Valeriya Naumova

Parameter Choice Strategies for Multipenalty Regularization

The widespread applicability of the multipenalty regularization is limited by the fact that theoretically optimal rate of reconstruction for a given problem can be realized by a one-parameter counterpart, provided that relevant information on the problem is available and taken into account in the regularization. In this paper, we explore the situation where no such information is given, but still accuracy of optimal order can be guaranteed by employing multipenalty regularization. Our focus is on the analysis and the justification of an a posteriori parameter choice rule for such a regularization scheme. First we present a modified version of the discrepancy principle within the multipenalty regularization framework. As a consequence we provide a theoretical justification to the multipenalty regularization scheme equipped with the a posteriori parameter choice rule. We then establish a fast numerical realization of the proposed discrepancy principle based on a model function approximation. Finally, we pro...

Fast dictionary learning from incomplete data

This paper extends the recently proposed and theoretically justified iterative thresholding and K residual means (ITKrM) algorithm to learning dictionaries from incomplete/masked training data (ITKrMM). It further adapts the algorithm to the presence of a low-rank component in the data and provides a strategy for recovering this low-rank component again from incomplete data. Several synthetic experiments show the advantages of incorporating information about the corruption into the algorithm. Further experiments on image data confirm the importance of considering a low-rank component in the data and show that the algorithm compares favourably to its closest dictionary learning counterparts, wKSVD and BPFA, either in terms of computational complexity or in terms of consistency between the dictionaries learned from corrupted and uncorrupted data. To further confirm the appropriateness of the learned dictionaries, we explore an application to sparsity-based image inpainting. There the ITKrMM dictionaries show a similar performance to other learned dictionaries like wKSVD and BPFA and a superior performance to other algorithms based on pre-defined/analytic dictionaries.

Minimization of multi-penalty functionals by alternating iterative thresholding and optimal parameter choices

Inspired by several recent developments in regularization theory, optimization, and signal processing, we present and analyze a numerical approach to multi-penalty regularization in spaces of sparsely represented functions. The sparsity prior is motivated by the largely expected geometrical/structured features of high-dimensional data, which may not be well-represented in the framework of typically more isotropic Hilbert spaces. In this paper, we are particularly interested in regularizers which are able to correctly model and separate the multiple components of additively mixed signals. This situation is rather common as pure signals may be corrupted by additive noise. To this end, we consider a regularization functional composed by a data-fidelity term, where signal and noise are additively mixed, a non-smooth and non-convex sparsity promoting term, and a penalty term to model the noise. We propose and analyze the convergence of an iterative alternating algorithm based on simple iterative thresholding steps to perform the minimization of the functional. By means of this algorithm, we explore the effect of choosing different regularization parameters and penalization norms in terms of the quality of recovering the pure signal and separating it from additive noise. For a given fixed noise level numerical experiments confirm a significant improvement in performance compared to standard one-parameter regularization methods. By using high-dimensional data analysis methods such as principal component analysis, we are able to show the correct geometrical clustering of regularized solutions around the expected solution. Eventually, for the compressive sensing problems considered in our experiments we provide a guideline for a choice of regularization norms and parameters.

Multi-Penalty Regularization for Detecting Relevant Variables

In this paper, we propose a new method for detecting relevant variables from a priori given high-dimensional data under the assumption that input-output relation is described by a nonlinear function depending on a few variables. The method is based on the inspection of the behavior of discrepancies of a multi-penalty regularization with a component-wise penalization for small and large values of regularization parameters. We provide a justification of the proposed method under a certain condition on sampling operators. The effectiveness of the method is demonstrated in an example with simulated data and in the reconstruction of a gene regulatory network. In the latter example, the obtained results provide clear evidence of the competitiveness of the proposed method with respect to the state-of-the-art approaches.

Meta-Learning Based Blood Glucose Predictor for Diabetic Smartphone App

The obvious and highly accepted convenience of smartphone apps will, already in the nearest future, bring new opportunities for diabetes therapy management. In particular, it is expected that smartphones will be able to read, store, and display the blood glucose concentration from the continuous glucose monitoring systems. Using our knowledge and experience gained in the framework of the large-scale European Union FP7 funded project “DIAdvisor: personal glucose predictive diabetes advisor” (2008–2012), we explore a possibility to develop a novel smartphone app for diabetes patients that provides estimations of the future blood glucose concentration from current and past blood glucose readings. In addition to reliable clinical accuracy, a prediction algorithm implemented in such an app should satisfy multiple requirements, such as easily and quickly implementable on any mobile operating system, portability from individual to individual without readjustment or retraining procedure, and a low battery usage feature. In this study, we present a description of the prediction algorithm, developed in the course of the DIAdvisor project, and its version on Android OS that meets the above-mentioned requirements. Additionally, we compare the clinical accuracy of the algorithm with the state of the art in terms of the “gold standard” metric, Clarke error grid analysis, and the recently introduced metric, prediction error grid analysis.

Conditions on optimal support recovery in unmixing problems by means of multi-penalty regularization

Inspired by several real-life applications in audio processing and medical image analysis, where the quantity of interest is generated by several sources to be accurately modeled and separated, as well as by recent advances in regularization theory and optimization, we study the conditions on optimal support recovery in inverse problems of unmixing type by means of multi-penalty regularization. nWe consider and analyze a regularization functional composed of a data-fidelity term, where signal and noise are additively mixed, a non-smooth, convex, sparsity promoting term, and a quadratic penalty term to model the noise. We prove not only that the well-established theory for sparse recovery in the single parameter case can be translated to the multi-penalty settings, but we also demonstrate the enhanced properties of multi-penalty regularization in terms of support identification compared to sole

Abstract 1296: CanPathPro—development of a platform for predictive pathway modelling using genetically engineered mouse models

ell^1

A-T-LAS

-minimization. We additionally confirm and support the theoretical results by extensive numerical simulations, which give a statistics of robustness of the multi-penalty regularization scheme with respect to the single-parameter counterpart. Eventually, we confirm a significant improvement in performance compared to standard

_{2,1}

ell^1

: A Multi-Penalty Approach to Compressed Sensing of Low-Rank Matrices with Sparse Decompositions

-regularization for compressive sensing problems considered in our experiments.