Oracle Based Active Set Algorithm for Scalable Elastic Net Subspace Clustering

State-of-the-art subspace clustering methods are based on expressing each data point as a linear combination of other data points while regularizing the matrix of coefficients with $\ell_1$, $\ell_2$ or nuclear norms. $\ell_1$ regularization is guaranteed to give a subspace-preserving affinity (i.e., there are no connections between points from different subspaces) under broad theoretical conditions, but the clusters may not be connected. $\ell_2$ and nuclear norm regularization often improve connectivity, but give a subspace-preserving affinity only for independent subspaces. Mixed $\ell_1$, $\ell_2$ and nuclear norm regularizations offer a balance between the subspace-preserving and connectedness properties, but this comes at the cost of increased computational complexity. This paper studies the geometry of the elastic net regularizer (a mixture of the $\ell_1$ and $\ell_2$ norms) and uses it to derive a provably correct and scalable active set method for finding the optimal coefficients. Our geometric analysis also provides a theoretical justification and a geometric interpretation for the balance between the connectedness (due to $\ell_2$ regularization) and subspace-preserving (due to $\ell_1$ regularization) properties for elastic net subspace clustering. Our experiments show that the proposed active set method not only achieves state-of-the-art clustering performance, but also efficiently handles large-scale datasets.Comment: 15 pages, 6 figures, accepted to CVPR 2016 for oral presentatio

Active-IRS Aided Wireless Network: System Modeling and Performance Analysis

Active intelligent reflecting surface (IRS) enables flexible signal reflection control with \emph{power amplification}, thus effectively compensating the product-distance path-loss in conventional passive-IRS aided systems. In this letter, we characterize the communication performance of an active-IRS aided single-cell wireless network. To this end, we first propose a \emph{customized} IRS deployment strategy, where the active IRSs are uniformly deployed within a ring concentric with the cell to serve the users far from the base station. Next, given the Nakagami-$m$ fading channel, we characterize the cascaded active-IRS channel by using the \emph{mixture Gamma distribution} approximation and derive a closed-form expression for the mean signal-to-noise ratio (SNR) at the user averaged over channel fading. Moreover, we numerically show that to maximize the system performance, it is necessary to choose a proper active-IRS density given a fixed number of total reflecting elements, which significantly differs from the passive-IRS case for which the centralized IRS deployment scheme is better. Furthermore, the active-IRS aided wireless network achieves higher spatial throughput than the passive-IRS counterpart when the total number of reflecting elements is small

Unconventional Superconducting Symmetry in a Checkerboard Antiferromagnet

We use a renormalized mean field theory to study the Gutzwiller projected BCS states of the extended Hubbard model in the large $U$ limit, or the $t$-$t'$-$J$-$J'$ model on a two-dimensional checkerboard lattice. At small $t'/t$, the frustration due to the diagonal terms of $t'$ and $J'$ does not alter the $d_{x^2-y^2}$-wave pairing symmetry, and the negative (positive) $t'/t$ enhances (suppresses) the pairing order parameter. At large $t'/t$, the ground state has an extended s-wave symmetry. At the intermediate $t'/t$, the ground state is $d+id$ or $d+is$-wave with time reversal symmetry broken.Comment: 6 pages, 6 figure

Charge Ordered RVB States in the Doped Cuprates

We study charge ordered d-wave resonating valence bond states (dRVB) in the doped cuprates, and estimate the energies of these states in a generalized $t-J$ model by using a renormalized mean field theory. The long range Coulomb potential tends to modulate the charge density in favor of the charge ordered RVB state. The possible relevance to the recently observed $4 \times 4$ checkerboard patterns in tunnelling conductance in high $T_c$ cuprates is discussed.Comment: 4 pages, 4 figures, 3 table

Strongly enhanced light-matter interaction in a hybrid photonic-plasmonic resonator

We propose a hybrid photonic-plasmonic resonant structure which consists of a metal nanoparticle (MNP) and a whispering gallery mode (WGM) microcavity. It is found that the hybrid mode enables a strong interaction between the light and matter, and the single-atom cooperativity is enhanced by more than two orders of magnitude compared to that in a bare WGM microcavity. This remarkable improvement originates from two aspects: (1) the MNP offers a highly enhanced local field in the vicinity of an emitter, and (2), surprisingly, the high-\textit{Q} property of WGMs can be maintained in the presence of the MNP. Thus the present system has great advantages over a single microcavity or a single MNP, and holds great potential in quantum optics, nonlinear optics and highly sensitive biosening.Comment: 5 pages, 4 figure