SparseJSR: A Fast Algorithm to Compute Joint Spectral Radius via Sparse SOS Decompositions

This paper deals with stability analysis of hybrid systems. Such systems are characterized by a combination of continuous dynamics and logic based switching between discrete modes. Lyapunov theory is a well known methodology for the stability analysis of linear and nonlinear systems in control system literature. Construction of Lyapunov functions for hybrid systems is generally a difficult task, but once these functions are defined, stabilization of the system is straight-forward. The sum of squares (SOS) decomposition and semidefinite programming has also provided an efficient methodology for analysis of nonlinear systems. The computational method used in this paper relies on the SOS decomposition of multivariate polynomials. By using SOS, we construct a (some) Lyapunov function(s) for the hybrid system. The reduction techniques provide numerical solution of large-scale instances; otherwise they will be practically unsolvable. The introduced method can be used for hybrid systems with linear or nonlinear vector fields. Some examples are given to demonstrate the capabilities of the proposed approach.

Sums of squares II: Matrix functions

This paper examines the problem of designing a nonlinear static output feedback controller for uncertain polynomial systems via an iterative sums of squares approach. The derivation of the static output feedback controller is given in terms of the solvability conditions of state dependent bilinear matrix inequalities (BMIs). An iterative algorithm based on the sum of squares (SOS) decomposition is proposed to solve these state-dependent BMIs. Finally, numerical examples are provided at the end of the paper as to demonstrate the validity of the proposed design technique.

The essence of distantly supervised relation extraction is that it is an incomplete multi-label classification problem with sparse and noisy features. To tackle the sparsity and noise challenges, we propose solving the classification problem using matrix completion on factorized matrix of minimized rank. We formulate relation classification as completing the unknown labels of testing items (entity pairs) in a sparse matrix that concatenates training and testing textual features with training labels. Our algorithmic framework is based on the assumption that the rank of item-byfeature and item-by-label joint matrix is low. We apply two optimization models to recover the underlying low-rank matrix leveraging the sparsity of feature-label matrix. The matrix completion problem is then solved by the fixed point continuation (FPC) algorithm, which can find the global optimum. Experiments on two widely used datasets with different dimensions of textual features demonstrate that our low-rank matrix completion approach significantly outperforms the baseline and the state-of-the-art methods.

We present a procedure that gives us an SOS (sum of squares) decomposition of a given real polynomial in variables, if there exists such decomposition. For the case of real polynomials in non-commutative variables we extend this procedure to obtain a sum of hermitian squares SOHS) decomposition whenever there exists any. This extended procedure is the main scientific contribution of the paper.

Establishing robust stability of discrete-time systems with time-varying uncertainty: The Gram-SOS approach

Regression methods are employed in analyzing a Space Shuttle failure data set. A class of models based on transforms of cumulative time and cumulative failures is considered. It is used to predict both current as well as future log transformed failure intensity. This class includes several popular software reliability models including the exponential, logarithmic, and power models. It also includes models based on transforms of the time per failure, the time-varying analogue to the mean time between failures. Models are compared on the basis of their predictive performance as measured by the predicted residual sum of squares (PRESS) criterion. Time per failure is included as one of the independent variables in the models identified as having the lowest PRESS score, both when predicting current as well as future log failure intensity. Analyses are conducted using failures within a final subset of the observation interval. This subset is chosen through inspection of the plot of time per failure in terms of cumulative time. The impact of the choice of this subset is assessed by comparing results for analyses of that subset with analyses of the data in the complete observation interval.

Bad-data detection in pre-estimation can help to improve state estimation [Teeuwsen & Erlich, 2006]. Since transient and abnormal conditions may occur in a power system, measurements may be polluted by bad data to cause estimated errors. Hence, it is necessary to accurately detect the bad data. To ensure the reliability of the measured data, a practical state estimator should have an ability to detect and identify the bad data as well as to eliminate their effects on the estimation [Zhang & Lo, 1991]. Generally, bad-date detection is important to guarantee the reliability of the measured data. If one or more errors occur in power system measurements, the states of the estimated system may be biased and the safety of power supply may be potentially dangerous. To avoid this situation, several bad-data detection and identification schemes have been presented. For example, WLS (weighted least squares) was proposed in 1989 [EI-Keib et al., 1989]. The weighted sum of squares of the measurement residuals was chosen as the objective function to be minimized. But, WLS-based state estimators were only developed by using a linearized measurement function [Huang & Lin, 2003] with complicated computations. Then, linear programming (LP) was proposed to improve the identification method [Peterson & Girgis, 1988]. However, the LP estimator may fail to reject the bad data and it can be attributed to the existence of leverage in the power system model [Ali Abur, 1990]. Thus, Ali Abur had proposed hypothesis testing identification (HTI) to extend the case of the LP estimator. Nevertheless, it had caused computational burdens with taking the special properties of the LP estimation equations into account. Huang [Huang & Lin, 2003] proposed a changeable weighting matrix to identify the bad data but it only can apply for static state estimations. Nevertheless, Zhang had proposed recursive measurement error estimation identification (RMEEI) for bad-data identification [Zhang et al., 1992]. State variables, residuals and their parameters can be updated after removing a measurement from the suspected data set to the remaining data set by using a set of linear recursive equations. With splitting the raw measurements into some parts, a set of residual equations used by the traditional methods can only apply to linear systems and it may result the operation of calculation burden and complexity because each part consists of some measurements [Zhang et al., 1992]. Source: Kalman Filter, Book edited by: Vedran Kordic, ISBN 978-953-307-094-0, pp. 390, May 2010, INTECH, Croatia, downloaded from SCIYO.COM

