1,720,978 research outputs found
Linear inverse problems with Hessian–Schatten total variation
Full text linkAbstract. In this paper, we characterize the class of extremal points of the unit ball of the Hessian-Schatten total variation (HTV) functional. The underlying motivation for our work stems from a general representer theorem that characterizes the solution set of regularized linear inverse problems in terms of the extremal points of the regularization ball. Our analysis is mainly based on studying the class of continuous and piecewise linear (CPWL) functions. In particular, we show that in dimension d = 2, CPWL functions are dense in the unit ball of the HTV functional. Moreover, we prove that a CPWL function is extremal if and only if its Hessian is minimally supported. For the converse, we prove that the density result (which we have only proven for dimension d = 2) implies that the closure of the CPWL extreme points contains all extremal points
Optimization Over Banach Spaces: A Unified View on Supervised Learning and Inverse Problems
No full textIn this thesis, we reveal that supervised learning and inverse problems share similar mathematical foundations. Consequently, we are able to present a unified variational view of these tasks that we formulate as optimization problems posed over infinite-dimensional Banach spaces. Throughout the thesis, we study this class of optimization problems from a mathematical perspective. We start by specifying adequate search spaces and loss functionals that are derived from applications. Next, we identify conditions that guarantee the existence of a solution and we provide a (finite) parametric form for the optimal solution. Finally, we utilize these theoretical characterizations to derive numerical solvers.
The thesis is divided into five parts. The first part is devoted to the theory of splines, a large class of continuous-domain models that are optimal in many of the studied frameworks. Our contributions in this part include the introduction of the notion of multi-splines, their theoretical properties, and shortest-support generators. In the second part, we study a broad class of optimization problems over Banach spaces and we prove a general representer theorem that characterizes their solution sets. The third and fourth parts of the thesis invoke the applicability of our general framework to supervised learning and inverse problems, respectively. Specifically, we derive various learning schemes from our variational framework that inherit a certain notion of "sparsity" and we establish the connection between our theory and deep neural networks, which are state-of-the-art in supervised learning. Moreover, we deploy our general theory to study continuous-domain inverse problems with multicomponent models, which can be applied to various signal and image processing tasks, in particular, curve fitting. Finally, we revisit the notions of splines and sparsity in the last part of the thesis, this time, from a stochastic perspective.LI
Wavelet analysis of the Besov regularity of Levy white noise
No full textWe characterize the local smoothness and the asymptotic growth rate of the Levy white noise. We do so by characterizing the weighted Besov spaces in which it is located. We extend known results in two ways. First, we obtain new bounds for the local smoothness via the Blumenthal-Getoor indices of the Levy white noise. We also deduce the critical local smoothness when the two indices coincide, which is true for symmetric-a-stable, compound Poisson, and symmetric-gamma white noises to name a few. Second, we express the critical asymptotic growth rate in terms of the moment properties of the Levy white noise. Previous analyses only provided lower bounds for both the local smoothness and the asymptotic growth rate. Showing the sharpness of these bounds requires us to determine in which Besov spaces a given Levy white noise is (almost surely) not. Our methods are based on the wavelet-domain characterization of Besov spaces and precise moment estimates for the wavelet coefficients of the Levy white noise.LI
Duality Mapping for Schatten Matrix Norms
No full textIn this paper, we fully characterize the duality mapping over the space of matrices that are equipped with Schatten norms. Our approach is based on the analysis of the saturation of the Holder inequality for Schatten norms. We prove in our main result that, for p is an element of (1, infinity), the duality mapping over the space of real-valued matrices with Schatten-p norm is a continuous and single-valued function and provide an explicit form for its computation. For the special case p = 1, the mapping is set-valued; by adding a rank constraint, we show that it can be reduced to a Borel-measurable single-valued function for which we also provide a closed-form expression.LI
Multikernel Regression with Sparsity Constraint
No full textIn this paper, we provide a Banach-space formulation of supervised learning with generalized total-variation (gTV) regularization. We identify the class of kernel functions that are admissible in this framework. Then, we propose a variation of supervised learning in a continuous-domain hybrid search space with gTV regularization. We show that the solution admits a multikernel expansion with adaptive positions. In this representation, the number of active kernels is upper-bounded by the number of data points while the gTV regularization imposes an l(1) penalty on the kernel coefficients. Finally, we illustrate numerically the outcome of our theory.LI
Convex optimization in sums of Banach spaces
No full textWe characterize the solution of a broad class of convex optimization problems that address the reconstruction of a function from a finite number of linear measurements. The underlying hypothesis is that the solution is decomposable as a finite sum of components, where each component belongs to its own prescribed Banach space; moreover, the problem is regularized by penalizing some composite norm of the solution. We establish general conditions for existence and derive the generic parametric representation of the solution components. These representations fall into three categories depending on the underlying regularization norm: (i) a linear expansion in terms of predefined “kernels” when the component space is a reproducing kernel Hilbert space (RKHS), (ii) a non-linear (duality) mapping of a linear combination of measurement functionals when the component Banach space is strictly convex, and, (iii) an adaptive expansion in terms of a small number of atoms within a larger dictionary when the component Banach space is not strictly convex. Our approach generalizes and unifies a number of multi-kernel (RKHS) and sparse-dictionary learning techniques for compressed sensing available in the literature. It also yields the natural extension of the classical spline-fitting techniques in (semi-)RKHS to the abstract Banach-space setting.LIBThis is an Open Access article under the terms of the Creative Commons Attribution Licens
The Wavelet Compressibility of Compound Poisson Processes
No full textIn this paper, we precisely quantify the wavelet compressibility of compound Poisson processes. To that end, we expand the given random process over the Haar wavelet basis and we analyse its asymptotic approximation properties. By only considering the nonzero wavelet coefficients up to a given scale, what we call the greedy approximation, we exploit the extreme sparsity of the wavelet expansion that derives from the piecewise-constant nature of compound Poisson processes. More precisely, we provide lower and upper bounds for the mean squared error of greedy approximation of compound Poisson processes. We are then able to deduce that the greedy approximation error has a sub-exponential and super-polynomial asymptotic behavior. Finally, we provide numerical experiments to highlight the remarkable ability of wavelet-based dictionaries in achieving highly compressible approximations of compound Poisson processes.LI
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
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