Learning Mixtures of Gaussians in High Dimensions

Efficiently learning mixture of Gaussians is a fundamental problem in statistics and learning theory. Given samples coming from a random one out of k Gaussian distributions in Rn, the learning problem asks to estimate the means and the covariance matrices of these Gaussians. This learning problem arises in many areas ranging from the natural sciences to the social sciences, and has also found many machine learning applications. Unfortunately, learning mixture of Gaussians is an information theoretically hard problem: in order to learn the parameters up to a reasonable accuracy, the number of samples required is exponential in the number of Gaussian components in the worst case. In this work, we show that provided we are in high enough dimensions, the class of Gaussian mixtures is learnable in its most general form under a smoothed analysis framework, where the parameters are randomly perturbed from an adversarial starting point. In particular, given samples from a mixture of Gaussians with randomly perturbed parameters, when n > {\Omega}(k^2), we give an algorithm that learns the parameters with polynomial running time and using polynomial number of samples. The central algorithmic ideas consist of new ways to decompose the moment tensor of the Gaussian mixture by exploiting its structural properties. The symmetries of this tensor are derived from the combinatorial structure of higher order moments of Gaussian distributions (sometimes referred to as Isserlis' theorem or Wick's theorem). We also develop new tools for bounding smallest singular values of structured random matrices, which could be useful in other smoothed analysis settings

Reconsidering the New Normal: Trauma, Vulnerability & Resilience in Post-Katrina New Orleans

Traumatic anthropogenic or natural disasters can redefine the ecological and social diversity of cities, with "new normal" conditions often emerging in post-trauma urban landscapes. The objective of our ULTRA project is to examine how the pace and trajectory of recovery in post-Katrina New Orleans reflect ecological and social diversity. Specifically, we are examining potential parallels and interactions between ecological and social diversity within and among neighborhoods, and across the New Orleans metropolitan area. We are doing so by (1) organizing and coordinating a network of scholars and practitioners to exchange experience and knowledge and thereby advance understanding of connections between diversity and recovery in post-trauma urban ecosystems; (2) assembling a central data archive on the structure and diversity of ecological communities of New Orleans, which involves conducting an inventory of the post-Katrina urban forest; (4) and conducting a GIS-based spatial analysis of pre- and post-trauma landscape and social metrics derived from satellite imagery and the 2000 and 2010 federal census, analyzed for diversification and compared to stabilization metrics. This citywide study is being supplemented with three fine-grained studies in the neighborhoods of the Lower Ninth Ward, Hollygrove, and Pontchartrain Park. Qualitative data collected in these neighborhoods provides insight into the relationships between trauma and ecological and social diversity, and identify variation in the timing, pace, and trajectory of neighborhood recovery. In the future, we will expand our efforts to consider how diversity reflects the availability and valuation of ecosystem services in post-trauma urban landscapes. Our intent is to develop New Orleans as a natural laboratory for the study of ecological and community resiliency

Long-Term Pyrene Exposure of Grass Shrimp, \u3ci\u3ePalaemonetes pugio\u3c/i\u3e, Affects Molting and Reproduction of Exposed Males and Offspring of Exposed Females

The objective of this study was to investigate the impact of long-term pyrene exposure on molting and reproduction in the model estuarine invertebrate, the grass shrimp (Palaemonetes pugio). Grass shrimp were exposed to measured concentrations of 5.1, 15.0, and 63.4 ppb (mu g/L) pyrene for 6 weeks, during which time we determined molting and survivorship. At the end of the exposure, we immediately sacrificed some of the shrimp for biomarker (CYP1A and vitellin) analyses. The remaining shrimp were used to analyze fecundity and embryo survivorship during an additional 6 weeks after termination of pyrene exposure. Male shrimp at the highest pyrene dose (63 ppb) experienced a significant delay in molting and in time until reproduction, and showed elevated ethoxycoumarin o-deethylase (ECOD) activity immediately after the 6-week exposure period. In contrast, 63 ppb pyrene did not affect these parameters in female shrimp. Females produced the same number of eggs per body weight, with high egg viability (98-100%) at all exposure levels, but with decreased survival for the offspring of the 63-ppb pyrene-exposed females. In addition, vitellin levels were elevated only in females at 63 ppb pyrene after the 6-week exposure. We hypothesize that the elevated vitellin binds pyrene and keeps it biologically unavailable to adult females, resulting in maternal transfer of pyrene to the embryos. This would account for the lack of effect of pyrene exposure on ECOD activity, molting, and reproduction in the adult females, and for reduced survival of their offspring

