A Random Matrix Approach to Differential Privacy and Structure Preserved Social Network Graph Publishing

Online social networks are being increasingly used for analyzing various societal phenomena such as epidemiology, information dissemination, marketing and sentiment flow. Popular analysis techniques such as clustering and influential node analysis, require the computation of eigenvectors of the real graph's adjacency matrix. Recent de-anonymization attacks on Netflix and AOL datasets show that an open access to such graphs pose privacy threats. Among the various privacy preserving models, Differential privacy provides the strongest privacy guarantees. In this paper we propose a privacy preserving mechanism for publishing social network graph data, which satisfies differential privacy guarantees by utilizing a combination of theory of random matrix and that of differential privacy. The key idea is to project each row of an adjacency matrix to a low dimensional space using the random projection approach and then perturb the projected matrix with random noise. We show that as compared to existing approaches for differential private approximation of eigenvectors, our approach is computationally efficient, preserves the utility and satisfies differential privacy. We evaluate our approach on social network graphs of Facebook, Live Journal and Pokec. The results show that even for high values of noise variance sigma=1 the clustering quality given by normalized mutual information gain is as low as 0.74. For influential node discovery, the propose approach is able to correctly recover 80 of the most influential nodes. We also compare our results with an approach presented in [43], which directly perturbs the eigenvector of the original data by a Laplacian noise. The results show that this approach requires a large random perturbation in order to preserve the differential privacy, which leads to a poor estimation of eigenvectors for large social networks

Connection between in-plane upper critical field Hc2H_{c2} and gap symmetry in layered dd-wave superconductors revisited

Angle-resolved upper critical field $H_{c2}$ provides an efficient tool to probe the gap symmetry of unconventional superconductors. We revisit the behavior of in-plane $H_{c2}$ in $d$-wave superconductors by considering both the orbital effect and Pauli paramagnetic effect. After carrying out systematic analysis, we show that the maxima of $H_{c2}$ could be along either nodal or antinodal directions of a $d$-wave superconducting gap, depending on the specific values of a number of tuning parameters. This behavior is in contrast to the common belief that the maxima of in-plane $H_{c2}$ are along the direction where the superconducting gap takes its maximal value. Therefore, identifying the precise $d$-wave gap symmetry through fitting experiments results of angle-resolved $H_{c2}$ with model calculations at a fixed temperature, as widely used in previous studies, is difficult and practically unreliable. However, our extensive analysis of angle-resolved $H_{c2}$ show that there is a critical temperature $T^{*}$: in-plane $H_{c2}$ exhibits its maxima along nodal directions at $T < T^{*}$ and along antinodal directions at $T^{*} < T < T_c$. The concrete value of $T^{*}$ may change as other parameters vary, but the existence of $\pi/4$ shift of $H_{c2}$ at $T^{\ast}$ appears to be a general feature. Thus a better method to identify the precise $d$-wave gap symmetry is to measure $H_{c2}$ at a number of different temperatures, and examine whether there is a $\pi/4$ shift in its angular dependence at certain $T^{*}$. We further show that Landau level mixing does not change this general feature. However, in the presence of Fulde-Ferrell-Larkin-Ovchinnikov state, the angular dependence of $H_{c2}$ becomes quite complicated, which makes it more difficult to determine the gap symmetry by measuring $H_{c2}$.Comment: 12 pages, 11 figure

Polarization, plasmon, and Debye screening in doped 3D ani-Weyl semimetal

We compute the polarization function in a doped three-dimensional anisotropic-Weyl semimetal, in which the fermion energy dispersion is linear in two components of the momenta and quadratic in the third. Through detailed calculations, we find that the long wavelength plasmon mode depends on the fermion density $n_e$ in the form $\Omega_{p}^{\bot}\propto n_{e}^{3/10}$ within the basal plane and behaves as $\Omega_{p}^{z}\propto n_{e}^{1/2}$ along the third direction. This unique characteristic of the plasmon mode can be probed by various experimental techniques, such as electron energy-loss spectroscopy. The Debye screening at finite chemical potential and finite temperature is also analyzed based on the polarization function.Comment: 11 page

