Parameter Estimation with the Ordered ℓ2\ell_{2} Regularization via an Alternating Direction Method of Multipliers

Regularization is a popular technique in machine learning for model estimation and avoiding overfitting. Prior studies have found that modern ordered regularization can be more effective in handling highly correlated, high-dimensional data than traditional regularization. The reason stems from the fact that the ordered regularization can reject irrelevant variables and yield an accurate estimation of the parameters. How to scale up the ordered regularization problems when facing the large-scale training data remains an unanswered question. This paper explores the problem of parameter estimation with the ordered $\ell_{2}$-regularization via Alternating Direction Method of Multipliers (ADMM), called ADMM-O$\ell_{2}$. The advantages of ADMM-O$\ell_{2}$ include (i) scaling up the ordered $\ell_{2}$ to a large-scale dataset, (ii) predicting parameters correctly by excluding irrelevant variables automatically, and (iii) having a fast convergence rate. Experiment results on both synthetic data and real data indicate that ADMM-O$\ell_{2}$ can perform better than or comparable to several state-of-the-art baselines

Sparse and Redundant Representations for Inverse Problems and Recognition

Sparse and redundant representation of data enables the description of signals as linear combinations of a few atoms from a dictionary. In this dissertation, we study applications of sparse and redundant representations in inverse problems and object recognition. Furthermore, we propose two novel imaging modalities based on the recently introduced theory of Compressed Sensing (CS). This dissertation consists of four major parts. In the first part of the dissertation, we study a new type of deconvolution algorithm that is based on estimating the image from a shearlet decomposition. Shearlets provide a multi-directional and multi-scale decomposition that has been mathematically shown to represent distributed discontinuities such as edges better than traditional wavelets. We develop a deconvolution algorithm that allows for the approximation inversion operator to be controlled on a multi-scale and multi-directional basis. Furthermore, we develop a method for the automatic determination of the threshold values for the noise shrinkage for each scale and direction without explicit knowledge of the noise variance using a generalized cross validation method. In the second part of the dissertation, we study a reconstruction method that recovers highly undersampled images assumed to have a sparse representation in a gradient domain by using partial measurement samples that are collected in the Fourier domain. Our method makes use of a robust generalized Poisson solver that greatly aids in achieving a significantly improved performance over similar proposed methods. We will demonstrate by experiments that this new technique is more flexible to work with either random or restricted sampling scenarios better than its competitors. In the third part of the dissertation, we introduce a novel Synthetic Aperture Radar (SAR) imaging modality which can provide a high resolution map of the spatial distribution of targets and terrain using a significantly reduced number of needed transmitted and/or received electromagnetic waveforms. We demonstrate that this new imaging scheme, requires no new hardware components and allows the aperture to be compressed. Also, it presents many new applications and advantages which include strong resistance to countermesasures and interception, imaging much wider swaths and reduced on-board storage requirements. The last part of the dissertation deals with object recognition based on learning dictionaries for simultaneous sparse signal approximations and feature extraction. A dictionary is learned for each object class based on given training examples which minimize the representation error with a sparseness constraint. A novel test image is then projected onto the span of the atoms in each learned dictionary. The residual vectors along with the coefficients are then used for recognition. Applications to illumination robust face recognition and automatic target recognition are presented

Compressive Wave Computation

This paper considers large-scale simulations of wave propagation phenomena. We argue that it is possible to accurately compute a wavefield by decomposing it onto a largely incomplete set of eigenfunctions of the Helmholtz operator, chosen at random, and that this provides a natural way of parallelizing wave simulations for memory-intensive applications. This paper shows that L1-Helmholtz recovery makes sense for wave computation, and identifies a regime in which it is provably effective: the one-dimensional wave equation with coefficients of small bounded variation. Under suitable assumptions we show that the number of eigenfunctions needed to evolve a sparse wavefield defined on N points, accurately with very high probability, is bounded by C log(N) log(log(N)), where C is related to the desired accuracy and can be made to grow at a much slower rate than N when the solution is sparse. The PDE estimates that underlie this result are new to the authors' knowledge and may be of independent mathematical interest; they include an L1 estimate for the wave equation, an estimate of extension of eigenfunctions, and a bound for eigenvalue gaps in Sturm-Liouville problems. Numerical examples are presented in one spatial dimension and show that as few as 10 percents of all eigenfunctions can suffice for accurate results. Finally, we argue that the compressive viewpoint suggests a competitive parallel algorithm for an adjoint-state inversion method in reflection seismology.Comment: 45 pages, 4 figure

ON SOME COMMON COMPRESSIVE SENSING RECOVERY ALGORITHMS AND APPLICATIONS

Compressive Sensing, as an emerging technique in signal processing is reviewed in this paper together with its’ common applications. As an alternative to the traditional signal sampling, Compressive Sensing allows a new acquisition strategy with significantly reduced number of samples needed for accurate signal reconstruction. The basic ideas and motivation behind this approach are provided in the theoretical part of the paper. The commonly used algorithms for missing data reconstruction are presented. The Compressive Sensing applications have gained significant attention leading to an intensive growth of signal processing possibilities. Hence, some of the existing practical applications assuming different types of signals in real-world scenarios are described and analyzed as well

