On the growth of von Neumann dimension of harmonic spaces of semipositive line bundles over covering manifolds

We study the harmonic space of line bundle valued forms over a covering manifold with a discrete group action $\Gamma$, and obtain an asymptotic estimate for the $\Gamma$-dimension of the harmonic space with respect to the tensor times $k$ in the holomorphic line bundle $L^{k}\otimes E$ and the type $(n,q)$ of the differential form, when $L$ is semipositive. In particular, we estimate the $\Gamma$-dimension of the corresponding reduced $L^2$-Dolbeault cohomology group. Essentially, we obtain a local estimate of the pointwise norm of harmonic forms with valued in semipositive line bundles over Hermitian manifolds

SOLUTIONS OF FRATIONAL EMDEN-FOWLER EQUATIONS BY HOMOTOPY ANALYSIS METHOD

In this paper, we have solved the singular initial value problems of fractional Emden-Fowler type equations by using the homotopy analysis method. The approximate analytical solution of this type equations are obtained

Stability of matrix factorization for collaborative filtering

We study the stability vis a vis adversarial noise of matrix factorization algorithm for matrix completion. In particular, our results include: (I) we bound the gap between the solution matrix of the factorization method and the ground truth in terms of root mean square error; (II) we treat the matrix factorization as a subspace fitting problem and analyze the difference between the solution subspace and the ground truth; (III) we analyze the prediction error of individual users based on the subspace stability. We apply these results to the problem of collaborative filtering under manipulator attack, which leads to useful insights and guidelines for collaborative filtering system design.Comment: ICML201

Constrained spline regression and hypothesis tests in the presence of correlation

2013 Fall.Includes bibliographical references.Extracting the trend from the pattern of observations is always difficult, especially when the trend is obscured by correlated errors. Often, prior knowledge of the trend does not include a parametric family, and instead the valid assumption are vague, such as "smooth" or "monotone increasing," Incorrectly specifying the trend as some simple parametric form can lead to overestimation of the correlation, and conversely, misspecifying or ignoring the correlation leads to erroneous inference for the trend. In this dissertation, we explore spline regression with shape constraints, such as monotonicity or convexity, for estimation and inference in the presence of stationary AR(p) errors. Standard criteria for selection of penalty parameter, such as Akaike information criterion (AIC), cross-validation and generalized cross-validation, have been shown to behave badly when the errors are correlated and in the absence of shape constraints. In this dissertation, correlation structure and penalty parameter are selected simultaneously using a correlation-adjusted AIC. The asymptotic properties of unpenalized spline regression in the presence of correlation are investigated. It is proved that even if the estimation of the correlation is inconsistent, the corresponding projection estimation of the regression function can still be consistent and have the optimal asymptotic rate, under appropriate conditions. The constrained spline fit attains the convergence rate of unconstrained spline fit in the presence of AR(p) errors. Simulation results show that the constrained estimator typically behaves better than the unconstrained version if the true trend satisfies the constraints. Traditional statistical tests for the significance of a trend rely on restrictive assumptions on the functional form of the relationship, e.g. linearity. In this dissertation, we develop testing procedures that incorporate shape restrictions on the trend and can account for correlated errors. These tests can be used in checking whether the trend is constant versus monotone, linear versus convex/concave and any combinations such as, constant versus increase and convex. The proposed likelihood ratio test statistics have an exact null distribution if the covariance matrix of errors is known. Theorems are developed for the asymptotic distributions of test statistics if the covariance matrix is unknown but the test statistics use a consistent estimator of correlation into their estimation. The comparisons of the proposed test with the F-test with the unconstrained alternative fit and the one-sided t-test with simple regression alternative fit are conducted through intensive simulations. Both test size and power of the proposed test are favorable, smaller test size and greater power in general, comparing to the F-test and t-test