Low-cost heuristics for matrix bandwidth reduction combined with a Hill-Climbing strategy

This paper studies heuristics for the bandwidth reduction of large-scale matrices in serial computations. Bandwidth optimization is a demanding subject for a large number of scientific and engineering applications. A heuristic for bandwidth reduction labels the rows and columns of a given sparse matrix. The algorithm arranges entries with a nonzero coefficient as close to the main diagonal as possible. This paper modifies an ant colony hyper-heuristic approach to generate expert-level heuristics for bandwidth reduction combined with a Hill-Climbing strategy when applied to matrices arising from specific application areas. Specifically, this paper uses low-cost state-of-the-art heuristics for bandwidth reduction in tandem with a Hill-Climbing procedure. The results yielded on a wide-ranging set of standard benchmark matrices showed that the proposed strategy outperformed low-cost state-of-the-art heuristics for bandwidth reduction when applied to matrices with symmetric sparsity patterns.

Generalized normal forms of the systems of ordinary differential equations with a quasi-homogeneous polynomial (ax1^2 + x2, x1x2) in the unperturbed part

In this paper a study on constructive construction of the generalized normal forms (GNF) is continued. The planar real-analytical at the origin system is considered. Its unperturbed part forms a first degree quasi-homogeneous first degree polynomial (αx21 + x2, x1x2) of type (1, 2) where parameter α ∈ 2 (-1/2, 0)[(0, 1/2]. For given value of this polynomial is a canonical form, that is an element of a class of equivalence relative to quasi-homogeneous substitutions of zero order into which any first order quasi-homogeneous polynomial of type (1, 2) is divided in accordance with the chosen structural principles due to it only making sense to reduce the systems with the various canonical forms in their unperturbed part to GNF. Based on the constructive method of resonance equations and sets, the resonance equations are derived. Perturbations of the acquired system satisfies these equations if an almost identity quasi-homogeneous substitution in the given system is applied. Their validity guarantees formal equivalence of the systems. Besides, resonance sets of coefficients are specified that allows to get all possible GNF structures and prove reducibility of the given system to a GNF with any of specified structures. In addition, some examples of characteristic GNFs are provided including that with the parameter leading to appearance of an additional resonance equation and the second nonzero coefficient of the appropriate orders in GNFs.

Hilbert series associated to symplectic quotients by SU2

We compute the Hilbert series of the graded algebra of real regular functions on the symplectic quotient associated to an [Formula: see text]-module and give an explicit expression for the first nonzero coefficient of the Laurent expansion of the Hilbert series at [Formula: see text]. Our expression for the Hilbert series indicates an algorithm to compute it, and we give the output of this algorithm for all representations of dimension at most [Formula: see text]. Along the way, we compute the Hilbert series of the module of covariants of an arbitrary [Formula: see text]- or [Formula: see text]-module as well as its first three Laurent coefficients.

Image Segmentation Based on Supervised Discriminative Learning

In view of the complex background of images and the segmentation difficulty, a sparse representation and supervised discriminative learning were applied to image segmentation. The sparse and over-complete representation can represent images in a compact and efficient manner. Most atom coefficients are zero, only a few coefficients are large, and the nonzero coefficient can reveal the intrinsic structures and essential properties of images. Therefore, sparse representations are beneficial to subsequent image processing applications. We first described the sparse representation theory. This study mainly revolved around three aspects, namely a trained dictionary, greedy algorithms, and the application of the sparse representation model in image segmentation based on supervised discriminative learning. Finally, we performed an image segmentation experiment on standard image datasets and natural image datasets. The main focus of this thesis was supervised discriminative learning, and the experimental results showed that the proposed algorithm was optimal, sparse, and efficient.

A Format-Compliant Encryption for Secure HEVC Video Sharing in Multimedia Social Network

Aiming at secure video sharing in multimedia social network, a format-compliant encryption scheme for high efficiency video coding (HEVC) based on sigh data hiding (SDH) is proposed. The encryption is tightly integrated with the encoding/decoding processes. For each coding unit (CU), the sign of the nonzero coefficient and the first hiding nonzero coefficient are both encrypted with key stream. Meanwhile, one of merging index, motion vector prediction index, sign of motion vector difference and reference frame index is chosen for encryption according to a control factor. As it is explored in this article, experimental results and analysis indicate that it can effectively resist brute-force attack, difference attack and replacement attack. Also, it can keep a good balance in encryption space, computation complexity and security. Based on the encryption scheme, a framework of its implementation in multimedia social network is presented. It has great potential to be implemented for secure video sharing in multimedia social network.

Iterative Forward-Backward Pursuit Algorithm for Compressed Sensing

It has been shown that iterative reweighted strategies will often improve the performance of many sparse reconstruction algorithms. Iterative Framework for Sparse Reconstruction Algorithms (IFSRA) is a recently proposed method which iteratively enhances the performance of any given arbitrary sparse reconstruction algorithm. However, IFSRA assumes that the sparsity level is known. Forward-Backward Pursuit (FBP) algorithm is an iterative approach where each iteration consists of consecutive forward and backward stages. Based on the IFSRA, this paper proposes the Iterative Forward-Backward Pursuit (IFBP) algorithm, which applies the iterative reweighted strategies to FBP without the need for the sparsity level. By using an approximate iteration strategy, IFBP gradually iterates to approach the unknown signal. Finally, this paper demonstrates that IFBP significantly improves the reconstruction capability of the FBP algorithm, via simulations including recovery of random sparse signals with different nonzero coefficient distributions in addition to the recovery of a sparse image.

Differential resolvents of minimal order and weight

We will determine the number of powers ofαthat appear with nonzero coefficient in anα-power linear differential resolvent of smallest possible order of a univariate polynomialP(t)whose coefficients lie in an ordinary differential field and whose distinct roots are differentially independent over constants. We will then give an upper bound on the weight of anα-resolvent of smallest possible weight. We will then compute the indicial equation, apparent singularities, and Wronskian of the Cockleα-resolvent of a trinomial and finish with a related determinantal formula.