128,623 research outputs found
Norms and numerical radii inequalities for ( ) - normal transaloid operators
The studies on Hilbert spaces for the last decade has been of great interest to many mathematicians and researchers, especially on operator inequalities related to operator norms and numerical radii for a family of bounded linear operators acting on a Hilbert spaces. Results on some inequalities for normal operators in Hilbert spaces for instance numerical ranges W(T), numerical radii w(T) and norm ||.|| obtained by Dragomir and Moslehian among others due to some classical inequalities for vectors in Hilbert spaces. The techniques employed to prove the results are elementary with some special vector inequalities in inner product spaces due to Buzano, Goldstein, Ryff and Clarke as well as some reverse Schwarz inequalities. Recently, the new field of operator theory done by Dragomir and Moslehian on norms and numerical radii for ( ) - normal operators developed basic concepts for our Statement of the problem on normal transaloid operators. M. Fujii and R. Nakamoto characterize transaloid operators in terms of spectral sets and dilations and other non-normal operators such as normaloid, convexoid and spectroid. Furuta did also characterization of normaloid operators. Since none has done on norms and numerical radii inequalities for ( ) – normal transaloid operators, then our aim is to characterize ( )- normal transaloid operators, characterize norm inequalities for ( )- normal transaloid operators and to characterize numerical radii for ( )- normal transaloid operators.  We use the approach of the Cauchy-Schwarz inequalities, parallelogram law, triangle inequality and tensor products. The results obtained are useful in applications in quantum mechanics
Designing structured tight frames via an alternating projection method
Tight frames, also known as general Welch-bound- equality sequences, generalize orthonormal systems. Numerous applications - including communications, coding, and sparse approximation- require finite-dimensional tight frames that possess additional structural properties. This paper proposes an alternating projection method that is versatile enough to solve a huge class of inverse eigenvalue problems (IEPs), which includes the frame design problem. To apply this method, one needs only to solve a matrix nearness problem that arises naturally from the design specifications. Therefore, it is the fast and easy to develop versions of the algorithm that target new design problems. Alternating projection will often succeed even if algebraic constructions are unavailable. To demonstrate that alternating projection is an effective tool for frame design, the paper studies some important structural properties in detail. First, it addresses the most basic design problem: constructing tight frames with prescribed vector norms. Then, it discusses equiangular tight frames, which are natural dictionaries for sparse approximation. Finally, it examines tight frames whose individual vectors have low peak-to-average-power ratio (PAR), which is a valuable property for code-division multiple-access (CDMA) applications. Numerical experiments show that the proposed algorithm succeeds in each of these three cases. The appendices investigate the convergence properties of the algorithm
Approximate Near Neighbors for General Symmetric Norms
We show that every symmetric normed space admits an efficient nearest
neighbor search data structure with doubly-logarithmic approximation.
Specifically, for every , , and every -dimensional
symmetric norm , there exists a data structure for
-approximate nearest neighbor search over
for -point datasets achieving query time and
space. The main technical ingredient of the algorithm is a
low-distortion embedding of a symmetric norm into a low-dimensional iterated
product of top- norms.
We also show that our techniques cannot be extended to general norms.Comment: 27 pages, 1 figur
Simultaneously Structured Models with Application to Sparse and Low-rank Matrices
The topic of recovery of a structured model given a small number of linear
observations has been well-studied in recent years. Examples include recovering
sparse or group-sparse vectors, low-rank matrices, and the sum of sparse and
low-rank matrices, among others. In various applications in signal processing
and machine learning, the model of interest is known to be structured in
several ways at the same time, for example, a matrix that is simultaneously
sparse and low-rank.
Often norms that promote each individual structure are known, and allow for
recovery using an order-wise optimal number of measurements (e.g.,
norm for sparsity, nuclear norm for matrix rank). Hence, it is reasonable to
minimize a combination of such norms. We show that, surprisingly, if we use
multi-objective optimization with these norms, then we can do no better,
order-wise, than an algorithm that exploits only one of the present structures.
This result suggests that to fully exploit the multiple structures, we need an
entirely new convex relaxation, i.e. not one that is a function of the convex
relaxations used for each structure. We then specialize our results to the case
of sparse and low-rank matrices. We show that a nonconvex formulation of the
problem can recover the model from very few measurements, which is on the order
of the degrees of freedom of the matrix, whereas the convex problem obtained
from a combination of the and nuclear norms requires many more
measurements. This proves an order-wise gap between the performance of the
convex and nonconvex recovery problems in this case. Our framework applies to
arbitrary structure-inducing norms as well as to a wide range of measurement
ensembles. This allows us to give performance bounds for problems such as
sparse phase retrieval and low-rank tensor completion.Comment: 38 pages, 9 figure
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