In this paper, we fully characterize the duality mapping over space of matrices that are equipped with Schatten norms. Our approach is based on analysis saturation Hölder inequality for We prove in our main result that, p∈(1,∞), real-valued Schatten-p norm a continuous and single-valued function provide an explicit form its computation. For special case p = 1, set-valued; by adding rank constra...

Suppose that we observe entries or, more generally, linear combinations of entries of an unknown m×T -matrix A corrupted by noise. We are particularly interested in the high-dimensional setting where the numbermT of unknown entries can be much larger than the sample size N . Motivated by several applications, we consider estimation of matrix A under the assumption that it has small rank. This c...

Singular values of a data in a matrix form provide insights on the structure of the data, the effective dimensionality, and the choice of hyper-parameters on higher-level data analysis tools. However, in many practical applications such as collaborative filtering and network analysis, we only get a partial observation. Under such scenarios, we consider the fundamental problem of recovering spec...

The Schatten quasi-norm was introduced tobridge the gap between the trace norm andrank function. However, existing algorithmsare too slow or even impractical for large-scale problems. Motivated by the equivalencerelation between the trace norm and its bilin-ear spectral penalty, we define two tractableSchatten norms, i.e. the bi-trace and tri-tracenorms, and prov...

The goal of this review is to explain some recent results [5] regarding generalizations of the Stein-Tomas (and Strichartz) inequalities to the context of trace ideals (Schatten spaces).

2 Definitions and main results 3 2.1 Notations and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Graph norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Schatten-von Neumann classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Hölder and weakly Hölder graphs . . . . . . . . . . . . ....

A compactness criterion for Stieltjes transformation S λ : L 2 → L 2 of the form (1.1) is obtained. The main result is conditions for belonging S λ to Schatten-von Neumann class S p , 0 < p < ∞.

In this paper we analyse optimality of the stable fixed point of the double bracket equations. We introduce different types of optimality and prove local and global optimality results with respect to the Schatten p-norms.