50 research outputs found
Scalable Frames and Convex Geometry
The recently introduced and characterized scalable frames can be considered
as those frames which allow for perfect preconditioning in the sense that the
frame vectors can be rescaled to yield a tight frame. In this paper we define
-scalability, a refinement of scalability based on the number of non-zero
weights used in the rescaling process, and study the connection between this
notion and elements from convex geometry. Finally, we provide results on the
topology of scalable frames. In particular, we prove that the set of scalable
frames with "small" redundancy is nowhere dense in the set of frames.Comment: 14 pages, to appear in Contemporary Mat
A note on scalable frames
We study the problem of determining whether a given frame is scalable, and
when it is, understanding the set of all possible scalings. We show that for
most frames this is a relatively simple task in that the frame is either not
scalable or is scalable in a unique way, and to find this scaling we just have
to solve a linear system. We also provide some insight into the set of all
scalings when there is not a unique scaling. In particular, we show that this
set is a convex polytope whose vertices correspond to minimal scalings
Remarks on scalable frames
This paper investigates scalable frame in . We define the
reduced diagram matrix of a frame and use it to classify scalability of the
frame under some conditions. We give a new approach to the scaling problem by
breaking the problem into two smaller ones, each of which is easily solved,
giving a simple way to check scaling. Finally, we study the scalability of dual
frames
Measures of scalability
Scalable frames are frames with the property that the frame vectors can be
rescaled resulting in tight frames. However, if a frame is not scalable, one
has to aim for an approximate procedure. For this, in this paper we introduce
three novel quantitative measures of the closeness to scalability for frames in
finite dimensional real Euclidean spaces. Besides the natural measure of
scalability given by the distance of a frame to the set of scalable frames,
another measure is obtained by optimizing a quadratic functional, while the
third is given by the volume of the ellipsoid of minimal volume containing the
symmetrized frame. After proving that these measures are equivalent in a
certain sense, we establish bounds on the probability of a randomly selected
frame to be scalable. In the process, we also derive new necessary and
sufficient conditions for a frame to be scalable.Comment: 27 pages, 5 figure
On Optimal Frame Conditioners
A (unit norm) frame is scalable if its vectors can be rescaled so as to
result into a tight frame. Tight frames can be considered optimally conditioned
because the condition number of their frame operators is unity. In this paper
we reformulate the scalability problem as a convex optimization question. In
particular, we present examples of various formulations of the problem along
with numerical results obtained by using our methods on randomly generated
frames.Comment: 11 page
Minimal scalings and structural properties of scalable frames
For a unit-norm frame in , a scaling is a vector such that is a Parseval frame in . If such a scaling exists, is said to be scalable. A scaling is a minimal scaling if has no proper scalable subframe. It is known that the set of all scalings of is a convex polytope with vertices corresponding to minimal scalings. In this talk, we provide a method to find a subset of contact points which provides a decomposition of the identity, and an estimate of the number of minimal scalings of a scalable frame. We provide a characterization of when minimal scalings are affinely dependent. Using this characterization, we can conclude that all strict scalings of have the same structural property. We also present the uniqueness of orthogonal partitioning property of any set of minimal scalings, which provides all possible tight subframes of a given scaled fram