Infrared behavior of dynamical fermion mass generation in QED3_{3}

Extensive investigations show that QED$_{3}$ exhibits dynamical fermion mass generation at zero temperature when the fermion flavor $N$ is sufficiently small. However, it seems difficult to extend the theoretical analysis to finite temperature. We study this problem by means of Dyson-Schwinger equation approach after considering the effect of finite temperature or disorder-induced fermion damping. Under the widely used instantaneous approximation, the dynamical mass displays an infrared divergence in both cases. We then adopt a new approximation that includes an energy-dependent gauge boson propagator and obtain results for dynamical fermion mass that do not contain infrared divergence. The validity of the new approximation is examined by comparing to the well-established results obtained at zero temperature.Comment: 15 pages, 6 figures, to appear on Phys. Rev.

Robust Metric Learning by Smooth Optimization

Most existing distance metric learning methods assume perfect side information that is usually given in pairwise or triplet constraints. Instead, in many real-world applications, the constraints are derived from side information, such as users' implicit feedbacks and citations among articles. As a result, these constraints are usually noisy and contain many mistakes. In this work, we aim to learn a distance metric from noisy constraints by robust optimization in a worst-case scenario, to which we refer as robust metric learning. We formulate the learning task initially as a combinatorial optimization problem, and show that it can be elegantly transformed to a convex programming problem. We present an efficient learning algorithm based on smooth optimization [7]. It has a worst-case convergence rate of O(1/{\surd}{\varepsilon}) for smooth optimization problems, where {\varepsilon} is the desired error of the approximate solution. Finally, our empirical study with UCI data sets demonstrate the effectiveness of the proposed method in comparison to state-of-the-art methods.Comment: Appears in Proceedings of the Twenty-Sixth Conference on Uncertainty in Artificial Intelligence (UAI2010

Quantum phase transition and unusual critical behavior in multi-Weyl semimetals

The low-energy behaviors of gapless double- and triple-Weyl fermions caused by the interplay of long-range Coulomb interaction and quenched disorder are studied by performing a renormalization group analysis. It is found that an arbitrarily weak disorder drives the double-Weyl semimetal to undergo a quantum phase transition into a compressible diffusive metal, independent of the disorder type and the Coulomb interaction strength. In contrast, the nature of the ground state of triple-Weyl fermion system relies sensitively on the specific disorder type in the noninteracting limit: The system is turned into a compressible diffusive metal state by an arbitrarily weak random scalar potential or $z$ component of random vector potential but exhibits stable critical behavior when there is only $x$ or $y$ component of random vector potential. In case the triple-Weyl fermions couple to random scalar potential, the system becomes a diffusive metal in the weak interaction regime but remains a semimetal if Coulomb interaction is sufficiently strong. Interplay of Coulomb interaction and $x$, or $y$, component of random vector potential leads to a stable infrared fixed point that is likely to be characterized by critical behavior. When Coulomb interaction coexists with the $z$ component of random vector potential, the system flows to the interaction-dominated strong coupling regime, which might drive a Mott insulating transition. It is thus clear that double- and triple-Weyl fermions exhibit distinct low-energy behavior in response to interaction and disorder. The physical explanation of such distinction is discussed in detail. The role played by long-range Coulomb impurity in triple-Weyl semimetal is also considered.Comment: 22 pages, 17 figure

Unconventional non-Fermi liquid state caused by nematic criticality in cuprates

At the nematic quantum critical point that exists in the $d_{x^2-y^2}$-wave superconducting dome of cuprates, the massless nodal fermions interact strongly with the quantum critical fluctuation of nematic order. We study this problem by means of renormalization group approach and show that, the fermion damping rate $\left|\mathrm{Im}\Sigma^R(\omega)\right|$ vanishes more rapidly than the energy $\omega$ and the quasiparticle residue $Z_f\rightarrow 0$ in the limit $\omega \rightarrow 0$. The nodal fermions thus constitute an unconventional non-Fermi liquid that represents an even weaker violation of Fermi liquid theory than a marginal Fermi liquid. We also investigate the interplay of quantum nematic critical fluctuation and gauge-potential-like disorder, and find that the effective disorder strength flows to the strong coupling regime at low energies. Therefore, even an arbitrarily weak disorder can drive the system to become a disorder controlled diffusive state. Based on these theoretical results, we are able to understand a number of interesting experimental facts observed in curpate superconductors.Comment: 29 pages, 6 figure

