is to demonstrate how a parsimonious deep representation can be constructed using domain knowledge.

/pdf/a-deep-learning-approach-to-diabetic-blood-glucose-1d4yjjqc7n.pdf

A Deep Learning Approach to Diabetic Blood Glucose Prediction

https://www.ricam.oeaw.ac.at/files/reports/11/rep11-31.pdf

A meta-learning approach to the regularized learning-Case study: Blood glucose prediction

The regularization parameter choice is a fundamental problem in Learning Theory since the performance of most supervised algorithms crucially depends on the choice of one or more of such parameters. In particular a main theoretical issue regards the amount of prior knowledge needed to choose the regularization parameter in order to obtain good learning rates. In this paper we present a parameter choice strategy, called the balancing principle, to choose the regularization parameter without knowledge of the regularity of the target function. Such a choice adaptively achieves the best error rate. Our main result applies to regularization algorithms in reproducing kernel Hilbert space with the square loss, though we also study how a similar principle can be used in other situations. As a straightforward corollary we can immediately derive adaptive parameter choices for various kernel methods recently studied. Numerical experiments with the proposed parameter choice rules are also presented.

/pdf/adaptive-kernel-methods-using-the-balancing-principle-d7dla9q4pe.pdf

Adaptive Kernel Methods Using the Balancing Principle

/pdf/a-deep-learning-approach-to-diabetic-blood-glucose-5b59fqnq20.pdf

A deep learning approach to diabetic blood glucose prediction

Determination of the distribution function of relaxation times (DFRT) is an approach that gives us more detailed insight into system processes, which are not observable by simple electrochemical impedance spectroscopy (EIS) measurements. DFRT maps EIS data into a function containing the timescale characteristics of the system under consideration. The extraction of such characteristics from noisy EIS measurements can be described by Fredholm integral equation of the first kind that is known to be ill-posed and can be treated only with regularization techniques. Moreover, since only a finite number of EIS data may actually be obtained, the above-mentioned equation appears as after application of a collocation method that needs to be combined with the regularization. In the present study, we discuss how a regularized collocation of DFRT problem can be implemented such that all appearing quantities allow symbolic computations as sums of table integrals. The proposed implementation of the regularized collocation is treated as a multi-parameter regularization. Another contribution of the present work is the adjustment of the previously proposed multiple parameter choice strategy to the context of DFRT problem. The resulting strategy is based on the aggregation of all computed regularized approximants, and can be in principle used in synergy with other methods for solving DFRT problem. We also report the results from the experiments that apply the synthetic data showing that the proposed technique successfully reproduced known exact DFRT. The data obtained by our techniques is also compared to data obtained by well-known DFRT software (DRTtools).

/pdf/adaptive-multi-parameter-regularization-approach-to-2shkqczoei.pdf

Sergei V. Pereverzyev

Papers

A Deep Learning Approach to Diabetic Blood Glucose Prediction

A meta-learning approach to the regularized learning-Case study: Blood glucose prediction

Adaptive Kernel Methods Using the Balancing Principle

A deep learning approach to diabetic blood glucose prediction

Adaptive multi-parameter regularization approach to construct the distribution function of relaxation times