Unicity conditions for low-rank matrix recovery

Low-rank matrix recovery addresses the problem of recovering an unknown low-rank matrix from few linear measurements. Nuclear-norm minimization is a tractible approach with a recent surge of strong theoretical backing. Analagous to the theory of compressed sensing, these results have required random measurements. For example, m >= Cnr Gaussian measurements are sufficient to recover any rank-r n x n matrix with high probability. In this paper we address the theoretical question of how many measurements are needed via any method whatsoever --- tractible or not. We show that for a family of random measurement ensembles, m >= 4nr - 4r^2 measurements are sufficient to guarantee that no rank-2r matrix lies in the null space of the measurement operator with probability one. This is a necessary and sufficient condition to ensure uniform recovery of all rank-r matrices by rank minimization. Furthermore, this value of $m$ precisely matches the dimension of the manifold of all rank-2r matrices. We also prove that for a fixed rank-r matrix, m >= 2nr - r^2 + 1 random measurements are enough to guarantee recovery using rank minimization. These results give a benchmark to which we may compare the efficacy of nuclear-norm minimization

Preasymptotic Convergence of Randomized Kaczmarz Method

Kaczmarz method is one popular iterative method for solving inverse problems, especially in computed tomography. Recently, it was established that a randomized version of the method enjoys an exponential convergence for well-posed problems, and the convergence rate is determined by a variant of the condition number. In this work, we analyze the preasymptotic convergence behavior of the randomized Kaczmarz method, and show that the low-frequency error (with respect to the right singular vectors) decays faster during first iterations than the high-frequency error. Under the assumption that the inverse solution is smooth (e.g., sourcewise representation), the result explains the fast empirical convergence behavior, thereby shedding new insights into the excellent performance of the randomized Kaczmarz method in practice. Further, we propose a simple strategy to stabilize the asymptotic convergence of the iteration by means of variance reduction. We provide extensive numerical experiments to confirm the analysis and to elucidate the behavior of the algorithms.Comment: 20 page

Structured Sparsity: Discrete and Convex approaches

Compressive sensing (CS) exploits sparsity to recover sparse or compressible signals from dimensionality reducing, non-adaptive sensing mechanisms. Sparsity is also used to enhance interpretability in machine learning and statistics applications: While the ambient dimension is vast in modern data analysis problems, the relevant information therein typically resides in a much lower dimensional space. However, many solutions proposed nowadays do not leverage the true underlying structure. Recent results in CS extend the simple sparsity idea to more sophisticated {\em structured} sparsity models, which describe the interdependency between the nonzero components of a signal, allowing to increase the interpretability of the results and lead to better recovery performance. In order to better understand the impact of structured sparsity, in this chapter we analyze the connections between the discrete models and their convex relaxations, highlighting their relative advantages. We start with the general group sparse model and then elaborate on two important special cases: the dispersive and the hierarchical models. For each, we present the models in their discrete nature, discuss how to solve the ensuing discrete problems and then describe convex relaxations. We also consider more general structures as defined by set functions and present their convex proxies. Further, we discuss efficient optimization solutions for structured sparsity problems and illustrate structured sparsity in action via three applications.Comment: 30 pages, 18 figure

Formação dos partidos brasileiros: questões de ideologia, rótulos partidários, lideranças e prática política, 1831-1888

This is a response to comments by R. Salles and M. Dantas, and discusses the use of Gramscian terminology, ideological differences between the parties, party names used during the Regency and Second Reign, and political practice at the provincial and national levels. It argues that the saquaremas were not a hegemonic party, that their leaders were organic, that the differences between the parties were fundamental on certain points, and that the use of party names in the text debated derive from contemporary usage and meaning. The response also comments on the fundamental differences involved in the Additional Act, on the significance of the reactionary centralizing legislation, and, finally, on the success and limitations of both State power and of provincial political mobilization in affecting provincial government, national policy, and imperial political practice.Apresento aqui uma resposta aos comentários de R. Salles e M. Dantas, em que se discutem o uso da terminologia gramsciana, as diferenças ideológicas entre os partidos, os nomes dos partidos durante a Regência e o Segundo Reinado e a prática política nos âmbitos provincial e nacional. Argumento que os saquaremas não eram um partido hegemônico, seus líderes eram orgânicos, as diferenças entre os partidos eram essenciais em certos pontos e o uso dos nomes dos partidos no texto discutido decorre da utilização e significado coevos. Esta réplica também aborda as divergências fundamentais que envolveram o Ato Adicional, o significado da legislação centralizadora do Regresso e, por fim, os êxitos e limitações tanto do poder do Estado como da mobilização política provincial em influir no governo provincial, na política nacional e na prática política imperial

