We show that matrix completion with trace-norm regularization can be significantly hurt when entries of the matrix are sampled non-uniformly, but that a properly weighted version of the trace-norm regularizer works well with non-uniform sampling. We show that the weighted trace-norm regularization indeed yields significant gains on the highly non-uniformly sampled Netflix dataset.

in the present paper, we study some properties of fuzzy norm of linear operators. at first the bounded inverse theorem on fuzzy normed linear spaces is investigated. then, we prove hahn banach theorem, uniform boundedness theorem and closed graph theorem on fuzzy normed linear spaces. finally the set of all compact operators on these spaces is studied.

in this paper, a parameter uniform numerical method based on shishkin mesh is suggested to solve a system of second order singularly perturbed differential equations with a turning point exhibiting boundary layers. it is assumed that both equations have a turning point at the same point. an appropriate piecewise uniform mesh is considered and a classical finite difference scheme is applied on t...

With the previously proposed non-uniform norm called lN -norm, which consists of a sequence of l1-norm or l0-norm elements according to relative magnitude, a novel lN-norm sparse recovery algorithm can be derived by projecting the gradient descent solution to the reconstruction feasible set. In order to gain analytical insights into the performance of this algorithm, in this letter we analyze t...

This paper studies the matrix completion problem under arbitrary sampling schemes. We propose a new estimator incorporating both max-norm and nuclear-norm regularization, based on which we can conduct efficient low-rank matrix recovery using a random subset of entries observed with additive noise under general non-uniform and unknown sampling distributions. This method significantly relaxes the...

For gasless combustion in a one-dimensional solid, we show a type of nonlinear stability of the physical combustion front: if a perturbation of the front is small in both a spatially uniform norm and an exponentially weighted norm, then the perturbation stays small in the spatially uniform norm and decays in the exponentially weighted norm, provided the linearized operator has no eigenvalues in...

In this paper we present results on asymptotic characteristics of multivariate function classes in the uniform norm. Our main interest is approximation functions with mixed smoothness parameter not larger than 1/2. focus will be behavior best m-term trigonometric as well decay Kolmogorov and entropy numbers It turns out that these quantities share a few fundamental abstract properties like thei...