Fast Global Convergence via Landscape of Empirical Loss

While optimizing convex objective (loss) functions has been a powerhouse for machine learning for at least two decades, non-convex loss functions have attracted fast growing interests recently, due to many desirable properties such as superior robustness and classification accuracy, compared with their convex counterparts. The main obstacle for non-convex estimators is that it is in general intractable to find the optimal solution. In this paper, we study the computational issues for some non-convex M-estimators. In particular, we show that the stochastic variance reduction methods converge to the global optimal with linear rate, by exploiting the statistical property of the population loss. En route, we improve the convergence analysis for the batch gradient method in \cite{mei2016landscape}

Critical nuclear charge and shape resonances for the two-electron systems

The hydrogen negative ion H$^-$ is the simplest two-electron system that exists in nature. This system is not only important in astrophysics but it also serves as an ideal ground to study electron-electron correlations. The peculiar balance of the correlations between the two electrons with the interaction of electron-nucleus in H$^-$ makes this system to have only two bound states, one being the ground state $1s^2\,^{1}\!S^e$ and the other the doubly-excited metastable state $2p^2\,^{3}\!P^e$ embedded below the hydrogen $n=2$ threshold. Here we report a calculation for the $2p^2\,^{3}\!P^e$ state of H$^-$ that yields the energy eigenvalue $E=-0.125\,355\,451\,242\,864\,058\,376\,012\,313\,25(2)$, in atomic units. Our result substantially improves the best available result by 16 orders of magnitude. We further study the critical nuclear charge $Z_{\rm cr}$, the minimum value of nuclear charge $Z$ that is required to bind a nucleus and two electrons. Our determination of $Z_{\rm cr}$ for the $2p^2\,^{3}\!P^e$ state of two-electron systems is $Z_{\rm cr}=0.994\,781\,292\,240\,366\,246\,3(1)$, corresponding to $1/Z_{\rm cr}= 1.005\,246\,085\,546\,985\,509\,4(1)$, which improves the best published value of $Z_{\rm cr}$ by about 10 orders of magnitude. We further investigate in a definitive way the unexplored regime of $Z < Z_{\rm cr}$ using the method of complex scaling and establish precise shape resonance poles for the state of $2p^2\,^{3}\!P^e$ in the complex energy plane

Non-convex Conditional Gradient Sliding

We investigate a projection free method, namely conditional gradient sliding on batched, stochastic and finite-sum non-convex problem. CGS is a smart combination of Nesterov's accelerated gradient method and Frank-Wolfe (FW) method, and outperforms FW in the convex setting by saving gradient computations. However, the study of CGS in the non-convex setting is limited. In this paper, we propose the non-convex conditional gradient sliding (NCGS) which surpasses the non-convex Frank-Wolfe method in batched, stochastic and finite-sum setting

The rotational invariants constructed by the products of three spherical harmonic polynomials

The rotational invariants constructed by the products of three spherical harmonic polynomials are expressed generally as homogeneous polynomials with respect to the three coordinate vectors, where the coefficients are calculated explicitly in this paper

Spatial solitons and stability in self-focusing and defocusing Kerr nonlinear media with generalized PT-symmetric Scarff-II potentials

We present a unified theoretical study of the bright solitons governed by self-focusing and defocusing nonlinear Schrodinger (NLS) equations with generalized parity-time (PT)-symmetric Scarff II potentials. Particularly, a PT-symmetric k-wavenumber Scarff II potential and a multi-well Scarff II potential are considered, respectively. For the k-wavenumber Scarff II potential, the parameter space can be divided into different regions, corresponding to unbroken and broken PT-symmetry and the bright solitons for self-focusing and defocusing Kerr nonlinearities. For the multi-well Scarff II potential the bright solitons can be obtained by using a periodically space-modulated Kerr nonlinearity. The linear stability of bright solitons with PT-symmetric k-wavenumber and multi-well Scarff II potentials is analyzed in details using numerical simulations. Stable and unstable bright solitons are found in both regions of unbroken and broken PT-symmetry due to the existence of the nonlinearity. Furthermore, the bright solitons in three-dimensional self-focusing and defocusing NLS equations with a generalized PT-symmetric Scarff II potential are explored. This may have potential applications in the field of optical information transmission and processing based on optical solitons in nonlinear dissipative but PT-symmetric systems.Comment: 11 pages, 13 figure

The Contributions of Neutral Higgs Bosons to Charmless Nonleptonic B Decays in MSSM

We investigate the contributions of neutral Higgs bosons to nonleptonic transition $b \to q \bar{s} s, q=d, s$ under the supersymmetric context. Their effects to decay width and CP violation in corresponding exclusive decays are explored. The anomalous dimension matrices of the operators which have to be incorporated to include the contributions of neutral Higgs bosons are given. We find that when tan$\beta$ is large (say, 50) and neutral Higgs bosons are not too heavy (say, 100 GeV), contributions of neutral Higgs penguin can dominate electroweak penguin contributions, and for some processes, they can greatly modify both decay width and CP asymmetry.Comment: 11 pages, typo corrected, references added, minor revisions mad

