Fast global convergence of gradient methods for high-dimensional statistical recovery

Many statistical $M$-estimators are based on convex optimization problems formed by the combination of a data-dependent loss function with a norm-based regularizer. We analyze the convergence rates of projected gradient and composite gradient methods for solving such problems, working within a high-dimensional framework that allows the data dimension \pdim to grow with (and possibly exceed) the sample size \numobs. This high-dimensional structure precludes the usual global assumptions---namely, strong convexity and smoothness conditions---that underlie much of classical optimization analysis. We define appropriately restricted versions of these conditions, and show that they are satisfied with high probability for various statistical models. Under these conditions, our theory guarantees that projected gradient descent has a globally geometric rate of convergence up to the \emph{statistical precision} of the model, meaning the typical distance between the true unknown parameter $\theta^*$ and an optimal solution $\hat{\theta}$. This result is substantially sharper than previous convergence results, which yielded sublinear convergence, or linear convergence only up to the noise level. Our analysis applies to a wide range of $M$-estimators and statistical models, including sparse linear regression using Lasso ($\ell_1$-regularized regression); group Lasso for block sparsity; log-linear models with regularization; low-rank matrix recovery using nuclear norm regularization; and matrix decomposition. Overall, our analysis reveals interesting connections between statistical precision and computational efficiency in high-dimensional estimation

Asymptotic silence-breaking singularities

We discuss three complementary aspects of scalar curvature singularities: asymptotic causal properties, asymptotic Ricci and Weyl curvature, and asymptotic spatial properties. We divide scalar curvature singularities into two classes: so-called asymptotically silent singularities and non-generic singularities that break asymptotic silence. The emphasis in this paper is on the latter class which have not been previously discussed. We illustrate the above aspects and concepts by describing the singularities of a number of representative explicit perfect fluid solutions.Comment: 25 pages, 6 figure

The interpretation and delivery of the Welsh Foundation Phase and its contribution to physical literacy

The introduction of the Foundation Phase gave a unique opportunity to study the interpretation and delivery of a play-based early childhood curriculum. This new curriculum saw the disappearance of Physical Education for pupils under the age of seven in Wales. Physical Education is acknowledged as more than the development of physical competence, being part of a process concerned with lifelong physical, intellectual, social and emotional learning accrued through a range of physical activities, in a variety of contexts (Doherty and Brennan, 2008). As such a goal of Physical Education is physical literacy, (Hardman, 2011; Talbot, 2007). In light of this, this research set out to explore the contribution of the Foundation Phase to the development of children’s physical literacy. In order to achieve this, a three-phase complementarity mixed-methods design (Greene et al., 1989) was used to generate data over two years in selected schools in Wales. The schools were found to be enacting the Foundation Phase with fidelity to the original aims of the policy makers by demonstrating the key features of play-based active learning, focused adult-led sessions, child-initiated learning, and use of the outdoors for learning. In so doing they were deemed to be successful in achieving the aim of the Foundation Phase of developing independent, motivated active learners. The Foundation Phase was also found to be supporting the development of children’s cognitive development with good levels of achievement in literacy and numeracy assessments. The playful pedagogy observed in the schools enabled the pupils to have autonomy in their learning. Pupils were motivated, active and engaged in embodied learning both indoors and outdoors. The findings indicated that the Foundation Phase was making a positive contribution to the development of children’s physical literacy

Noisy matrix decomposition via convex relaxation: Optimal rates in high dimensions

We analyze a class of estimators based on convex relaxation for solving high-dimensional matrix decomposition problems. The observations are noisy realizations of a linear transformation $\mathfrak{X}$ of the sum of an approximately) low rank matrix $\Theta^\star$ with a second matrix $\Gamma^\star$ endowed with a complementary form of low-dimensional structure; this set-up includes many statistical models of interest, including factor analysis, multi-task regression, and robust covariance estimation. We derive a general theorem that bounds the Frobenius norm error for an estimate of the pair $(\Theta^\star, \Gamma^\star)$ obtained by solving a convex optimization problem that combines the nuclear norm with a general decomposable regularizer. Our results utilize a "spikiness" condition that is related to but milder than singular vector incoherence. We specialize our general result to two cases that have been studied in past work: low rank plus an entrywise sparse matrix, and low rank plus a columnwise sparse matrix. For both models, our theory yields non-asymptotic Frobenius error bounds for both deterministic and stochastic noise matrices, and applies to matrices $\Theta^\star$ that can be exactly or approximately low rank, and matrices $\Gamma^\star$ that can be exactly or approximately sparse. Moreover, for the case of stochastic noise matrices and the identity observation operator, we establish matching lower bounds on the minimax error. The sharpness of our predictions is confirmed by numerical simulations.Comment: 41 pages, 2 figure

Super coset space geometry

Super coset spaces play an important role in the formulation of supersymmetric theories. The aim of this paper is to review and discuss the geometry of super coset spaces with particular focus on the way the geometrical structures of the super coset space G/H are inherited from the super Lie group G. The isometries of the super coset space are discussed and a definition of Killing supervectors - the supervectors associated with infinitesimal isometries - is given that can be easily extended to spaces other than coset spaces.Comment: 49 pages, 1 figure, AFK previously published under the name A. F. Schunc

