1,438 research outputs found
Nullspaces and frames
In this paper we give new characterizations of Riesz and conditional Riesz
frames in terms of the properties of the nullspace of their synthesis
operators. On the other hand, we also study the oblique dual frames whose
coefficients in the reconstruction formula minimize different weighted norms.Comment: 16 page
The Paulsen Problem, Continuous Operator Scaling, and Smoothed Analysis
The Paulsen problem is a basic open problem in operator theory: Given vectors
that are -nearly satisfying the
Parseval's condition and the equal norm condition, is it close to a set of
vectors that exactly satisfy the Parseval's
condition and the equal norm condition? Given , the squared
distance (to the set of exact solutions) is defined as where the infimum is over the set of exact solutions.
Previous results show that the squared distance of any -nearly
solution is at most and there are
-nearly solutions with squared distance at least .
The fundamental open question is whether the squared distance can be
independent of the number of vectors .
We answer this question affirmatively by proving that the squared distance of
any -nearly solution is . Our approach is based
on a continuous version of the operator scaling algorithm and consists of two
parts. First, we define a dynamical system based on operator scaling and use it
to prove that the squared distance of any -nearly solution is . Then, we show that by randomly perturbing the input vectors, the
dynamical system will converge faster and the squared distance of an
-nearly solution is when is large enough
and is small enough. To analyze the convergence of the dynamical
system, we develop some new techniques in lower bounding the operator capacity,
a concept introduced by Gurvits to analyze the operator scaling algorithm.Comment: Added Subsection 1.4; Incorporated comments and fixed typos; Minor
changes in various place
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