Quantifying the Borderline Candidate in Standard Setting

Background: Conceptualising the Borderline candidate is one of the most difficult tasks in standard setting. However, it is also central to the process. Here we set out to develop a methodology by which the score of borderline candidates can be retrospectively calculated from the Facility index (the percentage of items answered correctly) of assessment items. Methods: We explored performance of all candidates in an academic year in one UK medical school, covering 26 separate assessments. Each assessment had previously been standard set by either Angoff or Borderline Regressions methods. We identified Borderline candidates by reviewing their performance across all assessments in their year. A student was classed as 'Borderline' if they were within 1 Standard Error of Measurement above the pass score, or below the pass score, when a variety of cut-off points were explored experimentally. We plotted the item scores of the Borderline candidates as calculated by each method in comparison with Facility for the whole cohort, and fitted curves to the resulting distributions. Results: Borderline candidate scores intercepted the self-plot of all candidate scores at two places - at a facility of 100% and a facility of 20%. These correspond to all candidates getting the item correct and all candidates guessing the outcome. We observed a strong curvilinear distribution showed by Borderline candidates compared to the whole cohort. This relationship was well described by an exponential of the form y ≈ C·exp(F·x), where y is the Facility of Borderline candidates on that Item, x is the observed Item Facility of the whole cohort, and C and F are constants. We found C and F had similar values under different conditions. Using the typical values for C and F and the observed cohort facility, we could predict the probable Facility for Borderline candidates over the test: in other words, we could calculate the appropriate cut score for Borderline candidates. Differentiating the equation indicates where the assessment ought to be most sensitive. Conclusions: This approach can be used to standard-set assessments in their entirety when they are low stakes or norm referenced, in preference to Cohen methods. Where Cohen methods are based on the performance of one candidate (or a very small number of candidates), this exponential method is based on all candidates and all items and is therefore more robust. In high stakes assessments, it can be used to correct values where the Facility is very different from the standard-set value, and its use in this context for the UK General Medical Council proposed national exam. It could also be used to standard set novel items such as Very Short Answer formats, where standard setting panels are unfamiliar with the expected performance of these items

Symbiotic Gene Activation is Interrupted by Endocrine Disrupting Chemicals

Endocrine disrupting chemicals (EDCs) include organochlorine pesticides, plastics manufacturing by-products, and certain herbicides[1]. These chemicals have been shown to disrupt hormonal signaling in exposed wildlife, lab animals, and mammalian cell culture by binding to estrogen receptors (ER-α and ER-β) and affecting the expression of estrogen responsive genes[2,3]. Additionally, certain plant chemicals, termed phytoestrogens, are also able to bind to estrogen receptors and modulate gene expression, and as such also may be considered EDCs[4]. One example of phytoestrogen action is genistein, a phytochemical produced by soybeans, binding estrogen receptors, and changing expression of estrogen responsive genes which certain studies have linked to a lower incidence of hormonally related cancers in Japanese populations[5]. Why would plants make compounds that are able to act as estrogens in the human body? Obviously, soybeans do not intentionally produce phytoestrogens to prevent breast cancer in Japanese women

Long-range three-body atom-diatom potential for doublet Li3{}_3

An accurate long-range {\em ab initio} potential energy surface has been calculated for the ground state ${}^2A'$ lithium trimer in the frozen diatom approximation using all electron RCCSD(T). The {\em ab initio} energies are corrected for basis set superposition error and extrapolated to the complete basis limit. Molecular van der Waals dispersion coefficients and three-body dispersion damping terms for the atom-diatomic dissociation limit are presented from a linear least squares fit and shown to be an essentially exact representation of the {\em ab initio} surface at large range

Altered Breast Development in Young Girls from an Agricultural Environment

In several human populations, the age at which female breast development begins is reported to have declined over the last five decades. Much debate has occurred over whether this reported decline has actually occurred and what factors contribute to it. However, geographical patterns reflecting earlier developmental onset in some human populations suggest environmental factors influence this phenomenon. These factors include interactions between genetic makeup, nutrition, and possible cumulative exposure to estrogens, both endogenous as well as environmental beginning during in utero development. We examined the onset of breast development in a group of peripubertal girls from the Yaqui Valley of Sonora, Mexico. We observed that girls from valley towns, areas using modern agricultural practices, exhibited larger breast fields than those of girls living in the foothills who exhibited similar stature [e.g., weight, height, body mass index (BMI)], and genetic background. Further, girls from valley towns displayed a poorly defined relationship between breast size and mammary gland development, whereas girls from the Yaqui foothills, where traditional ranching occurs, show a robust positive relationship between breast size and mammary size. The differences noted were obtained by a medically based exam involving morphometric analysis and palpation of tissues, in contrast to visual staging alone. In fact, use of the Tanner scale, involving visual staging of breast development for puberty, detected no differences between the study populations. Mammary tissue, determined by palpation, was absent in 18.5% of the girls living in agricultural areas, although palpable breast adipose tissue was present. No relationship was seen between mammary diameter and weight or BMI in either population. These data suggest that future in-depth studies examining mammary tissue growth and fat deposition in breast tissue are required if we are to understand environmental influences on these phenomena

Shape selection in non-Euclidean plates

We investigate isometric immersions of disks with constant negative curvature into $\mathbb{R}^3$, and the minimizers for the bending energy, i.e. the $L^2$ norm of the principal curvatures over the class of $W^{2,2}$ isometric immersions. We show the existence of smooth immersions of arbitrarily large geodesic balls in $\mathbb{H}^2$ into $\mathbb{R}^3$. In elucidating the connection between these immersions and the non-existence/singularity results of Hilbert and Amsler, we obtain a lower bound for the $L^\infty$ norm of the principal curvatures for such smooth isometric immersions. We also construct piecewise smooth isometric immersions that have a periodic profile, are globally $W^{2,2}$, and have a lower bending energy than their smooth counterparts. The number of periods in these configurations is set by the condition that the principal curvatures of the surface remain finite and grows approximately exponentially with the radius of the disc. We discuss the implications of our results on recent experiments on the mechanics of non-Euclidean plates