25 research outputs found
Duality-based Higher-order Non-smooth Optimization on Manifolds
We propose a method for solving non-smooth optimization problems on
manifolds. In order to obtain superlinear convergence, we apply a Riemannian
Semi-smooth Newton method to a non-smooth non-linear primal-dual optimality
system based on a recent extension of Fenchel duality theory to Riemannian
manifolds. We also propose an inexact version of the Riemannian Semi-smooth
Newton method and prove conditions for local linear and superlinear
convergence. Numerical experiments on l2-TV-like problems confirm superlinear
convergence on manifolds with positive and negative curvature
Recommended from our members
Learning-based Optimization for Signal and Image Processing
Incorporating machine learning techniques into optimization problems and solvers attracts increasing attention. Given a particular type of optimization problem that needs to be solved repeatedly, machine learning techniques can find some features for this category of optimization and develop algorithms with excellent performance. This thesis deals with algorithms and convergence analysis in learning-based optimization in three aspects: learning dictionaries, learning optimization solvers and learning regularizers.Learning dictionaries for sparse coding is significant for signal processing. Convolutional sparse coding is a form of sparse coding with a structured, translation invariant dictionary. Most convolutional dictionary learning algorithms to date operate in the batch mode, requiring simultaneous access to all training images during the learning process, which results in very high memory usage, and severely limits the training data size that can be used. I proposed two online convolutional dictionary learning algorithms that offered far better scaling of memory and computational cost than batch methods and provided a rigorous theoretical analysis of these methods.Learning fast solvers for optimization is a rising research topic. In recent years, unfolding iterative algorithms as neural networks has become an empirical success in solving sparse recovery problems. However, its theoretical understanding is still immature, which prevents us from fully utilizing the power of neural networks. I studied unfolded ISTA (Iterative Shrinkage Thresholding Algorithm) for sparse signal recovery and established its convergence. Based on the properties of parameters required by convergence, the model can be significantly simplified and, consequently, has much less training cost and better recovery performance.Learning regularizers or priors improves the performance of optimization solvers, especially for signal and image processing tasks. Plug-and-play (PnP) is a non-convex framework that integrates modern priors, such as BM3D or deep learning-based denoisers, into ADMM or other proximal algorithms. Although PnP has been recently studied extensively with great empirical success, theoretical analysis addressing even the most basic question of convergence has been insufficient. In this thesis, the theoretical convergence of PnP-FBS and PnP-ADMM was established, without using diminishing stepsizes, under a certain Lipschitz condition on the denoisers. Furthermore, real spectral normalization was proposed for training deep learning-based denoisers to satisfy the proposed Lipschitz condition
Generalized averaged Gaussian quadrature and applications
A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal
MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications
Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described
Structured Learning with Parsimony in Measurements and Computations: Theory, Algorithms, and Applications
University of Minnesota Ph.D. dissertation. July 2018. Major: Electrical Engineering. Advisor: Jarvis Haupt. 1 computer file (PDF); xvi, 289 pages.In modern ``Big Data'' applications, structured learning is the most widely employed methodology. Within this paradigm, the fundamental challenge lies in developing practical, effective algorithmic inference methods. Often (e.g., deep learning) successful heuristic-based approaches exist but theoretical studies are far behind, limiting understanding and potential improvements. In other settings (e.g., recommender systems) provably effective algorithmic methods exist, but the sheer sizes of datasets can limit their applicability. This twofold challenge motivates this work on developing new analytical and algorithmic methods for structured learning, with a particular focus on parsimony in measurements and computation, i.e., those requiring low storage and computational costs. Toward this end, we make efforts to investigate the theoretical properties of models and algorithms that present significant improvement in measurement and computation requirement. In particular, we first develop randomized approaches for dimensionality reduction on matrix and tensor data, which allow accurate estimation and inference procedures using significantly smaller data sizes that only depend on the intrinsic dimension (e.g., the rank of matrix/tensor) rather than the ambient ones. Our next effort is to study iterative algorithms for solving high dimensional learning problems, including both convex and nonconvex optimization. Using contemporary analysis techniques, we demonstrate guarantees of iteration complexities that are analogous to the low dimensional cases. In addition, we explore the landscape of nonconvex optimizations that exhibit computational advantages over their convex counterparts and characterize their properties from a general point of view in theory
Learning to compress and search visual data in large-scale systems
The problem of high-dimensional and large-scale representation of visual data
is addressed from an unsupervised learning perspective. The emphasis is put on
discrete representations, where the description length can be measured in bits
and hence the model capacity can be controlled. The algorithmic infrastructure
is developed based on the synthesis and analysis prior models whose
rate-distortion properties, as well as capacity vs. sample complexity
trade-offs are carefully optimized. These models are then extended to
multi-layers, namely the RRQ and the ML-STC frameworks, where the latter is
further evolved as a powerful deep neural network architecture with fast and
sample-efficient training and discrete representations. For the developed
algorithms, three important applications are developed. First, the problem of
large-scale similarity search in retrieval systems is addressed, where a
double-stage solution is proposed leading to faster query times and shorter
database storage. Second, the problem of learned image compression is targeted,
where the proposed models can capture more redundancies from the training
images than the conventional compression codecs. Finally, the proposed
algorithms are used to solve ill-posed inverse problems. In particular, the
problems of image denoising and compressive sensing are addressed with
promising results.Comment: PhD thesis dissertatio
Abstracts on Radio Direction Finding (1899 - 1995)
The files on this record represent the various databases that originally composed the CD-ROM issue of "Abstracts on Radio Direction Finding" database, which is now part of the Dudley Knox Library's Abstracts and Selected Full Text Documents on Radio Direction Finding (1899 - 1995) Collection. (See Calhoun record https://calhoun.nps.edu/handle/10945/57364 for further information on this collection and the bibliography).
