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

Global Solution to the Three-Dimensional Incompressible Flow of Liquid Crystals

The equations for the three-dimensional incompressible flow of liquid crystals are considered in a smooth bounded domain. The existence and uniqueness of the global strong solution with small initial data are established. It is also proved that when the strong solution exists, all the global weak solutions constructed in [16] must be equal to the unique strong solution

Orientability and energy minimization in liquid crystal models

Uniaxial nematic liquid crystals are modelled in the Oseen-Frank theory through a unit vector field $n$. This theory has the apparent drawback that it does not respect the head-to-tail symmetry in which $n$ should be equivalent to -$n$. This symmetry is preserved in the constrained Landau-de Gennes theory that works with the tensor $Q=s\big(n\otimes n- \frac{1}{3} Id\big)$.We study the differences and the overlaps between the two theories. These depend on the regularity class used as well as on the topology of the underlying domain. We show that for simply-connected domains and in the natural energy class $W^{1,2}$ the two theories coincide, but otherwise there can be differences between the two theories, which we identify. In the case of planar domains we completely characterise the instances in which the predictions of the constrained Landau-de Gennes theory differ from those of the Oseen-Frank theory

Lower bounds for nodal sets of Dirichlet and Neumann eigenfunctions

Let \phi\ be a Dirichlet or Neumann eigenfunction of the Laplace-Beltrami operator on a compact Riemannian manifold with boundary. We prove lower bounds for the size of the nodal set {\phi=0}.Comment: 7 page

Testing Students with Special Educational Needs in Large-Scale Assessments – Psychometric Properties of Test Scores and Associations with Test Taking Behavior

Assessing competencies of students with special educational needs in learning (SEN-L) poses a challenge for large-scale assessments (LSAs). For students with SEN-L, the available competence tests may fail to yield test scores of high psychometric quality, which are—at the same time—measurement invariant to test scores of general education students. We investigated whether we can identify a subgroup of students with SEN-L, for which measurement invariant competence measures of adequate psychometric quality may be obtained with tests available in LSAs. We furthermore investigated whether differences in test-taking behavior may explain dissatisfying psychometric properties and measurement non-invariance of test scores within LSAs. We relied on person fit indices and mixture distribution models to identify students with SEN-L for whom test scores with satisfactory psychometric properties and measurement invariance may be obtained. We also captured differences in test-taking behavior related to guessing and missing responses. As a result we identified a subgroup of students with SEN-L for whom competence scores of adequate psychometric quality that are measurement invariant to those of general education students were obtained. Concerning test taking behavior, there was a small number of students who unsystematically picked response options. Removing these students from the sample slightly improved item fit. Furthermore, two different patterns of missing responses were identified that explain to some extent problems in the assessments of students with SEN-L.Peer Reviewe

Grid services for the MAGIC experiment

Exploring signals from the outer space has become an observational science under fast expansion. On the basis of its advanced technology the MAGIC telescope is the natural building block for the first large scale ground based high energy gamma-ray observatory. The low energy threshold for gamma-rays together with different background sources leads to a considerable amount of data. The analysis will be done in different institutes spread over Europe. Therefore MAGIC offers the opportunity to use the Grid technology to setup a distributed computational and data intensive analysis system with the nowadays available technology. Benefits of Grid computing for the MAGIC telescope are presented.Comment: 5 pages, 1 figures, to be published in the Proceedings of the 6th International Symposium ''Frontiers of Fundamental and Computational Physics'' (FFP6), Udine (Italy), Sep. 26-29, 200

HISTORIC AND RECENT WINTER SANDHILL CRANE DISTRIBUTION IN CALIFORNIA

Understanding the geographic distribution and long-term dynamics of winter foraging areas and night roost sites of sandhill cranes (Grus canadensis) is important to their conservation and management. We studied sandhill crane distribution in California’s Central Valley from December 2012 through February 2013. We mapped observed flock and night roost locations. Flock locations occurred between Tehama County in the north and Kern County in the south. Flocks were concentrated in the northern Sacramento Valley, the Sacramento-San Joaquin Delta, the northern San Joaquin Valley south of Tracy to Mendota (including the lower Stanislaus and Tuolumne River floodplains and the Grasslands Region), and the southern San Joaquin Valley in the vicinity of Pixley in Tulare County. We also reviewed records of historic occurrences of cranes in California to interpret the importance of our flock and night roost locations. Although cranes wintered in the Los Angeles, San Diego, and San Francisco Bay metropolitan areas in the 19th and early 20th centuries, they no longer occur in significant numbers in these areas due to widespread habitat loss. Three additional areas which were used in the mid-20th century have apparently been abandoned or are being used only infrequently: the Red Bluff area (along the Sacramento River between Red Bluff and Anderson, Tehama County), the Goose Lake area (Kern County), and the Carrizo Plain (San Luis Obispo County). The primary cause of site abandonment at these sites is loss of suitable foraging habitat (small grain crops). With the exception of the Southern San Joaquin region, crane winter range has expanded in the Central Valley since the 1960s. Range expansion has principally been due to expansion of public wildlife refuges and private sanctuaries, plus improvements in their management (including reductions in hunting disturbance). To improve habitat conditions for cranes across their Central Valley wintering range, we recommend that management be focused on protection, enhancement, and creation of crane habitat complexes, each of which should contain 1 or more roost sites surrounded by sufficient well-managed foraging habitat. The following conservation strategies (listed in order of priority) should be implemented for each major crane wintering region: 1) protect existing, unprotected roost sites by fee-title acquisition or conservation easements (prioritize among sites according to their importance to greater sandhill cranes; G. c. tabida); 2) protect foraging landscapes around existing roosts, primarily through easements restricting development and crop types that are incompatible to cranes; 3) enhance food availability within those landscapes by improving foraging conditions on conservation lands and providing annual incentives for improvements on private lands; and 4) create additional protected roost sites toward the edge of their existing range where birds can access additional foraging areas