The Augmented Lagrange Multiplier Method for Exact Recovery of Corrupted Low-Rank Matrices

This paper proposes scalable and fast algorithms for solving the Robust PCA problem, namely recovering a low-rank matrix with an unknown fraction of its entries being arbitrarily corrupted. This problem arises in many applications, such as image processing, web data ranking, and bioinformatic data analysis. It was recently shown that under surprisingly broad conditions, the Robust PCA problem can be exactly solved via convex optimization that minimizes a combination of the nuclear norm and the $\ell^1$-norm . In this paper, we apply the method of augmented Lagrange multipliers (ALM) to solve this convex program. As the objective function is non-smooth, we show how to extend the classical analysis of ALM to such new objective functions and prove the optimality of the proposed algorithms and characterize their convergence rate. Empirically, the proposed new algorithms can be more than five times faster than the previous state-of-the-art algorithms for Robust PCA, such as the accelerated proximal gradient (APG) algorithm. Moreover, the new algorithms achieve higher precision, yet being less storage/memory demanding. We also show that the ALM technique can be used to solve the (related but somewhat simpler) matrix completion problem and obtain rather promising results too. We further prove the necessary and sufficient condition for the inexact ALM to converge globally. Matlab code of all algorithms discussed are available at http://perception.csl.illinois.edu/matrix-rank/home.htmlComment: Please cite "Zhouchen Lin, Risheng Liu, and Zhixun Su, Linearized Alternating Direction Method with Adaptive Penalty for Low Rank Representation, NIPS 2011." (available at arXiv:1109.0367) instead for a more general method called Linearized Alternating Direction Method This manuscript first appeared as University of Illinois at Urbana-Champaign technical report #UILU-ENG-09-2215 in October 2009 Zhouchen Lin, Risheng Liu, and Zhixun Su, Linearized Alternating Direction Method with Adaptive Penalty for Low Rank Representation, NIPS 2011. (available at http://arxiv.org/abs/1109.0367

Modelo de factores que afectan el interés individual de los estudiantes en las ciencias básicas: diseño y validación de un cuestionario: Model of factors affecting the individual interest of students in basic sciences: design and validation of a questionnaire

The purpose was to design and validate an instrument to measure a set of endogenous and exogenous factors that affect the individual interest of students in learning basic science in secondary school. In the first stage, a multivariate factor model was developed using the grounded theory method. Based on the model, the questionnaire was designed and then content validation was carried out, with the participation of judges and a pilot test. The initial version of the instrument contains 21 items, each of them represents an endogenous or exogenous causal factor; the construct validation was done with students from 1st to 5th year of secondary school. The exploratory factor analysis was carried out with a sample of 274 students, identifying 6 latent dimensions and the confirmatory factor analysis with a sample of 336 students. The results show optimal evidences of the questionnaire that corroborate its validity. It is a practical instrument that, through the perceptual evaluations of the students, empirical inquiries can be made about the effects of each of the factors that affect the individual interest of the students, giving rise to multivariate diagnoses. Keywords: Endogenous and exogenous factors, interest, basic sciences, validation, questionnaire.Resumen: El propósito fue diseñar y validar un instrumento para medir a un conjunto de factores endógenos y exógenos que afectan el interés individual de los estudiantes en el aprendizaje de las ciencias básicas. Primeramente, se elaboró un modelo multivariado de factores y luego se diseñó el cuestionario, que fue sometido a validación de contenido con la participación de jueces y una prueba piloto. La validación de constructo se hizo mediante el análisis factorial exploratorio, con una muestra de 274 alumnos de educación secundaria, se identificaron cinco dimensiones latentes. El análisis factorial confirmatorio se realizó con una muestra de 336 alumnos. Los resultados muestran evidencias óptimas del cuestionario que corroboran su validez con 18 ítems. Es un instrumento práctico que permite realizar indagaciones empíricas sobre los efectos de cada uno de los factores que afectan el interés individual mediante las valoraciones perceptivas de los alumnos, dando lugar a diagnósticos multivariados. Palabras clave: factores endógenos y exógenos, interés hacia las ciencias básicas, validación.     Abstract: The purpose was to design and validate an instrument to measure a set of endogenous and exogenous factors that affect the individual interest of students in learning basic sciences. First, a multivariate factor model was developed and then the questionnaire was designed, which was submitted to content validation with the participation of judges and a pilot test. The construct validation was done through exploratory factor analysis, with a sample of 274 secondary school students, five latent dimensions were identified. The confirmatory factor analysis was carried out with a sample of 336 students. The results show optimal evidence of the questionnaire that corroborates its validity with 18 items. It is a practical instrument that allows empirical inquiries to be made about the effects of each of the factors that affect individual interest through perceptual evaluations of the students, giving rise to multivariate diagnoses. Keywords: endogenous and exogenous factors, interest in basic sciences, validation

Modelo de factores que afectan el interés individual de los estudiantes en las ciencias básicas: diseño y validación de un cuestionario.

