36 research outputs found
Construction of equiangular signatures for synchronous CDMA systems
Welch bound equality (WBE) signature sequences maximize the uplink sum capacity in direct-spread synchronous code division multiple access (CDMA) systems. WBE sequences have a nice interference invariance property that typically holds only when the system is fully loaded, and, to maintain this property, the signature set must be redesigned and reassigned as the number of active users changes. An additional equiangular constraint on the signature set, however, maintains interference invariance. Finding such signatures requires equiangular side constraints to be imposed on an inverse eigenvalue problem. The paper presents an alternating projection algorithm that can design WBE sequences that satisfy equiangular side constraints. The proposed algorithm can be used to find Grassmannian frames as well as equiangular tight frames. Though one projection is onto a closed, but non-convex, set, it is shown that this algorithm converges to a fixed point, and these fixed points are partially characterized
Non-optimality of the Greedy Algorithm for subspace orderings in the method of alternating projections
The method of alternating projections involves projecting an element of a
Hilbert space cyclically onto a collection of closed subspaces. It is known
that the resulting sequence always converges in norm and that one can obtain
estimates for the rate of convergence in terms of quantities describing the
geometric relationship between the subspaces in question, namely their pairwise
Friedrichs numbers. We consider the question of how best to order a given
collection of subspaces so as to obtain the best estimate on the rate of
convergence. We prove, by relating the ordering problem to a variant of the
famous Travelling Salesman Problem, that correctness of a natural form of the
Greedy Algorithm would imply that , before presenting a
simple example which shows that, contrary to a claim made in the influential
paper [Kayalar-Weinert, Math. Control Signals Systems, vol. 1(1), 1988], the
result of the Greedy Algorithm is not in general optimal. We go on to establish
sharp estimates on the degree to which the result of the Greedy Algorithm can
differ from the optimal result. Underlying all of these results is a
construction which shows that for any matrix whose entries satisfy certain
natural assumptions it is possible to construct a Hilbert space and a
collection of closed subspaces such that the pairwise Friedrichs numbers
between the subspaces are given precisely by the entries of that matrix.Comment: To appear in Results in Mathematic
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
Variable selection in high-dimensional additive models based on norms of projections
We consider the problem of variable selection in high-dimensional sparse
additive models. We focus on the case that the components belong to
nonparametric classes of functions. The proposed method is motivated by
geometric considerations in Hilbert spaces and consists of comparing the norms
of the projections of the data onto various additive subspaces. Under minimal
geometric assumptions, we prove concentration inequalities which lead to new
conditions under which consistent variable selection is possible. As an
application, we establish conditions under which a single component can be
estimated with the rate of convergence corresponding to the situation in which
the other components are known.Comment: 27 page
Functions with Prescribed Best Linear Approximations
A common problem in applied mathematics is to find a function in a Hilbert
space with prescribed best approximations from a finite number of closed vector
subspaces. In the present paper we study the question of the existence of
solutions to such problems. A finite family of subspaces is said to satisfy the
\emph{Inverse Best Approximation Property (IBAP)} if there exists a point that
admits any selection of points from these subspaces as best approximations. We
provide various characterizations of the IBAP in terms of the geometry of the
subspaces. Connections between the IBAP and the linear convergence rate of the
periodic projection algorithm for solving the underlying affine feasibility
problem are also established. The results are applied to problems in harmonic
analysis, integral equations, signal theory, and wavelet frames