Fast Rates of ERM and Stochastic Approximation: Adaptive to Error Bound Conditions

Error bound conditions (EBC) are properties that characterize the growth of an objective function when a point is moved away from the optimal set. They have recently received increasing attention in the field of optimization for developing optimization algorithms with fast convergence. However, the studies of EBC in statistical learning are hitherto still limited. The main contributions of this paper are two-fold. First, we develop fast and intermediate rates of empirical risk minimization (ERM) under EBC for risk minimization with Lipschitz continuous, and smooth convex random functions. Second, we establish fast and intermediate rates of an efficient stochastic approximation (SA) algorithm for risk minimization with Lipschitz continuous random functions, which requires only one pass of $n$ samples and adapts to EBC. For both approaches, the convergence rates span a full spectrum between $\widetilde O(1/\sqrt{n})$ and $\widetilde O(1/n)$ depending on the power constant in EBC, and could be even faster than $O(1/n)$ in special cases for ERM. Moreover, these convergence rates are automatically adaptive without using any knowledge of EBC. Overall, this work not only strengthens the understanding of ERM for statistical learning but also brings new fast stochastic algorithms for solving a broad range of statistical learning problems

On the Local Minima of the Empirical Risk

Population risk is always of primary interest in machine learning; however, learning algorithms only have access to the empirical risk. Even for applications with nonconvex nonsmooth losses (such as modern deep networks), the population risk is generally significantly more well-behaved from an optimization point of view than the empirical risk. In particular, sampling can create many spurious local minima. We consider a general framework which aims to optimize a smooth nonconvex function $F$ (population risk) given only access to an approximation $f$ (empirical risk) that is pointwise close to $F$ (i.e., $\|F-f\|_{\infty} \le \nu$). Our objective is to find the $\epsilon$-approximate local minima of the underlying function $F$ while avoiding the shallow local minima---arising because of the tolerance $\nu$---which exist only in $f$. We propose a simple algorithm based on stochastic gradient descent (SGD) on a smoothed version of $f$ that is guaranteed to achieve our goal as long as $\nu \le O(\epsilon^{1.5}/d)$. We also provide an almost matching lower bound showing that our algorithm achieves optimal error tolerance $\nu$ among all algorithms making a polynomial number of queries of $f$. As a concrete example, we show that our results can be directly used to give sample complexities for learning a ReLU unit.Comment: To appear in NIPS 201

Chiral symmetry-breaking corrections to strong decays of D*s0(2317) and D's1(2460) in HH\c{hi}PT

The strong decays of two narrow mesons $D_{s0}^{*}(2317)$ and $D_{s1}^{'}(2460)$ are studied within the framework of heavy hadron chiral perturbation theory. Up to next-to-leading order in $1/\Lambda_{\chi}$, by a fit to the experimental widths of their nonstrange partners, the chiral symmetry-breaking coupling constants are extracted. The single-pion decay widths are estimated to be $\Gamma(D_{s0}^{*}(2317)\to D_{s}^{+}\pi^{0})=9.2\pm2.3$ KeV and $\Gamma(D_{s1}^{'}(2460)\to D_{s}^{*+}\pi^{0})=9.0\pm2.1$ KeV, respectively, which are consistent with the experimental constraints and comparable with other theoretical predictions. The numerical analysis shows that chiral-symmetry corrections to the decay widths are significant. Applications and predictions for the corresponding beauty mesons are also provided.Comment: 16 pages, 3figures, LaTe