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

CoSaMP: Iterative signal recovery from incomplete and inaccurate samples

Compressive sampling offers a new paradigm for acquiring signals that are compressible with respect to an orthonormal basis. The major algorithmic challenge in compressive sampling is to approximate a compressible signal from noisy samples. This paper describes a new iterative recovery algorithm called CoSaMP that delivers the same guarantees as the best optimization-based approaches. Moreover, this algorithm offers rigorous bounds on computational cost and storage. It is likely to be extremely efficient for practical problems because it requires only matrix-vector multiplies with the sampling matrix. For many cases of interest, the running time is just O(N*log^2(N)), where N is the length of the signal.Comment: 30 pages. Revised. Presented at Information Theory and Applications, 31 January 2008, San Dieg

Greedy Signal Recovery Review

The two major approaches to sparse recovery are L1-minimization and greedy methods. Recently, Needell and Vershynin developed Regularized Orthogonal Matching Pursuit (ROMP) that has bridged the gap between these two approaches. ROMP is the first stable greedy algorithm providing uniform guarantees. Even more recently, Needell and Tropp developed the stable greedy algorithm Compressive Sampling Matching Pursuit (CoSaMP). CoSaMP provides uniform guarantees and improves upon the stability bounds and RIC requirements of ROMP. CoSaMP offers rigorous bounds on computational cost and storage. In many cases, the running time is just O(NlogN), where N is the ambient dimension of the signal. This review summarizes these major advances

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