Model Hamiltonian and Time Reversal Breaking Topological Phases of Anti-ferromagnetic Half-Heusler Materials

In this work, we construct a generalized Kane model with a new coupling term between itinerant electron spins and local magnetic moments of anti-ferromagnetic ordering in order to describe the low energy effective physics in a large family of anti-ferromagnetic half-Heusler materials. Topological properties of this generalized Kane model is studied and a large variety of topological phases, including Dirac semimetal phase, Weyl semimetal phase, nodal line semimetal phase, type-B triple point semimetal phase, topological mirror (or glide) insulating phase and anti-ferromagnetic topological insulating phase, are identified in different parameter regions of our effective models. In particular, we find that the system is always driven into the anti-ferromagnetic topological insulator phase once a bulk band gap is open, irrespective of the magnetic moment direction, thus providing a robust realization of anti-ferromagentic topological insulators. Furthermore, we discuss the possible realization of these topological phases in realistic anti-ferromagnetic half-Heusler materials. Our effective model provides a basis for the future study of physical phenomena in this class of materials.Comment: 16 pages, 10 figure

On characterization of Poisson integrals of Schrodinger operators with BMO traces

Let L be a Schrodinger operator of the form L=-\Delta+V acting on L^2(Rn) where the nonnegative potential V belongs to the reverse Holder class Bq for some q>= n. Let BMO_L(Rn) denote the BMO space on Rn associated to the Schrodinger operator L. In this article we will show that a function f in BMO_L(Rn) is the trace of the solution of L'u=-u_tt+Lu=0, u(x,0)= f(x), where u satisfies a Carleson condition. Conversely, this Carleson condition characterizes all the L-harmonic functions whose traces belong to the space BMO_L(Rn). This result extends the analogous characterization founded by Fabes, Johnson and Neri for the classical BMO space of John and Nirenberg.Comment: 25 pages, to appear in Journal of Functional Analysi

Properties of Interstellar Medium In Infrared Bright QSOs Probed by [O I]63 micron and [C II]158 micron Emission Lines

We present a study of interstellar medium in the host galaxies of 9 QSOs at 0.1<z<0.2 with blackhole masses of $3\times10^7\,M_\odot$ to $3\times10^9\,M_\odot$ based on the far-IR spectroscopy taken with {\it Herschel Space Observatory}. We detect the [OI]63$\mu$m ([CII]158$\mu$m) emission in 6(8) out of 8(9) sources. Our QSO sample has far-infrared luminosities (LFIR)~several times $10^{11}L_\odot$. The observed line-to-LFIR ratios (LOI/LFIR and LCII/LFIR) are in the ranges of $2.6\times10^{-4}$-$10^{-2}$ and $2.8\times10^{-4}$-$2\times10^{-3}$ respectively (including upper limits). These ratios are comparable to the values found in local ULIRGs, but higher than the average value published so far for $z$$>$1 IR bright QSOs. One target, W0752+19, shows an additional broad velocity component (~720 km/s), and exceptionally strong [OI]63$\mu$m emission with LOI/LFIR of $10^{-2}$, an order of magnitude higher than that of average value found among local (U)LIRGs. Combining with the analyses of the {\it SDSS} optical spectra, we conclude that the [OI]63$\mu$m emission in these QSOs is unlikely excited by shocks. We infer that the broad [OI]63 micron emission in W0752+19 could arise from the warm and dense ISM in the narrow line region of the central AGN. Another possible explanation is the existence of a dense gas outflow with $n_{\rm H}\sim10^4$\,cm$^{-3}$, where the corresponding broad [CII] emission is suppressed. Based on the far-IR [OI] and [CII] line ratios, we estimate the constraints on the ISM density and UV radiation field intensity of $n_{\rm H} \lesssim 10^{3.3}$ cm$^{-3}$ and $10^3<G_0 \lesssim 10^{4.2}$, respectively. These values are consistent with those found in local Seyfert 1 ULIRGs. In contrast, the gas with broad velocity width in W0752+19 has $n_{\rm H} \gtrsim 10^{4.3}$ cm$^{-3}$ and $G_0>10^4$.Comment: Accepted for publication in Ap

Orthogonal Voronoi Diagram and Treemap

In this paper, we propose a novel space partitioning strategy for implicit hierarchy visualization such that the new plot not only has a tidy layout similar to the treemap, but also is flexible to data changes similar to the Voronoi treemap. To achieve this, we define a new distance function and neighborhood relationship between sites so that space will be divided by axis-aligned segments. Then a sweepline+skyline based heuristic algorithm is proposed to allocate the partitioned spaces to form an orthogonal Voronoi diagram with orthogonal rectangles. To the best of our knowledge, it is the first time to use a sweepline-based strategy for the Voronoi treemap. Moreover, we design a novel strategy to initialize the diagram status and modify the status update procedure so that the generation of our plot is more effective and efficient. We show that the proposed algorithm has an O(nlog(n)) complexity which is the same as the state-of-the-art Voronoi treemap. To this end, we show via experiments on the artificial dataset and real-world dataset the performance of our algorithm in terms of computation time, converge rate, and aspect ratio. Finally, we discuss the pros and cons of our method and make a conclusion