Low rank representation on Riemannian manifold of symmetric positive definite matrices

Sparse coding aims to find a more compact representation based on a set of dictionary atoms. A well-known technique looking at 2D sparsity is the low rank representation (LRR). However, in many computer vision applications, data often originate from a manifold, which is equipped with some Riemannian geometry. In this case, the existing LRR becomes inappropriate for modeling and incorporating the intrinsic geometry of the manifold that is potentially important and critical to applications. In this paper, we generalize the LRR over the Euclidean space to the LRR model over a specific Rimannian manifold—the manifold of symmetric positive matrices (SPD). Experiments on several computer vision datasets showcase its noise robustness and superior performance on classification and segmentation compared with state-of-the-art approaches.

On the cardinality and complexity of the set of codings for self-similar sets with positive Lebesgue measure

Let λ1,…,λn be real numbers in (0,1) and p1,…,pn be points in Rd. Consider the collection of maps fj:Rd→Rd given by fj(x)=λjx+(1−λj)pj. It is a well known result that there exists a unique nonempty compact set Λ⊂Rd satisfying Λ=∪nj=1fj(Λ). Each x∈Λ has at least one coding, that is a sequence (ϵi)∞i=1 ∈{1,…,n}N that satisfies limN→∞fϵ1…fϵN(0)=x. We study the size and complexity of the set of codings of a generic x∈Λ when Λ has positive Lebesgue measure. In particular, we show that under certain natural conditions almost every x∈Λ has a continuum of codings. We also show that almost every x∈Λ has a universal coding. Our work makes no assumptions on the existence of holes in Λ and improves upon existing results when it is assumed Λ contains no holes.

The problem of two fixed centers: bifurcation diagram for positive energies

We give a comprehensive analysis of the Euler-Jacobi problem of motion in the field of two fixed centers with arbitrary relative strength and for positive values of the energy. These systems represent nontrivial examples of integrable dynamics and are analysed from the point of view of the energy-momentum mapping from the phase space to the space of the integration constants. In this setting, we describe the structure of the scattering trajectories in phase space and derive an explicit description of the bifurcation diagram, i.e., the set of critical value of the energy-momentum map.

Positive phenotypic selection inferred from phylogenies

Rates of phenotypic evolution vary widely in nature and these rates may often reflect the intensity of natural selection. Here we outline an approach for detecting exceptional shifts in the rate of phenotypic evolution across phylogenies. We introduce a simple new branch-specific metric ∆V/∆B that divides observed phenotypic change along a branch into two components: (1) that attributable to the background rate (∆B), and (2) that attributable to departures from the background rate (∆V). Where the amount of expected change derived from variation in the rate of morphological evolution doubles that explained by to the background rate (∆V/∆B > 2), we identify this as positive phenotypic selection. We apply our approach to six datasets, finding multiple instances of positive selection in each. Our results support the growing appreciation that the traditional gradual view of phenotypic evolution is rarely upheld, with a more episodic view taking its place. This moves focus away from viewing phenotypic evolution as a simple homogeneous process and facilitates reconciliation with macroevolutionary interpretations from a genetic perspective, paving the way to novel insights into the link between genotype and phenotype. The ability to detect positive selection when genetic data are unavailable or unobtainable represents an attractive prospect for extant species, but when applied to fossil data it can reveal patterns of natural selection in deep time that would otherwise be impossible.

Long-term positive effects of repeating a year in school: Six-year longitudinal study of self-beliefs, anxiety, social relations, school grades, and test scores

Consistently with a priori predictions, school retention (repeating a year in school) had largely positive effects for a diverse range of 10 outcomes (e.g., math self-concept, self-efficacy, anxiety, relations with teachers, parents and peers, school grades, and standardized achievement test scores). The design, based on a large, representative sample of German students (N = 1,325, M age = 11.75 years) measured each year during the first five years of secondary school, was particularly strong. It featured four independent retention groups (different groups of students, each repeating one of the four first years of secondary school, total N = 103), with multiple post-test waves to evaluate short- and long-term effects, controlling for covariates (gender, age, SES, primary school grades, IQ) and one or more sets of 10 outcomes realised prior to retention. Tests of developmental invariance demonstrated that the effects of retention (controlling for covariates and pre-retention outcomes) were highly consistent across this potentially volatile early-to-middle adolescent period; largely positive effects in the first year following retention were maintained in subsequent school years following retention. Particularly considering that these results are contrary to at least some of the accepted wisdom about school retention, the findings have important implications for educational researchers, policymakers and parents.