Sparse Recovery via Differential Inclusions

In this paper, we recover sparse signals from their noisy linear measurements by solving nonlinear differential inclusions, which is based on the notion of inverse scale space (ISS) developed in applied mathematics. Our goal here is to bring this idea to address a challenging problem in statistics, \emph{i.e.} finding the oracle estimator which is unbiased and sign-consistent using dynamics. We call our dynamics \emph{Bregman ISS} and \emph{Linearized Bregman ISS}. A well-known shortcoming of LASSO and any convex regularization approaches lies in the bias of estimators. However, we show that under proper conditions, there exists a bias-free and sign-consistent point on the solution paths of such dynamics, which corresponds to a signal that is the unbiased estimate of the true signal and whose entries have the same signs as those of the true signs, \emph{i.e.} the oracle estimator. Therefore, their solution paths are regularization paths better than the LASSO regularization path, since the points on the latter path are biased when sign-consistency is reached. We also show how to efficiently compute their solution paths in both continuous and discretized settings: the full solution paths can be exactly computed piece by piece, and a discretization leads to \emph{Linearized Bregman iteration}, which is a simple iterative thresholding rule and easy to parallelize. Theoretical guarantees such as sign-consistency and minimax optimal $l_2$-error bounds are established in both continuous and discrete settings for specific points on the paths. Early-stopping rules for identifying these points are given. The key treatment relies on the development of differential inequalities for differential inclusions and their discretizations, which extends the previous results and leads to exponentially fast recovering of sparse signals before selecting wrong ones.Comment: In Applied and Computational Harmonic Analysis, 201

Geometry Method: An Architectural Education Experiment of Crossing Boundaries of Chinese and Western

Western architecture has a clear system and context. One important reason is that geometry is the foundation and core of Western classical architecture. Looking back at the traditional Chinese architecture, the mentoring method of oral communication is the only way of passing on knowledge. It is almost impossible to learn from a systematic model. One of my experiments in architectural education is to re-recognize the connotation of Chinese native architecture with the method of geometry as a core, to explore the potential geometric and mathematical prototypes of Chinese traditional architectural space and aesthetics, and to translate abstract concepts and intentions into visual image and digital relationships. While discussing the spatial formal logically, it sorts out the context and classification structure of traditional Chinese space to help understand the essence of architectural design and its significance. This pedagogy is unfoldedasa”trilogy”system:(1)ResearchandanalysisoftraditionalChineseclassical architecture, and establishing an analytical model; (2) Build relationships between geometric analysis and space; (3) Design/ Model making

Review of data-driven methods used to control the normalization of the top quark background contribution in the H --> WW(*) --> {\ellν\ellν} analyses at the LHC

A few data-driven methods used in deriving the normalization of the dominant top quark background contribution in the $H → WW(*) → {\ellν\ellν}$ analyses by the ATLAS and CMS experiments at the LHC are reviewed and compared. Additional information, justi fication or modi fication to some of the methods is provided. These methods have also been or can be applied to other analyses such as cross section measurements of the Standard Model WW process and searches for new physics in channels with similar final states

Silicon nitride metalenses for unpolarized high-NA visible imaging

As one of nanoscale planar structures, metasurface has shown excellent superiorities on manipulating light intensity, phase and/or polarization with specially designed nanoposts pattern. It allows to miniature a bulky optical lens into the chip-size metalens with wavelength-order thickness, playing an unprecedented role in visible imaging systems (e.g. ultrawide-angle lens and telephoto). However, a CMOS-compatible metalens has yet to be achieved in the visible region due to the limitation on material properties such as transmission and compatibility. Here, we experimentally demonstrate a divergent metalens based on silicon nitride platform with large numerical aperture (NA~0.98) and high transmission (~0.8) for unpolarized visible light, fabricated by a 695-nm-thick hexagonal silicon nitride array with a minimum space of 42 nm between adjacent nanoposts. Nearly diffraction-limit virtual focus spots are achieved within the visible region. Such metalens enables to shrink objects into a micro-scale size field of view as small as a single-mode fiber core. Furthermore, a macroscopic metalens with 1-cm-diameter is also realized including over half billion nanoposts, showing a potential application of wide viewing-angle functionality. Thanks to the high-transmission and CMOS-compatibility of silicon nitride, our findings may open a new door for the miniaturization of optical lenses in the fields of optical fibers, microendoscopes, smart phones, aerial cameras, beam shaping, and other integrated on-chip devices.Comment: 16 pages, 7 figure

Observing the Cell in Its Native State: Imaging Subcellular Dynamics in Multicellular Organisms

True physiological imaging of subcellular dynamics requires studying cells within their parent organisms, where all the environmental cues that drive gene expression, and hence the phenotypes that we actually observe, are present. A complete understanding also requires volumetric imaging of the cell and its surroundings at high spatiotemporal resolution, without inducing undue stress on either. We combined lattice light-sheet microscopy with adaptive optics to achieve, across large multicellular volumes, noninvasive aberration-free imaging of subcellular processes, including endocytosis, organelle remodeling during mitosis, and the migration of axons, immune cells, and metastatic cancer cells in vivo. The technology reveals the phenotypic diversity within cells across different organisms and developmental stages and may offer insights into how cells harness their intrinsic variability to adapt to different physiological environments

Dimension reduction and parameter estimation for additive index models *

In this paper, we consider simultaneous model selection and estimation for the additive index model. The additive index model is a class of structured nonparametric models that can be expressed as additive models of a set of unknown linear transformation of the original predictor variables. We introduce a penalized least squares estimator and discuss how it can be efficiently computed in practice. Both theoretical and empirical properties of the estimate are presented to demonstrate its merits. Extensions to more general prediction framework are also discussed