Envelopes and principal component regression

Envelope methods offer targeted dimension reduction for various models. The overarching goal is to improve efficiency in multivariate parameter estimation by projecting the data onto a lower-dimensional subspace known as the envelope. Envelope approaches have advantages in analyzing data with highly correlated variables, but their iterative Grassmannian optimization algorithms do not scale very well with ultra high-dimensional data. While the connections between envelopes and partial least squares in multivariate linear regression have promoted recent progress in high-dimensional studies of envelopes, we propose a more straightforward way of envelope modeling from a novel principal components regression perspective. The proposed procedure, Non-Iterative Envelope Component Estimation (NIECE), has excellent computational advantages over the iterative Grassmannian optimization alternatives in high dimensions. We develop a unified NIECE theory that bridges the gap between envelope methods and principal components in regression. The new theoretical insights also shed light on the envelope subspace estimation error as a function of eigenvalue gaps of two symmetric positive definite matrices used in envelope modeling. We apply the new theory and algorithm to several envelope models, including response and predictor reduction in multivariate linear models, logistic regression, and Cox proportional hazard model. Simulations and illustrative data analysis show the potential for NIECE to improve standard methods in linear and generalized linear models significantly

Remote information concentration and multipartite entanglement in multilevel systems

Remote information concentration (RIC) in $d$-level systems (qudits) is studied. It is shown that the quantum information initially distributed in three spatially separated qudits can be remotely and deterministically concentrated to a single qudit via an entangled channel without performing any global operations. The entangled channel can be different types of genuine multipartite pure entangled states which are inequivalent under local operations and classical communication. The entangled channel can also be a mixed entangled state, even a bound entangled state which has a similar form to the Smolin state, but has different features from the Smolin state. A common feature of all these pure and mixed entangled states is found, i.e., they have $d^2$ common commuting stabilizers. The differences of qudit-RIC and qubit-RIC ($d=2$) are also analyzed.Comment: 10 pages, 3 figure

Molecular Lines of 13 Galactic Infrared Bubble Regions

We investigated the physical properties of molecular clouds and star formation processes around infrared bubbles which are essentially expanding HII regions. We performed observations of 13 galactic infrared bubble fields containing 18 bubbles. Five molecular lines, 12CO (J=1-0), 13CO (J=1-0), C18O(J=1-0), HCN (J=1-0), and HCO+ (J=1-0), were observed, and several publicly available surveys, GLIMPSE, MIPSGAL, ATLASGAL, BGPS, VGPS, MAGPIS, and NVSS, were used for comparison. We find that these bubbles are generally connected with molecular clouds, most of which are giant. Several bubble regions display velocity gradients and broad shifted profiles, which could be due to the expansion of bubbles. The masses of molecular clouds within bubbles range from 100 to 19,000 solar mass, and their dynamic ages are about 0.3-3.7 Myr, which takes into account the internal turbulence pressure of surrounding molecular clouds. Clumps are found in the vicinity of all 18 bubbles, and molecular clouds near four of these bubbles with larger angular sizes show shell-like morphologies, indicating that either collect-and-collapse or radiation-driven implosion processes may have occurred. Due to the contamination of adjacent molecular clouds, only six bubble regions are appropriate to search for outflows, and we find that four of them have outflow activities. Three bubbles display ultra-compact HII regions at their borders, and one of them is probably responsible for its outflow. In total, only six bubbles show star formation activities in the vicinity, and we suggest that star formation processes might have been triggered.Comment: 55 Pages, 32 figures. Accepted for publication in A