75,413 research outputs found
Approximate Nonnegative Matrix Factorization via Alternating Minimization
In this paper we consider the Nonnegative Matrix Factorization (NMF) problem:
given an (elementwise) nonnegative matrix find, for
assigned , nonnegative matrices and
such that . Exact, non trivial, nonnegative
factorizations do not always exist, hence it is interesting to pose the
approximate NMF problem. The criterion which is commonly employed is
I-divergence between nonnegative matrices. The problem becomes that of finding,
for assigned , the factorization closest to in I-divergence. An
iterative algorithm, EM like, for the construction of the best pair
has been proposed in the literature. In this paper we interpret the algorithm
as an alternating minimization procedure \`a la Csisz\'ar-Tusn\'ady and
investigate some of its stability properties. NMF is widespreading as a data
analysis method in applications for which the positivity constraint is
relevant. There are other data analysis methods which impose some form of
nonnegativity: we discuss here the connections between NMF and Archetypal
Analysis. An interesting system theoretic application of NMF is to the problem
of approximate realization of Hidden Markov Models
Joint Source and Relay Precoding Designs for MIMO Two-Way Relaying Based on MSE Criterion
Properly designed precoders can significantly improve the spectral efficiency
of multiple-input multiple-output (MIMO) relay systems. In this paper, we
investigate joint source and relay precoding design based on the
mean-square-error (MSE) criterion in MIMO two-way relay systems, where two
multi-antenna source nodes exchange information via a multi-antenna
amplify-and-forward relay node. This problem is non-convex and its optimal
solution remains unsolved. Aiming to find an efficient way to solve the
problem, we first decouple the primal problem into three tractable
sub-problems, and then propose an iterative precoding design algorithm based on
alternating optimization. The solution to each sub-problem is optimal and
unique, thus the convergence of the iterative algorithm is guaranteed.
Secondly, we propose a structured precoding design to lower the computational
complexity. The proposed precoding structure is able to parallelize the
channels in the multiple access (MAC) phase and broadcast (BC) phase. It thus
reduces the precoding design to a simple power allocation problem. Lastly, for
the special case where only a single data stream is transmitted from each
source node, we present a source-antenna-selection (SAS) based precoding design
algorithm. This algorithm selects only one antenna for transmission from each
source and thus requires lower signalling overhead. Comprehensive simulation is
conducted to evaluate the effectiveness of all the proposed precoding designs.Comment: 32 pages, 10 figure
A fast algorithm for matrix balancing
As long as a square nonnegative matrix A contains sufficient nonzero elements, then the matrix can be balanced, that is we can find a diagonal scaling of A that is doubly stochastic. A number of algorithms have been proposed to achieve the balancing, the most well known of these being Sinkhorn-Knopp. In this paper we derive new algorithms based on inner-outer iteration schemes. We show that Sinkhorn-Knopp belongs to this family, but other members can converge much more quickly. In particular, we show that while stationary iterative methods offer little or no improvement in many cases, a scheme using a preconditioned conjugate gradient method as the inner iteration can give quadratic convergence at low cost
Error estimators and their analysis for CG, Bi-CG and GMRES
We present an analysis of the uncertainty in the convergence of iterative
linear solvers when using relative residue as a stopping criterion, and the
resulting over/under computation for a given tolerance in error. This shows
that error estimation is indispensable for efficient and accurate solution of
moderate to high conditioned linear systems (), where is
the condition number of the matrix. An error estimator for
iterations of the CG (Conjugate Gradient) algorithm was proposed more than two
decades ago. Recently, an error estimator was described for
the GMRES (Generalized Minimal Residual) algorithm which allows for
non-symmetric linear systems as well, where is the iteration number. We
suggest a minor modification in this GMRES error estimation for increased
stability. In this work, we also propose an error estimator
for A-norm and norm of the error vector in Bi-CG (Bi-Conjugate
Gradient) algorithm. The robust performance of these estimates as a stopping
criterion results in increased savings and accuracy in computation, as
condition number and size of problems increase
- …