4 research outputs found
Coherence Optimization and Best Complex Antipodal Spherical Codes
Vector sets with optimal coherence according to the Welch bound cannot exist
for all pairs of dimension and cardinality. If such an optimal vector set
exists, it is an equiangular tight frame and represents the solution to a
Grassmannian line packing problem. Best Complex Antipodal Spherical Codes
(BCASCs) are the best vector sets with respect to the coherence. By extending
methods used to find best spherical codes in the real-valued Euclidean space,
the proposed approach aims to find BCASCs, and thereby, a complex-valued vector
set with minimal coherence. There are many applications demanding vector sets
with low coherence. Examples are not limited to several techniques in wireless
communication or to the field of compressed sensing. Within this contribution,
existing analytical and numerical approaches for coherence optimization of
complex-valued vector spaces are summarized and compared to the proposed
approach. The numerically obtained coherence values improve previously reported
results. The drawback of increased computational effort is addressed and a
faster approximation is proposed which may be an alternative for time critical
cases
On block coherence of frames
Block coherence of matrices plays an important role in analyzing the
performance of block compressed sensing recovery algorithms (Bajwa and Mixon,
2012). In this paper, we characterize two block coherence metrics: worst-case
and average block coherence. First, we present lower bounds on worst-case block
coherence, in both the general case and also when the matrix is constrained to
be a union of orthobases. We then present deterministic matrix constructions
based upon Kronecker products which obtain these lower bounds. We also
characterize the worst-case block coherence of random subspaces. Finally, we
present a flipping algorithm that can improve the average block coherence of a
matrix, while maintaining the worst-case block coherence of the original
matrix. We provide numerical examples which demonstrate that our proposed
deterministic matrix construction performs well in block compressed sensing
Density of Spherically-Embedded Stiefel and Grassmann Codes
The density of a code is the fraction of the coding space covered by packing
balls centered around the codewords. This paper investigates the density of
codes in the complex Stiefel and Grassmann manifolds equipped with the chordal
distance. The choice of distance enables the treatment of the manifolds as
subspaces of Euclidean hyperspheres. In this geometry, the densest packings are
not necessarily equivalent to maximum-minimum-distance codes. Computing a
code's density follows from computing: i) the normalized volume of a metric
ball and ii) the kissing radius, the radius of the largest balls one can pack
around the codewords without overlapping. First, the normalized volume of a
metric ball is evaluated by asymptotic approximations. The volume of a small
ball can be well-approximated by the volume of a locally-equivalent tangential
ball. In order to properly normalize this approximation, the precise volumes of
the manifolds induced by their spherical embedding are computed. For larger
balls, a hyperspherical cap approximation is used, which is justified by a
volume comparison theorem showing that the normalized volume of a ball in the
Stiefel or Grassmann manifold is asymptotically equal to the normalized volume
of a ball in its embedding sphere as the dimension grows to infinity. Then,
bounds on the kissing radius are derived alongside corresponding bounds on the
density. Unlike spherical codes or codes in flat spaces, the kissing radius of
Grassmann or Stiefel codes cannot be exactly determined from its minimum
distance. It is nonetheless possible to derive bounds on density as functions
of the minimum distance. Stiefel and Grassmann codes have larger density than
their image spherical codes when dimensions tend to infinity. Finally, the
bounds on density lead to refinements of the standard Hamming bounds for
Stiefel and Grassmann codes.Comment: Two-column version (24 pages, 6 figures, 4 tables). To appear in IEEE
Transactions on Information Theor