Digital volume correlation (DVC) is a powerful experimental tool for quantitative 3D internal deformation measurement throughout the interior of materials or biological tissues. By comparing the volumetric images acquired at the reference and deformed states by a volumetric imaging facility (e.g., X-ray CT), DVC provides full-field displacements with subvoxel accuracy and full-field strain maps. When using DVC method for internal deformation measurement, the quality of internal speckle pattern of a test sample has an important impact on the accuracy and precision of the measured displacements. In this work, theoretical analysis and numerical simulation experiments are firstly performed to assess the quality of internal speckle patterns. The results show that the displacement measurement error of DVC is negative correlation with the sum of square of subvolume intensity gradient (SSSIG) of a subvolume. Thus, the SSSIG value of a subvolume can be used as a simple and effective metric for quality assessment of its internal speckle pattern. In practical DVC analyses, increasing the SSSIG parameter of a subvolume helps to improve the measurement accuracy and precision of DVC. Despite the SSSIG in a subvolume can be increased by enlarging its subvolume size, this simple approach, however, would lead to increased calculation burden. With a purpose to increase SSSIG without significantly increase the amount of calculation, this paper further proposes a method for optimal selection of calculation voxel points in subvolumes. Specifically, the voxel points with small gray gradients in a subvolume are excluded from DVC calculation to decrease the amount of calculation when increasing the size of the subvolume. The efficacy of the presented voxel point optimal selection method is validated by analyzing computer simulated and experimentally obtained volume images. Experimental results reveal that the proposed voxel point selection method is capable of reducing the calculation burden when increasing the subvolume size to increase the SSSIG parameters.

Latent factor (LF) models are highly effective in extracting useful knowledge from High-Dimensional and Sparse (HiDS) matrices which are commonly seen in various industrial applications. An LF model usually adopts iterative optimizers, which may consume many iterations to achieve a local optima, resulting in considerable time cost. Hence, how to accelerate the training process of an LF model becomes a highly significant issue. To address it, this work innovatively proposes a randomized latent factor (RLF) model. It incorporates the principle of randomized learning techniques for neural networks into the LF analysis on HiDS matrices to alleviate the computational burden greatly. It also extends the standard learning process for randomized neural networks in context of LF analysis to make the resulting model represent an HiDS matrix correctly. Experimental results on three HiDS matrices from industrial applications demonstrate that compared with state-of-the-art LF models, RLF is able to achieve significantly higher computational efficiency and comparable prediction accuracy for missing data. More importantly, it provides a novel, effective, and efficient approach to LF analysis on HiDS matrices.

Artifact reduction or removal is a challenging task when the artifact creation physics are not well modeled mathematically. One of such situations is metal artifacts in x-ray CT when the metallic material is unknown, and the x-ray spectrum is wide. A neural network is used to act as the objective function for iterative artifact reduction when the artifact model is unknown. A hypothetical unpredictable projection data distortion model is used to illustrate the proposed approach. The model is unpredictable, because it is controlled by a random variable. A convolutional neural network is trained to recognize the artifacts. The trained network is then used to compute the objective function for an iterative algorithm, which tries to reduce the artifacts in a computed tomography (CT) task. The objective function is evaluated in the image domain. The iterative algorithm for artifact reduction is in the projection domain. A gradient descent algorithm is used for the objective function optimization. The associated gradient is calculated with the chain rule. The learning curves illustrate the decreasing treads of the objective function as the number of iterations increases. The images after the iterative treatment show the reduction of artifacts. A quantitative metric, the Sum Square Difference (SSD), also indicates the effectiveness of the proposed method. The methodology of using a neural network as an objective function has potential value for situations where a human developed model is difficult to describe the underlying physics. Real-world applications are expected to be benefit from this methodology.