Multi-Modal Similarity Learning for 3D Deformable Registration of Medical Images

Alors que la perspective de la fusion d images médicales capturées par des systèmes d imageries de type différent est largement contemplée, la mise en pratique est toujours victime d un obstacle théorique : la définition d une mesure de similarité entre les images. Des efforts dans le domaine ont rencontrés un certain succès pour certains types d images, cependant la définition d un critère de similarité entre les images quelle que soit leur origine et un des plus gros défis en recalage d images déformables. Dans cette thèse, nous avons décidé de développer une approche générique pour la comparaison de deux types de modalités donnés. Les récentes avancées en apprentissage statistique (Machine Learning) nous ont permis de développer des solutions innovantes pour la résolution de ce problème complexe. Pour appréhender le problème de la comparaison de données incommensurables, nous avons choisi de le regarder comme un problème de plongement de données : chacun des jeux de données est plongé dans un espace commun dans lequel les comparaisons sont possibles. A ces fins, nous avons exploré la projection d un espace de données image sur l espace de données lié à la seconde image et aussi la projection des deux espaces de données dans un troisième espace commun dans lequel les calculs sont conduits. Ceci a été entrepris grâce à l étude des correspondances entre les images dans une base de données images pré-alignées. Dans la poursuite de ces buts, de nouvelles méthodes ont été développées que ce soit pour la régression d images ou pour l apprentissage de métrique multimodale. Les similarités apprises résultantes sont alors incorporées dans une méthode plus globale de recalage basée sur l optimisation discrète qui diminue le besoin d un critère différentiable pour la recherche de solution. Enfin nous explorons une méthode qui permet d éviter le besoin d une base de données pré-alignées en demandant seulement des données annotées (segmentations) par un spécialiste. De nombreuses expériences sont conduites sur deux bases de données complexes (Images d IRM pré-alignées et Images TEP/Scanner) dans le but de justifier les directions prises par nos approches.Even though the prospect of fusing images issued by different medical imagery systems is highly contemplated, the practical instantiation of it is subject to a theoretical hurdle: the definition of a similarity between images. Efforts in this field have proved successful for select pairs of images; however defining a suitable similarity between images regardless of their origin is one of the biggest challenges in deformable registration. In this thesis, we chose to develop generic approaches that allow the comparison of any two given modality. The recent advances in Machine Learning permitted us to provide innovative solutions to this very challenging problem. To tackle the problem of comparing incommensurable data we chose to view it as a data embedding problem where one embeds all the data in a common space in which comparison is possible. To this end, we explored the projection of one image space onto the image space of the other as well as the projection of both image spaces onto a common image space in which the comparison calculations are conducted. This was done by the study of the correspondences between image features in a pre-aligned dataset. In the pursuit of these goals, new methods for image regression as well as multi-modal metric learning methods were developed. The resulting learned similarities are then incorporated into a discrete optimization framework that mitigates the need for a differentiable criterion. Lastly we investigate on a new method that discards the constraint of a database of images that are pre-aligned, only requiring data annotated (segmented) by a physician. Experiments are conducted on two challenging medical images data-sets (Pre-Aligned MRI images and PET/CT images) to justify the benefits of our approach.CHATENAY MALABRY-Ecole centrale (920192301) / SudocSudocFranceF

Image Processing and Machine Learning for Hyperspectral Unmixing: An Overview and the HySUPP Python Package

Spectral pixels are often a mixture of the pure spectra of the materials, called endmembers, due to the low spatial resolution of hyperspectral sensors, double scattering, and intimate mixtures of materials in the scenes. Unmixing estimates the fractional abundances of the endmembers within the pixel. Depending on the prior knowledge of endmembers, linear unmixing can be divided into three main groups: supervised, semi-supervised, and unsupervised (blind) linear unmixing. Advances in Image processing and machine learning substantially affected unmixing. This paper provides an overview of advanced and conventional unmixing approaches. Additionally, we draw a critical comparison between advanced and conventional techniques from the three categories. We compare the performance of the unmixing techniques on three simulated and two real datasets. The experimental results reveal the advantages of different unmixing categories for different unmixing scenarios. Moreover, we provide an open-source Python-based package available at https://github.com/BehnoodRasti/HySUPP to reproduce the results

ИНТЕЛЛЕКТУАЛЬНЫЙ числовым программным ДЛЯ MIMD-компьютер

For most scientific and engineering problems simulated on computers the solving of problems of the computational mathematics with approximately given initial data constitutes an intermediate or a final stage. Basic problems of the computational mathematics include the investigating and solving of linear algebraic systems, evaluating of eigenvalues and eigenvectors of matrices, the solving of systems of non-linear equations, numerical integration of initial- value problems for systems of ordinary differential equations.Для більшості наукових та інженерних задач моделювання на ЕОМ рішення задач обчислювальної математики з наближено заданими вихідними даними складає проміжний або остаточний етап. Основні проблеми обчислювальної математики відносяться дослідження і рішення лінійних алгебраїчних систем оцінки власних значень і власних векторів матриць, рішення систем нелінійних рівнянь, чисельного інтегрування початково задач для систем звичайних диференціальних рівнянь.Для большинства научных и инженерных задач моделирования на ЭВМ решение задач вычислительной математики с приближенно заданным исходным данным составляет промежуточный или окончательный этап. Основные проблемы вычислительной математики относятся исследования и решения линейных алгебраических систем оценки собственных значений и собственных векторов матриц, решение систем нелинейных уравнений, численного интегрирования начально задач для систем обыкновенных дифференциальных уравнений

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum