Today, we will continue with our discussion of scalar and matrix concentration, with a discussion of the matrix analogues of Markov’s, Chebychev’s, and Chernoff’s Inequalities. Then, we will return to bounding the error for our approximating matrix multiplication algorithm. We will start with using Hoeffding-Azuma bounds from last class to get improved Frobenius norm bounds, and then (next time...

and Hans Schneider· Department of Mathematics University of Wisconsin Madison, Wisconsin 53706 An elementary proof is given that a bounded multiplicative group of complex (real) n X n nonsingular matrices is similar to a unitary (orthogonal) group. Given a norm on a complex n-space, it follows that there exists a nonsingular n X n matrix L (the lAewner-John matrix for the norm) such that LHL -1...

In this paper, we conduct a thorough study on the first and second order properties of the Moreau-Yosida regularization of the vector k-norm function, the indicator function of its epigraph, and the indicator function of the vector k-norm ball. We start with settling the vector k-norm case via applying the existing breakpoint searching algorithms to the metric projector over its dual norm ball....

There has been much recent progress on the use of the nuclear norm for the so-called matrix completion problem (the problem of imputing missing values of a matrix). In this paper we investigate the use of the nuclear norm for modelling time series, with particular attention to imputing missing data and forecasting. We introduce a simple alternating projections type algorithm based on the nuclea...

The bilinear inequality is derived from the linear one with the help of an operatorvalued version of the Cauchy-Schwarz inequality. All these results, at least in their finite form, are obtained by simple and elegant methods well within the scope of a basic course on Hilbert spaces. (They can alternatively be obtained by tensor product techniques, but in the author’s view, these methods are les...

Fortran 77 codes for estimating the 1-norm of a real or complex matrix were presented by Higham in [ACM Trans. Math. Software, 14 (1988), pp. 381-396]. The codes have found use in various applications and have been adopted by two program libraries. Further observations about the norm estimation algorithm and experience in using it are reported here. In particular, an example is given where the ...

We present an iterative procedure which asymptotically scales the infinity norm of both rows and columns in a matrix to 1. This scaling strategy exhibits some optimality properties and additionally preserves symmetry. The algorithm also shows fast linear convergence with an asymptotic rate of 1=2. We discuss possible extensions of such an algorithm when considering the one-norm or the two norm ...

We give a systematic development of the application of matrix norms to rapid mixing in spin systems. We show that rapid mixing of both random update Glauber dynamics and systematic scan Glauber dynamics occurs if any matrix norm of the associated dependency matrix is less than 1. We give improved analysis for the case in which the diagonal of the dependency matrix is 0 (as in heat bath dynamics...

It is the main goal of this paper to propose a novel method to perform matrix completion on-line. Motivated by a wide variety of applications, ranging from the design of recommender systems to sensor network localization through seismic data reconstruction, we consider the matrix completion problem when entries of the matrix of interest are observed gradually. Precisely, we place ourselves in t...