The ratio between processor and main memory performance has been increasing since quite some time, and can safely be expected to do so throughout the oncoming years. In the era of single-core processors, this was mainly observable by increased latency, for example when measured in number of (possibly stalled) CPU clock cycles. Nowadays, with multicore chips, multiple cores share the same connection to off-chip main memory, which effectively reduces available bandwidth, as well. Caches help in both cases: they provide both a much lower latency and a much higher bandwidth by being located on-chip. By holding copies of least recently used memory blocks, caches exploit the fact that programs on the average access memory in ways that often access the same memory cell (temporal locality), or nearby memory cells (spatial locality). However, this natural locality is not enough for scientific computing in HPC. Further improving any existing access locality of given algorithms is very much wanted. In this talk, we present strategies to improve the locality of memory accesses for linear algebra problems occurring in different kinds of applications: (1) an algorithmic approach based on Peano spacefilling curves that leads to inherently cache efficient (cache oblivious) matrix algorithms, such as matrix multiplication or LU decomposition for dense and sparse matrices - on single-core CPUs, as well as in the context of shared-memory multicore platforms. (2) cache optimization strategies for matrix-vector multiplications with very large, sparse matrices, as they occur in the iterative MLEM algorithm, which is used for image reconstruction in nuclear medicine. Here, different cache-aware optimization strategies are combined in order to better exploit large caches, small caches, and single cache lines.

How to accurately estimate facial age is still an intractable task because of insufficiency of training data. In this paper, a hybrid model is proposed to estimate facial age by means of extreme learning machine (ELM) and label distribution support vector regressor (LDSVR). In the proposed method, the bio-inspired features are adopted to estimate the facial age due to its prominent performance. In order to improve the accuracy and decrease the computation burden, the ratio of feature’s between-category to within-category sums of squares (BW) is designed as a criterion to select features. To define the category of each sample, the training data is divided into several sets according to age group. Different virtual class labels are assigned to the samples of each set, respectively. Given the reduced data, a multiple-input-single-output ELM regression model is established to estimate the facial ages. Moreover, a label distribution support vector regressor is adopted to estimate facial age based on a multiple-input-multiple-output regression model. After obtaining the outputs of ELM and LDSVR, a linear weighting strategy is devised to compute the final estimation of facial age. Experimental results on a well known facial image database demonstrates the feasibility and efficiency of the proposed hybrid model.

ABSTRACTThe K-means clustering method is a widely adopted clustering algorithm in data mining and pattern recognition, where the partitions are made by minimizing the total within group sum of squares based on a given set of variables. Weighted K-means clustering is an extension of the K-means method by assigning nonnegative weights to the set of variables. In this paper, we aim to obtain more meaningful and interpretable clusters by deriving the optimal variable weights for weighted K-means clustering. Specifically, we improve the weighted k-means clustering method by introducing a new algorithm to obtain the globally optimal variable weights based on the Karush-Kuhn-Tucker conditions. We present the mathematical formulation for the clustering problem, derive the structural properties of the optimal weights, and implement an recursive algorithm to calculate the optimal weights. Numerical examples on simulated and real data indicate that our method is superior in both clustering accuracy and computational efficiency.

Joint Regression Analysis is a widely used technique for cultivar comparison. For each cultivar a linear regression is adjusted on a non-observable regressor: the environmental index. This index measures, for each block, the corresponding productivity. When all cultivars are present in all the blocks in the field trials, the series of experiments is complete. To carry out the minimization of the sum of sums of squares of residuals in order to estimate the coefficients of the regressions and the environmental indexes an iterative algorithm, the zigzag algorithm, was introduced, see Mexia et al. [J.T. Mexia, L 2 environmental indexes, Biom. Lett. 36 (1999), pp. 137–143.]. This algorithm performs well, see e.g. Mexia et al. [Weighted linear joint regression analysis, Biom. Lett. 38 (2001), pp. 33–40.] and Mexia and Pereira [J.T. Mexia, D.G. Pereira, Joint regression analysis for winter rye cultivars using L 2 indexes, J. Colloquium Biometryczne 31 (2001), pp. 207–212.] but it has not been shown that it converges to the absolute minimum of the goal function. We present an alternative algorithm and show that, in the complete case, it converges to the absolute minimum. Through an example it is shown that the results obtained using both algorithms agree. We analyse the reason behind the agreement between both algorithms.