On the Stability of Sequential Updates and Downdates

The updating and downdating of QR decompositions has important applications in a number of areas. There is essentially one standard updating algorithm, based on plane rotations, which is backwards stable. Three downdating algorithms have been treated in the literature: the LINPACK algorithm, the method of hyperbolic transformations, and Chambers' algorithm. Although none of these algorithms is backwards stable, the first and third satisfy a relational stability condition. In this paper, it is shown that relational stability extends to a sequence of updates and downdates. In consequence, other things being equal, if the final decomposition in the sequence is well conditioned, it will be accurately computed, even though intermediate decompositions may be almost completely inaccurate. These results are also applied to the two-sided orthogonal decompositions, such as the URV decomposition. (Also cross-referenced as UMIACS-TR-94-30

Square Root Propagation

We propose a message propagation scheme for numerically stable inference in Gaussian graphical models which can otherwise be susceptible to errors caused by finite numerical precision. We adapt square root algorithms, popular in Kalman filtering, to graphs with arbitrary topologies. The method consists of maintaining potentials and generating messages that involve the square root of precision matrices. Combining this with the machinery of the junction tree algorithm leads to an efficient and numerically stable algorithm. Experiments are presented to demonstrate the robustness of the method to numerical errors that can arise in complex learning and inference problems

Sliding window adaptive fast QR and QR-lattice algorithms

Published versio

Cooperative greedy pursuit strategies for sparse signal representation by partitioning

Cooperative Greedy Pursuit Strategies are considered for approximating a signal partition subjected to a global constraint on sparsity. The approach aims at producing a high quality sparse approximation of the whole signal, using highly coherent redundant dictionaries. The cooperation takes place by ranking the partition units for their sequential stepwise approximation, and is realized by means of i)forward steps for the upgrading of an approximation and/or ii) backward steps for the corresponding downgrading. The advantage of the strategy is illustrated by approximation of music signals using redundant trigonometric dictionaries. In addition to rendering stunning improvements in sparsity with respect to the concomitant trigonometric basis, these dictionaries enable a fast implementation of the approach via the Fast Fourier Transfor

Downdating a Rank-Revealing URV Decomposition

Abstract. The rank-revealing URV decomposition is a useful tool for the subspace tracking problem in digital signal processing. Updating the decomposition is a stable process. However, downdating a rank-revealing URV decomposition could be unstable because the R factor is ill-conditioned. In this paper, we review some existing downdating algorithms for the full-rank URV decomposition in the absence of U and develop a new combined algorithm. We also show that the combined algorithm has relational stability. For the rank-revealing URV decomposition, we review a two-step method that applies full-rank downdating algorithms to the signal and noise parts separately. We compare several combinations of the full-rank algorithms and demonstrate good performance of our combined algorithm

On the Stability of Sequential Updates and Downdates

The updating and downdating of Cholesky decompositions has important applications in a number of areas. There is essentially one standard updating algorithm, based on plane rotations, which is backwards stable. Three downdating algorithms have been treated in the literature: the LINPACK algorithm, the method of hyperbolic transformations, and Chambers' algorithm. Although none of these algorithms is backwards stable, the first and third satisfy a relational stability condition. In this paper, it is shown that relational stability extends to a sequence of updates and downdates. In consequence, other things being equal, if the final decomposition in the sequence is well conditioned, it will be accurately computed, even though intermediate decompositions may be almost completely inaccurate. These results are also applied to the two-sided orthogonal decompositions, such as the URV decomposition. This report is available by anonymous ftp from thales.cs.umd.edu in the directory pub/report..