Dating Early Archean partial melting events: insights from Re-Os dating of micrometric Os-minerals from Kalahari Craton mantle xenoliths

Our understanding on the formation, mechanism and timing of the formation of the Earth’s earliest continents hinges on obtaining robust and accurate ages of melt depletion and lithosphere stabilisation. The Re-Os geochronometer has been used to obtain such information. However, recent advances indicate that metasomatism can impact on the signatures obtained. This thesis utilises the combination of Re-Os and highly siderophile element (HSE) systematics to accurately assess the impact secondary mineralisation has on the Re-Os ages determined. A suite of samples from the Kalahari craton (Kaapvaal and Zimbabwe cratons) are investigated in terms of their metasomatic overprint and Re-Os systematics. The sub-suite of peridotites from the Kaapvaal craton are first analysed for whole-rock major and trace elements, Re-Os and HSE systematics. The peridotites were also analysed for the 187Os/188Os compositions at the single grain base metal sulphide (BMS) and platinum-group mineral (PGM) scale. These analyses indicate that the Kaapvaal peridotites have experienced high degrees of melt depletion followed by variable and significant enrichment in the incompatible trace elements and HSE. The enrichment of HSE led to the precipitation of metasomatic BMS which impact on the Re-Os ages determined. As such, the whole-rock and single grain BMS TRD (rhenium depletion model age) indicate that the mantle was pervasively metasomatised as early as 3.2 Ga. Nano-particle PGM (Pt-alloys) with radiogenic 187Os/188Os (0.1294-0.1342) were found included within unradiogenic BMS (187Os/188Os 0.1066-0.1084). This signifies that the PGM formed in the presence of Re and evolved to high 187Os/188Os compositions over a long time scale. The Os composition of the Pt-alloys, combined with their nano-particle nature and the Os dichotomy with the host BMS signifies that the Pt-alloys formed in a HSE-Si-rich melt. This provides further evidence for the metasomatic overprinting of the Kaapvaal peridotites. Despite the high degree of metasomatism experienced by the Letlhakane peridotites (Zimbabwe craton), as evidenced by their re-enriched HSE-Se-Te systematics, single grain BMS preserve evidence of partial melting events. The 187Os/188Os analyses of the BMS provide ages >2.5 Ga older than the whole-rock. The oldest BMS TRD age of 3.7 Ga is preserved in a metasomatic BMS associated with secondary clinopyroxene and phlogopite. The attainment of an Eoarchean age from a metasomatic BMS suggests that the metasomatic fluid is able to entrain or nucleate on residual BMS. As such, the obtained ages reflect a mixing between the two different Os signatures. Whereas older ages reflect the dominance of the residual PGM on the bulk Os composition, younger ages are due to the control from the metasomatic melt. The 3.7 Ga TRD obtained age also pushes the age of initial stabilisation of the Zimbabwe lithosphere to within the age of the oldest crustal rocks in this region. The combined results of the Letlhakane and Kaapvaal peridotites indicate that whilst the Re-Os system can be affected by the metasomatic addition of BMS, the combination of HSE and Re-Os at the whole-rock and micro-scale can still resolve geologically significant ages

Photovoice and photodocumentary for enhancing community partner engagement and student learning in a public health field school in Cape Town

Background: Field school research, which begins by considering community partners as pedagogues and thus exploring their perspectives on student learning, is uncommon. Photovoice is a method for self-expression of such marginalized voices. Purpose: Describe the photovoice to photodocumentary process and present results of its evaluation. Methodology/Approach: We employed photovoice with the local guides who accompany community health research field school students in Cape Town. Guides were prompted to take photographs of what students may not see or understand about their community. These were discussed at three workshops and developed into a photodocumentary for inclusion in the curriculum. Twenty-one students completed an open-ended questionnaire. These, and student/staff discussions, were thematically analyzed. Findings/Conclusion: Students reported learning about cultural practices which otherwise would not be visible to them. They felt greater connection to their guides who they saw in a new light, and became aware of how topic-specificity drew attention away from overarching characteristics of community life. Questions raised might be better explored through facilitated discussion rather than question-and-answer sessions alone. Implications: Field schools depend on academic–community partnerships. Photovoice can create space for community partner reflection on student learning and the co-creation of effective pedagogical tools valued by students

Global dynamics of the mixmaster model

The asymptotic behaviour of vacuum Bianchi models of class A near the initial singularity is studied, in an effort to confirm the standard picture arising from heuristic and numerical approaches by mathematical proofs. It is shown that for solutions of types other than VIII and IX the singularity is velocity dominated and that the Kretschmann scalar is unbounded there, except in the explicitly known cases where the spacetime can be smoothly extended through a Cauchy horizon. For types VIII and IX it is shown that there are at most two possibilities for the evolution. When the first possibility is realized, and if the spacetime is not one of the explicitly known solutions which can be smoothly extended through a Cauchy horizon, then there are infinitely many oscillations near the singularity and the Kretschmann scalar is unbounded there. The second possibility remains mysterious and it is left open whether it ever occurs. It is also shown that any finite sequence of distinct points generated by iterating the Belinskii-Khalatnikov-Lifschitz mapping can be realized approximately by a solution of the vacuum Einstein equations of Bianchi type IX.Comment: 16 page