Blind identification of under-determined mixtures (UDM) is involved in numerous applications, including multi-way factor analysis (MWA) and signal processing. In the latter case, the use of high-order statistics (HOS) like cumulants leads to the decomposition of symmetric tensors. Yet, little has been published about rank-revealing decompositions of symmetric tensors. Definitions of rank are discussed, and useful results on generic rank are proved, with the help of tools borrowed from algebraic geometry

In our article, only a set of random positions of missing values was used for each dataset. However, imputation methods may perform differently on the same dataset with different sets of random positions of missing values. In addition, the performance of BPCA of each figure was not correct due to a mistake in implementation. After correcting an error in the implementation and using ten different sets of random positions of missing values, we regenerated all figures. Thus, the results presented here are the fair comparisons between LLSimpute and BPCA. (top) and 5 (bottom). 193p. The NRMSEs of LLSk/L2 and BPCA were 0.6068 and 0.6064, respectively. In the SP.CYCLE dataset that has significant cluster structures, LLSimpute showed competitive performance with BPCA. The NRMSEs of LLSk/L2 and BPCA on the GA.ENV dataset were 0.5145 and 0.5232, respectively. 194p. LLSk/L2 showed the best performance when the number of samples was small since it uses local similarity structures and an optimization process via least squares. For various percentages of missing entries, LLSk/L2 showed competitive performance with BPCA. 195p. Figure 5 shows the NRMSEs with respect to noise levels. The performance sensitivity to noise of LLSk/L2 and LLSk/PC were similar to that of BPCA. 196p. All methods showed robustness against the noise when the SD was less than 0.1.

MOTIVATION: Gene expression data often contain missing expression values. Effective missing value estimation methods are needed since many algorithms for gene expression data analysis require a complete matrix of gene array values. In this paper, imputation methods based on the least squares formulation are proposed to estimate missing values in the gene expression data, which exploit local similarity structures in the data as well as least squares optimization process. RESULTS: The proposed local least squares imputation method (LLSimpute) represents a target gene that has missing values as a linear combination of similar genes. The similar genes are chosen by k-nearest neighbors or k coherent genes that have large absolute values of Pearson correlation coefficients. Non-parametric missing values estimation method of LLSimpute are designed by introducing an automatic k-value estimator. In our experiments, the proposed LLSimpute method shows competitive results when compared with other imputation methods for missing value estimation on various datasets and percentages of missing values in the data. AVAILABILITY: The software is available at http://www.cs.umn.edu/~hskim/tools.html CONTACT: [email protected]

Block coders are among the most common compression tools available for still images and video sequences. Their low computational complexity along with their good performance make them a popular choice for compression of natural images. Yet, at low bit-rates, block coders introduce visually annoying artifacts into the image. One approach that alleviates this problem is to downsample the image, apply the coding algorithm, and interpolate back to the original resolution. In this paper, we consider the use of optimal decimation and interpolation filters in this scheme. We first consider only optimization of the interpolation filter, by formulating the problem as least-squares minimization. We then consider the joint optimization over both the decimation and the interpolation filters, using the variable projection method. The experimental results presented clearly exhibit a significant improvement over other approaches.

We present a novel algorithm for the fast computation of PageRank, a hyperlink-based estimate of the ''importance'' of Web pages. The original PageRank algorithm uses the Power Method to compute successive iterates that converge to the principal eigenvector of the Markov matrix representing the Web link graph. The algorithm presented here, called Quadratic Extrapolation, accelerates the convergence of the Power Method by periodically subtracting off estimates of the nonprincipal eigenvectors from the current iterate of the Power Method. In Quadratic Extrapolation, we take advantage of the fact that the first eigenvalue of a Markov matrix is known to be 1 to compute the nonprincipal eigenvectors using successive iterates of the Power Method. Empirically, we show that using Quadratic Extrapolation speeds up PageRank computation by 25-300% on a Web graph of 80 million nodes, with minimal overhead. Our contribution is useful to the PageRank community and the numerical linear algebra community in general, as it is a fast method for determining the dominant eigenvector of a matrix that is too large for standard fast methods to be practical.