El propósito fue diseñar y validar un instrumento para medir a un conjunto de factores endógenos y exógenos que afectan el interés individual de los estudiantes en el aprendizaje de las ciencias básicas. Primeramente, se elaboró un modelo multivariado de factores y luego se diseñó el cuestionario, que fue sometido a validación de contenido con la participación de jueces y una prueba piloto. La validación de constructo se hizo mediante el análisis factorial exploratorio, con una muestra de 274 alumnos de educación secundaria, se identificaron cinco dimensiones latentes. El análisis factorial confirmatorio se realizó con una muestra de 336 alumnos. Los resultados muestran evidencias óptimas del cuestionario que corroboran su validez con 18 ítems. Es un instrumento práctico que permite realizar indagaciones empíricas sobre los efectos de cada uno de los factores que afectan el interés individual mediante las valoraciones perceptivas de los alumnos, dando lugar a diagnósticos multivariados

Electron Tomography of the Contact between T Cells and SIV/HIV-1: Implications for Viral Entry

The envelope glycoproteins of primate lentiviruses, including human and simian immunodeficiency viruses (HIV and SIV), are heterodimers of a transmembrane glycoprotein (usually gp41), and a surface glycoprotein (gp120), which binds CD4 on target cells to initiate viral entry. We have used electron tomography to determine the three-dimensional architectures of purified SIV virions in isolation and in contact with CD4+ target cells. The trimeric viral envelope glycoprotein surface spikes are heterogeneous in appearance and typically ∼120 Å long and ∼120 Å wide at the distal end. Docking of SIV or HIV-1 on the T cell surface occurs via a neck-shaped contact region that is ∼400 Å wide and consistently consists of a closely spaced cluster of five to seven rod-shaped features, each ∼100 Å long and ∼100 Å wide. This distinctive structure is not observed when viruses are incubated with T lymphocytes in the presence of anti-CD4 antibodies, the CCR5 antagonist TAK779, or the peptide entry inhibitor SIVmac251 C34. For virions bound to cells, few trimers were observed away from this cluster at the virion–cell interface, even in cases where virus preparations showing as many as 70 envelope glycoprotein trimers per virus particle were used. This contact zone, which we term the “entry claw”, provides a spatial context to understand the molecular mechanisms of viral entry. Determination of the molecular composition and structure of the entry claw may facilitate the identification of improved drugs for the inhibition of HIV-1 entry

Molecular Architectures of Trimeric SIV and HIV-1 Envelope Glycoproteins on Intact Viruses: Strain-Dependent Variation in Quaternary Structure

The initial step in target cell infection by human, and the closely related simian immunodeficiency viruses (HIV and SIV, respectively) occurs with the binding of trimeric envelope glycoproteins (Env), composed of heterodimers of the viral transmembrane glycoprotein (gp41) and surface glycoprotein (gp120) to target T-cells. Knowledge of the molecular structure of trimeric Env on intact viruses is important both for understanding the molecular mechanisms underlying virus-cell interactions and for the design of effective immunogen-based vaccines to combat HIV/AIDS. Previous analyses of intact HIV-1 BaL virions have already resulted in structures of trimeric Env in unliganded and CD4-liganded states at ∼20 Å resolution. Here, we show that the molecular architectures of trimeric Env from SIVmneE11S, SIVmac239 and HIV-1 R3A strains are closely comparable to that previously determined for HIV-1 BaL, with the V1 and V2 variable loops located at the apex of the spike, close to the contact zone between virus and cell. The location of the V1/V2 loops in trimeric Env was definitively confirmed by structural analysis of HIV-1 R3A virions engineered to express Env with deletion of these loops. Strikingly, in SIV CP-MAC, a CD4-independent strain, trimeric Env is in a constitutively “open” conformation with gp120 trimers splayed out in a conformation similar to that seen for HIV-1 BaL Env when it is complexed with sCD4 and the CD4i antibody 17b. Our findings suggest a structural explanation for the molecular mechanism of CD4-independent viral entry and further establish that cryo-electron tomography can be used to discover distinct, functionally relevant quaternary structures of Env displayed on intact viruses

Determination of Molecular Structures of HIV Envelope Glycoproteins using Cryo-Electron Tomography and Automated Sub-tomogram Averaging

Since its discovery nearly 30 years ago, more than 60 million people have been infected with the human immunodeficiency virus (HIV) (www.usaid.gov). The virus infects and destroys CD4+ T-cells thereby crippling the immune system, and causing an acquired immunodeficiency syndrome (AIDS) 2. Infection begins when the HIV Envelope glycoprotein "spike" makes contact with the CD4 receptor on the surface of the CD4+ T-cell. This interaction induces a conformational change in the spike, which promotes interaction with a second cell surface co-receptor 5,9. The significance of these protein interactions in the HIV infection pathway makes them of profound importance in fundamental HIV research, and in the pursuit of an HIV vaccine