Anisotropic dark energy model with a hybrid scale factor

Anisotropic dark energy model with dynamic pressure anisotropies along different spatial directions is constructed at the backdrop of a spatially homogeneous diagonal Bianchi type $V$ $(BV)$ space-time in the framework of General Relativity. A time varying deceleration parameter generating a hybrid scale factor is considered to simulate a cosmic transition from early deceleration to late time acceleration. We found that the pressure anisotropies along the $y-$ and $z-$ axes evolve dynamically and continue along with the cosmic expansion without being subsided even at late times. The anisotropic pressure along the $x-$axis becomes equal to the mean fluid pressure. At a late phase of cosmic evolution, the model enters into a phantom region. From a state finder diagnosis, it is found that the model overlaps with $Lambda$CDM at late phase of cosmic time.

Frequency spectrum of axisymmetric horizontal oscillations in accretion disks

We present the spectrum of eigenfrequencies of axisymmetric acoustic-inertial oscillations of thin accretion disks for a Schwarzschild black hole modeled with a pseudo-potential. There are nine discrete frequencies, corresponding to trapped modes. Eigenmodes with nine or more radial nodes in the inner disk belong to the continuum, whose frequency range starts somewhat below the maximum value of the radial epicyclic frequency. The results are derived under the assumption that the oscillatory motion is parallel to the midplane of the disk.

Geometrically thin accretion disk around Maclaurin spheroid

We investigated a semi-analytic and numerical model to study the geometrically thin and optically thick accretion disk around Maclaurin spheroid (MS). The main interest is in the inner region of the so called {alpha}-disk, {alpha} being the viscosity parameter. Analytical calculations are done assuming radiation pressure and gas pressure dominated for close to Eddington mass accretion rate and $dot{M}lesssim 0.1dot{M_{Edd}}$ respectively. We found that the change in eccentricity of MS gives a change at high frequency region in the emitted spectra. We found that disk parameters are dependent on eccentricity of MS. Our semi-analytic results show that qualitatively an increase in eccentricity of MS has same behavior as decrease in mass accretion rate. Numerical work has been carried out to see the viscous time evolution of the accretion disk around MS. In numerical model we showed that if the eccentricity of the object is high the matter will diffuse slowly during its viscous evolution. This gives a clue that how spin-up or spin-down can change the time evolution of the accretion disk using a simple Newtonian approach. The change in spectra can be used to determine the eccentricity of MS and thus period of the MS.

Relativistic effects on coronal ejection in variable X-ray sources

Optically thin coronae around neutron stars suffering an X-ray burst can be ejected as a result of rapid increase in stellar luminosity. In general relativity (GR), radiation pressure from the central luminous star counteracts gravitational attraction more strongly than in Newtonian physics. However, motion near the neutron star is very effectively impeded by the radiation field. We discuss coronal ejection in a general relativistic calculation of the motion of a test particle in a spherically symmetric radiation field. At every radial distance from the star larger than that of the ISCO, and any initial luminosity of the star, there exists a luminosity change which leads to coronal ejection. The luminosity required to eject from the system the inner parts of the optically thin neutron-star corona is very high in the presence of radiation drag and always close to the Eddington luminosity. Outer parts of the corona, at a distance of ~20 $R_G$ or more, will be ejected by a sub-Eddington outburst. Mildly fluctuating luminosity will lead to dissipation in the plasma and may explain the observed X-ray temperatures of coronae in low mass X-ray binaries (LMXBs). At large radial distances from the star ($3cdot 10^3 R_G$ or more) the results do not depend on whether or not Poynting-Robertson drag is included in the calculation.

R3MC: A Riemannian three-factor algorithm for low-rank matrix completion

We exploit the versatile framework of Riemannian optimization on quotient manifolds to develop R3MC, a nonlinear conjugate-gradient method for low-rank matrix completion. The underlying search space of fixed-rank matrices is endowed with a novel Riemannian metric that is tailored to the least-squares cost. Numerical comparisons suggest that R3MC robustly outperforms state-of-the-art algorithms across different problem instances, especially those that combine scarcely sampled and ill-conditioned data.

Low-rank optimization for distance matrix completion

This paper addresses the problem of low-rank distance matrix completion. This problem amounts to recover the missing entries of a distance matrix when the dimension of the data embedding space is possibly unknown but small compared to the number of considered data points. The focus is on high-dimensional problems. We recast the considered problem into an optimization problem over the set of low-rank positive semidefinite matrices and propose two efficient algorithms for low-rank distance matrix completion. In addition, we propose a strategy to determine the dimension of the embedding space. The resulting algorithms scale to high-dimensional problems and monotonically converge to a global solution of the problem. Finally, numerical experiments illustrate the good performance of the proposed algorithms on benchmarks.

A Riemannian geometry for low-rank matrix completion

We propose a new Riemannian geometry for fixed-rank matrices that is specifically tailored to the low-rank matrix completion problem. Exploiting the degree of freedom of a quotient space, we tune the metric on our search space to the particular least square cost function. At one level, it illustrates in a novel way how to exploit the versatile framework of optimization on quotient manifold. At another level, our algorithm can be considered as an improved version of LMaFit, the state-of-the-art Gauss-Seidel algorithm. We develop necessary tools needed to perform both first-order and second-order optimization. In particular, we propose gradient descent schemes (steepest descent and conjugate gradient) and trust-region algorithms. We also show that, thanks to the simplicity of the cost function, it is numerically cheap to perform an exact linesearch given a search direction, which makes our algorithms competitive with the state-of-the-art on standard low-rank matrix completion instances.

Fixed-rank matrix factorizations and Riemannian low-rank optimization

Motivated by the problem of learning a linear regression model whose parameter is a large fixed-rank non-symmetric matrix, we consider the optimization of a smooth cost function defined on the set of fixed-rank matrices. We adopt the geometric framework of optimization on Riemannian quotient manifolds. We study the underlying geometries of several well-known fixed-rank matrix factorizations and then exploit the Riemannian quotient geometry of the search space in the design of a class of gradient descent and trust-region algorithms. The proposed algorithms generalize our previous results on fixed-rank symmetric positive semidefinite matrices, apply to a broad range of applications, scale to high-dimensional problems and confer a geometric basis to recent contributions on the learning of fixed-rank non-symmetric matrices. We make connections with existing algorithms in the context of low-rank matrix completion and discuss relative usefulness of the proposed framework. Numerical experiments suggest that the proposed algorithms compete with the state-of-the-art and that manifold optimization offers an effective and versatile framework for the design of machine learning algorithms that learn a fixed-rank matrix.

Low-rank optimization with trace norm penalty

The paper addresses the problem of low-rank trace norm minimization. We propose an algorithm that alternates between fixed-rank optimization and rank-one updates. The fixed-rank optimization is characterized by an efficient factorization that makes the trace norm differentiable in the search space and the computation of duality gap numerically tractable. The search space is nonlinear but is equipped with a particular Riemannian structure that leads to efficient computations. We present a second-order trust-region algorithm with a guaranteed quadratic rate of convergence. Overall, the proposed optimization scheme converges super-linearly to the global solution while maintaining complexity that is linear in the number of rows and columns of the matrix. To compute a set of solutions efficiently for a grid of regularization parameters we propose a predictor-corrector approach that outperforms the naive warm-restart approach on the fixed-rank quotient manifold. The performance of the proposed algorithm is illustrated on problems of low-rank matrix completion and multivariate linear regression.