In clinical applications where structural asymmetries between homologous shapes have been correlated with pathology, the questions of definition and quantification of "asymmetry" arise naturally. When not only the degree but the position of deformity is thought relevant, asymmetry localization must also be addressed. Asymmetries between paired shapes have already been formulated in terms of (nonrigid) diffeomorphisms between the shapes. For the infinity of such maps possible for a given pair, we define optimality as the minimization of deviation from isometry under the constraint of piecewise deformation homogeneity. We propose a novel variational formulation for segmenting asymmetric regions from surface pairs based on the minimization of a functional of both the deformation map and the segmentation boundary, which defines the regions within which the homogeneity constraint is to be enforced. The functional minimization is achieved via a quasi-simultaneous evolution of the map and the segmenting curve, conducted on and between two-dimensional surface parametric domains. We present examples using both synthetic data and pairs of left and right hippocampal structures and demonstrate the relevance of the extracted features through a clinical epilepsy classification analysis.

Superresolution reconstruction produces a high-resolution image from a set of low-resolution images. Previous iterative methods for superresolution had not adequately addressed the computational and numerical issues for this ill-conditioned and typically underdetermined large scale problem. We propose efficient block circulant preconditioners for solving the Tikhonov-regularized superresolution problem by the conjugate gradient method. We also extend to underdetermined systems the derivation of the generalized cross-validation method for automatic calculation of regularization parameters. The effectiveness of our preconditioners and regularization techniques is demonstrated with superresolution results for a simulated sequence and a forward looking infrared (FLIR) camera image sequence.

In many image restoration/resolution enhancement applications, the blurring process, i.e., point spread function (PSF) of the imaging system, is not known or is known only to within a set of parameters. We estimate these PSF parameters for this ill-posed class of inverse problem from raw data, along with the regularization parameters required to stabilize the solution, using the generalized cross-validation method (GCV). We propose efficient approximation techniques based on the Lanczos algorithm and Gauss quadrature theory, reducing the computational complexity of the GCV. Data-driven PSF and regularization parameter estimation experiments with synthetic and real image sequences are presented to demonstrate the effectiveness and robustness of our method.

Superresolution reconstruction produces a high resolution image from a set of low resolution images. Previous work on superresolution had not adequately addressed the computational issues for this problem. In this paper, we propose efficient block circulant preconditioners for solving the regularized superresolution problem by conjugate gradients. The effectiveness of the preconditioners is demonstrated with superresolution results for a simulated image sequence and a FLIR image sequence.

We formulate and solve a new parameter estimation problem in the presence of bounded data uncertainties. The new method is suitable when a priori bounds on the uncertain data are available; its solution guarantees that the effect of the uncertainties will never be unnecessarily overestimated beyond what is reasonably assumed by the a priori bounds.

The paper introduces an assertion scheme based on the backward error analysis for error detection in algorithms that solve dense systems of linear equations, Ax=b. Unlike previous methods, this backward error assertion model is specifically designed to operate in an environment of floating point arithmetic subject to round-off errors, and it can be easily instrumented in a Watchdog processor environment. The complexity of verifying assertions is O(n/sup 2/), compared to the O(n/sup 3/) complexity of algorithms solving Ax=b. Unlike other proposed error detection methods, this assertion model does not require any encoding of the matrix A. Experimental results under various error models are presented to validate the effectiveness of this assertion scheme.>

In various applications it is necessary to keep track of a low-rank approximation of a covariance matrix, R(t), slowly varying with time. It is convenient to track the left singular vectors associated with the largest singular values of the triangular factor, L(t), of its Cholesky factorization. These algorithms are referred to as square-root. The drawback of the eigenvalue decomposition (EVD) or the singular value decompositions (SVD) is usually the volume of the computations. Various numerical methods for carrying out this task are surveyed, and it is shown why this heavy computational burden is questionable in numerous situations and should be revised. Indeed, the complexity per eigenpair is generally a quadratic function of the problem size, but there exist faster algorithms with linear complexity. Finally, in order to make a choice among the large and fuzzy set of available techniques, comparisons based on computer simulations in a relevant signal processing context are made.>

We present a block Lanczos method for computing the greatest singular values and associated vectors of a large and sparse matrix, say A. Our algorithm does not transform A but accesses it through a user-supplied routine that computes the product AX or A tX for a given matrix X.This paper includes a discussion of the various ways to compute the singular-value decomposition of an upper triangular band matrix, this problem arises as a subproblem to be solved in the block Lanczos procedure.