Due to issues of technological obsolescence preventing current and future audiences from accessing the bibliography, DKL exported and converted into the three files on this record the various databases contained in the CD-ROM.
The contents of these files are:
1) RDFA_CompleteBibliography_xls.zip [RDFA_CompleteBibliography.xls: Metadata for the complete bibliography, in Excel 97-2003 Workbook format; RDFA_Glossary.xls: Glossary of terms, in Excel 97-2003 Workbookformat; RDFA_Biographies.xls: Biographies of leading figures, in Excel 97-2003 Workbook format];
2) RDFA_CompleteBibliography_csv.zip [RDFA_CompleteBibliography.TXT: Metadata for the complete bibliography, in CSV format; RDFA_Glossary.TXT: Glossary of terms, in CSV format; RDFA_Biographies.TXT: Biographies of leading figures, in CSV format];
3) RDFA_CompleteBibliography.pdf: A human readable display of the bibliographic data, as a means of double-checking any possible deviations due to conversion
Variable metric line-search based methods for nonconvex optimization
L'obiettivo di questa tesi è quello di proporre nuovi metodi iterativi del prim'ordine per un'ampia classe di problemi di ottimizzazione non convessa, in cui la funzione obiettivo è data dalla somma di un termine differenziabile, eventualmente non convesso, e di uno convesso, eventualmente non differenziabile. Tali problemi sono frequenti in applicazioni scientifiche quali l'elaborazione numerica di immagini e segnali, in cui il primo termine gioca il ruolo di funzione di discrepanza tra il dato osservato e l'oggetto ricostruito, mentre il secondo è il termine di regolarizzazione, volto ad imporre alcune specifiche proprietà sull'oggetto desiderato. Il nostro approccio è duplice: da un lato, i metodi proposti vengono accelerati facendo uso di strategie adattive di selezione dei parametri coinvolti; dall'altro lato, la convergenza di tali metodi viene garantita imponendo, ad ogni iterazione, un'opportuna condizione di sufficiente decrescita della funzione obiettivo.
Il nostro primo contributo consiste nella messa a punto di un nuovo metodo di tipo proximal-gradient, che alterna un passo del gradiente sulla parte differenziabile ad uno proximal sulla parte convessa, denominato Variable Metric Inexact Line-search based Algorithm (VMILA). Tale metodo è innovativo da più punti di vista. Innanzitutto, a differenza della maggior parte dei metodi proximal-gradient, VMILA permette di adottare una metrica variabile nel calcolo dell'operatore proximal con estrema libertà di scelta, imponendo soltanto che i parametri coinvolti appartengano a sottoinsiemi limitati degli spazi in cui vengono definiti. In secondo luogo, in VMILA il calcolo del punto proximal viene effettuato tramite un preciso criterio di inesattezza, che può essere concretamente implementato in alcuni casi di interesse. Questo aspetto assume una rilevante importanza ogni qualvolta l'operatore proximal non sia calcolabile in forma chiusa. Infine, le iterate di VMILA sono calcolate tramite una ricerca di linea inesatta lungo la direzione ammissibile e secondo una specifica condizione di sufficiente decrescita di tipo Armijo.
Il secondo contributo di questa tesi è proposto in un caso particolare del problema di ottimizzazione precedentemente considerato, in cui si assume che il termine convesso sia dato dalla somma di un numero finito di funzioni indicatrici di insiemi chiusi e convessi. In altre parole, si considera il problema di minimizzare una funzione differenziabile in cui i vincoli sulle incognite hanno una struttura separabile. In letteratura, il metodo classico per affrontare tale problema è senza dubbio il metodo di Gauss-Seidel (GS) non lineare, dove la minimizzazione della funzione obiettivo è ciclicamente alternata su ciascun blocco di variabili del problema. In questa tesi, viene proposta una versione inesatta dello schema GS, denominata Cyclic Block Generalized Gradient Projection (CBGGP) method, in cui la minimizzazione parziale su ciascun blocco di variabili è realizzata mediante un numero finito di passi del metodo del gradiente proiettato. La novità nell'approccio proposto consiste nell'introduzione di metriche non euclidee nel calcolo del gradiente proiettato.