Formação dos partidos políticos no Brasil da Regência à Conciliação, 1831-1857

The parties derived from Chamber factions, led by orators representing the planting and commercial oligarchies and mobilized urban groups. The antecedents, clear in the 1823 Constituent Assembly, crystallize in the "liberal opposition" of 1826-31. The moderate majority dominated the first years of the Regency, but divided over more radical liberal reform. A reactionary movement led to a new majority party in 1837, emphasizing a strong state balanced by a representative parliament and cabinet. This party, eventually known as the Conservatives, faced an opposition, eventually known as the Liberals, who, while sharing some liberal beliefs, initially comprised an alliance of opportunity. After the emperor took power, he proved suspicious of partisan loyalties and ambitions, and increasingly dominated the cabinet, enhancing its power, undercutting the parties and parliament, and increasing state autonomy, as demonstrated in the Conciliação and its heir, the Liga Progressista. These tensions explain the meaning of the political crises of 1868 and the 1871 Lei de Ventre Livre and the legacy of cynicism over representative government which followed.Os partidos se originaram de facções da Câmara lideradas por oradores que representavam oligarquias rurais e comerciais, bem como grupos urbanos mobilizados. Suas origens, evidentes na Assembléia Constituinte de 1823, consolidaram-se na "oposição liberal" de 1826-31. A maioria moderada dominou os primeiros anos da Regência, mas dividiu-se a respeito do aprofundamento da reforma liberal. Um movimento de reação levou a um novo partido majoritário em 1837, privilegiando um estado forte equilibrado com parlamento e gabinete representativos. Esse partido, posteriormente conhecido como os Conservadores, enfrentou uma oposição, depois conhecida como os Liberais que, embora compartilhassem algumas crenças liberais, inicialmente compuseram uma aliança de ocasião. Após assumir o poder, o imperador, que se mostrou desconfiado das lealdades e ambições partidárias, passou a dominar progressivamente o gabinete, aumentando seu poder, limitando os partidos e o parlamento e aumentando a autonomia do Estado, como se percebe na Conciliação e em sua herdeira, a Liga Progressista. Essas tensões explicam o significado da crise política de 1868, da Lei do Ventre Livre de 1871 e do legado de ceticismo para com o governo representativo que se seguiu

Low Complexity Regularization of Linear Inverse Problems

Inverse problems and regularization theory is a central theme in contemporary signal processing, where the goal is to reconstruct an unknown signal from partial indirect, and possibly noisy, measurements of it. A now standard method for recovering the unknown signal is to solve a convex optimization problem that enforces some prior knowledge about its structure. This has proved efficient in many problems routinely encountered in imaging sciences, statistics and machine learning. This chapter delivers a review of recent advances in the field where the regularization prior promotes solutions conforming to some notion of simplicity/low-complexity. These priors encompass as popular examples sparsity and group sparsity (to capture the compressibility of natural signals and images), total variation and analysis sparsity (to promote piecewise regularity), and low-rank (as natural extension of sparsity to matrix-valued data). Our aim is to provide a unified treatment of all these regularizations under a single umbrella, namely the theory of partial smoothness. This framework is very general and accommodates all low-complexity regularizers just mentioned, as well as many others. Partial smoothness turns out to be the canonical way to encode low-dimensional models that can be linear spaces or more general smooth manifolds. This review is intended to serve as a one stop shop toward the understanding of the theoretical properties of the so-regularized solutions. It covers a large spectrum including: (i) recovery guarantees and stability to noise, both in terms of $\ell^2$-stability and model (manifold) identification; (ii) sensitivity analysis to perturbations of the parameters involved (in particular the observations), with applications to unbiased risk estimation ; (iii) convergence properties of the forward-backward proximal splitting scheme, that is particularly well suited to solve the corresponding large-scale regularized optimization problem