Per entrambi i metodi si dimostra, senza alcuna ipotesi di convessità sulla funzione obiettivo, che ciascun punto di accumulazione della successione delle iterate è stazionario. Nel caso di VMILA, è invece possibile dimostrare la convergenza forte delle iterate ad un punto stazionario quando la funzione obiettivo soddisfa la disuguaglianza di Kurdyka-Lojasiewicz. Numerosi test numerici in problemi di elaborazione di immagini, quali la ricostruzione di immagini sfocate e rumorose, la compressione di immagini, la stima di fase in microscopia e la deconvoluzione cieca di immagini in astronomia, danno prova della flessibilità ed efficacia dei metodi proposti.The aim of this thesis is to propose novel iterative first order methods tailored for a wide class of nonconvex nondifferentiable optimization problems, in which the objective function is given by the sum of a differentiable, possibly nonconvex function and a convex, possibly nondifferentiable term. Such problems have become ubiquitous in scientific applications such as image or signal processing, where the first term plays the role of the fit-to-data term, describing the relation between the desired object and the measured data, whereas the second one is the penalty term, aimed at restricting the search of the object itself to those satisfying specific properties. Our approach is twofold: on one hand, we accelerate the proposed methods by making use of suitable adaptive strategies to choose the involved parameters; on the other hand, we ensure convergence by imposing a sufficient decrease condition on the objective function at each iteration.
Our first contribution is the development of a novel proximal--gradient method denominated Variable Metric Inexact Line-search based Algorithm (VMILA). The proposed approach is innovative from several points of view. First of all, VMILA allows to adopt a variable metric in the computation of the proximal point with a relative freedom of choice. Indeed the only assumption that we make is that the parameters involved belong to bounded sets. This is unusual with respect to the state-of-the-art proximal-gradient methods, where the parameters are usually chosen by means of a fixed rule or tightly related to the Lipschitz constant of the problem. Second, we introduce an inexactness criterion for computing the proximal point which can be practically implemented in some cases of interest. This aspect assumes a relevant importance whenever the proximal operator is not available in a closed form, which is often the case. Third, the VMILA iterates are computed by performing a line-search along the feasible direction and according to a specific Armijo-like condition, which can be considered as an extension of the classical Armijo rule proposed in the context of differentiable optimization.
The second contribution is given for a special instance of the previously considered optimization problem, where the convex term is assumed to be a finite sum of the indicator functions of closed, convex sets. In other words, we consider a problem of constrained differentiable optimization in which the constraints have a separable structure. The most suited method to deal with this problem is undoubtedly the nonlinear Gauss-Seidel (GS) or block coordinate descent method, where the minimization of the objective function is cyclically alternated on each block of variables of the problem. In this thesis, we propose an inexact version of the GS scheme, denominated Cyclic Block Generalized Gradient Projection (CBGGP) method, in which the partial minimization over each block of variables is performed inexactly by means of a fixed number of gradient projection steps. The novelty of the proposed approach consists in the introduction of non Euclidean metrics in the computation of the gradient projection. As for VMILA, the sufficient decrease of the function is imposed by means of a block version of the Armijo line-search.
For both methods, we prove that each limit point of the sequence of iterates is stationary, without any convexity assumptions. In the case of VMILA, strong convergence of the iterates to a stationary point is also proved when the objective function satisfies the Kurdyka-Lojasiewicz property. Extensive numerical experience in image processing applications, such as image deblurring and denoising in presence of non-Gaussian noise, image compression, phase estimation and image blind deconvolution, shows the flexibility of our methods in addressing different nonconvex problems, as well as their ability to effectively accelerate the progress towards the solution of the treated problem
Tikhonov-type iterative regularization methods for ill-posed inverse problems: theoretical aspects and applications
Ill-posed inverse problems arise in many fields of science and engineering. The ill-conditioning and the big dimension make the task of numerically solving this kind of problems very challenging.
In this thesis we construct several algorithms for solving ill-posed inverse problems. Starting from the classical Tikhonov regularization method we develop iterative methods that enhance the performances of the originating method.
In order to ensure the accuracy of the constructed algorithms we insert a priori knowledge on the exact solution and empower the regularization term. By exploiting the structure of the problem we are also able to achieve fast computation even when the size of the problem becomes very big.
We construct algorithms that enforce constraint on the reconstruction, like nonnegativity or flux conservation and exploit enhanced version of the Euclidian norm using a regularization operator and different semi-norms, like the Total Variaton, for the regularization term.
For most of the proposed algorithms we provide efficient strategies for the choice of the regularization parameters, which, most of the times, rely on the knowledge of the norm of the noise that corrupts the data.
For each method we analyze the theoretical properties in the finite dimensional case or in the more general case of Hilbert spaces.
Numerical examples prove the good performances of the algorithms proposed in term of both accuracy and efficiency