Optimal solution of the nearest correlation matrix problem by minimization of the maximum norm

The nearest correlation matrix problem is to find a valid (positive semidefinite) correlation matrix, R(m,m), that is nearest to a given invalid (negative semidefinite) or pseudo-correlation matrix, Q(m,m); m larger than 2. In the literature on this problem, 'nearest' is invariably defined in the sense of the least Frobenius norm. Research works of Rebonato and Jaeckel (1999), Higham (2002), Anjos et al. (2003), Grubisic and Pietersz (2004), Pietersz, and Groenen (2004), etc. use Frobenius norm explicitly or implicitly. However, it is not necessary to define 'nearest' in this conventional sense. The thrust of this paper is to define 'nearest' in the sense of the least maximum norm (LMN) of the deviation matrix (R-Q), and to obtain R nearest to Q. The LMN provides the overall minimum range of deviation of the elements of R from those of Q. We also append a computer program (source codes in FORTRAN) to find the LMN R from a given Q. Presently we use the random walk search method for optimization. However, we suggest that more efficient methods based on the Genetic algorithms may replace the random walk algorithm of optimization.Nearest correlation matrix problem; Frobenius norm; maximum norm; LMN correlation matrix; positive semidefinite; negative semidefinite; positive definite; random walk algorithm; Genetic algorithm; computer program; source codes; FORTRAN; simulation

Multiself-loop Lackadaisical Quantum Walk with Partial Phase Inversion

Quantum walks are the quantum counterpart of classical random walks and provide an intuitive framework for building new quantum algorithms. The lackadaisical quantum walk, which is a quantum analog of the lazy random walk, is obtained by adding a self-loop transition to each state allowing the walker to stay stuck in the same state, being able to improve the performance of the quantum walks as search algorithms. However, the high dependence of a weight $l$ makes it a key parameter to reach the maximum probability of success in the search process. Although many advances have been achieved with search algorithms based on quantum walks, the number of self-loops can also be critical for search tasks. Believing that the multiple self-loops have not yet been properly explored, this article proposes the quantum search algorithm Multiself-loop Lackadaisical Quantum Walk with Partial Phase Inversion, which is based on a lackadaisical quantum walk with multiple self-loops where the target state phase is partially inverted. Each vertex has $m$ self-loops, with weights $l' = l/m$, where $l$ is a real parameter. The phase inversion is based on Grover's algorithm and acts partiality, modifying the phase of a given quantity $s \leqslant m$ of self-loops. On a hypercube structure, we analyzed the situation where $s=1$ and $1 \leqslant m \leqslant 30$ and investigated its effects in the search for 1 to 12 marked vertices. Based on two ideal weights $l$ used in the literature, we propose two new weight values. As a result, with the proposal of the Multiself-loop Lackadaisical Quantum Walk with partial phase inversion of target states and the new weight values for the self-loop, this proposal improved the maximum success probabilities to values close to 1. This article contributes with a new perspective on the use of quantum interferences in the construction of new quantum search algorithms.Comment: 16 pages, 4 figures, 3 table

A versatile stochastic model of a function of unknown and time varying form

AbstractProperties of a random walk model of an unknown function are studied. The model is suitable for use in the following (among others) problem. Given a system with a performance function of unknown, time varying, and possibly multipeak form (with respect to a single system parameter), and given that the only information available are noise perturbed samples of the function at selected parameter settings, then determine the successive parameter settings such that the sum of the values of the observations is maximum. An attempt to avoid the optimal search problem through the use of several intuitively reasonable heuristics is presented

Product Search Algorithm Based on Improved Ant Colony Optimization in a Distributed Network

The crowd intelligence-based e-commerce transaction network (CIeTN) is a distributed and unstructured network structure. Smart individuals, such as buyers, sellers, and third-party organizations, can store information in local nodes and connect and share information via moments. The purpose of this study is to design a product search algorithm on the basis of ant colony optimization (ACO) to achieve an efficient and accurate search for the product demand of a node in the network. We introduce the improved ideas of maximum and minimum ants to design a set of heuristic search algorithms on the basis of ACO. To reduce search blindness, additional relevant heuristic factors are selected to define the heuristic calculation equation. The pheromone update mechanism integrating into the product matching factor and forwarding probability is used to design the network search rules among nodes in the search algorithm. Finally, the search algorithm is facilitated by Java language programming and PeerSim software. Experimental results show that the algorithm has significant advantages over the flooding method and the random walk method in terms of search success rate, search time, product matching, search network consumption, and scalability. The search algorithm introduces the idea of improving the maximum and minimum ant colony system and proposes new ideas in the design of heuristic factors in the heuristic equation and the pheromone update strategy. The search algorithm can search for product information effectively

Optimal network topologies: Expanders, Cages, Ramanujan graphs, Entangled networks and all that

We report on some recent developments in the search for optimal network topologies. First we review some basic concepts on spectral graph theory, including adjacency and Laplacian matrices, and paying special attention to the topological implications of having large spectral gaps. We also introduce related concepts as ``expanders'', Ramanujan, and Cage graphs. Afterwards, we discuss two different dynamical feautures of networks: synchronizability and flow of random walkers and so that they are optimized if the corresponding Laplacian matrix have a large spectral gap. From this, we show, by developing a numerical optimization algorithm that maximum synchronizability and fast random walk spreading are obtained for a particular type of extremely homogeneous regular networks, with long loops and poor modular structure, that we call entangled networks. These turn out to be related to Ramanujan and Cage graphs. We argue also that these graphs are very good finite-size approximations to Bethe lattices, and provide almost or almost optimal solutions to many other problems as, for instance, searchability in the presence of congestion or performance of neural networks. Finally, we study how these results are modified when studying dynamical processes controlled by a normalized (weighted and directed) dynamics; much more heterogeneous graphs are optimal in this case. Finally, a critical discussion of the limitations and possible extensions of this work is presented.Comment: 17 pages. 11 figures. Small corrections and a new reference. Accepted for pub. in JSTA

Search for continuous gravitational waves from 20 accreting millisecond x-ray pulsars in O3 LIGO data

Results are presented of searches for continuous gravitational waves from 20 accreting millisecond x-ray pulsars with accurately measured spin frequencies and orbital parameters, using data from the third observing run of the Advanced LIGO and Advanced Virgo detectors. The search algorithm uses a hidden Markov model, where the transition probabilities allow the frequency to wander according to an unbiased random walk, while the J-statistic maximum-likelihood matched filter tracks the binary orbital phase. Three narrow subbands are searched for each target, centered on harmonics of the measured spin frequency. The search yields 16 candidates, consistent with a false alarm probability of 30% per subband and target searched. These candidates, along with one candidate from an additional target-of-opportunity search done for SAX J1808.4-3658, which was in outburst during one month of the observing run, cannot be confidently associated with a known noise source. Additional follow-up does not provide convincing evidence that any are a true astrophysical signal. When all candidates are assumed nonastrophysical, upper limits are set on the maximum wave strain detectable at 95% confidence, h095%. The strictest constraint is h095%=4.7×10-26 from IGR J17062-6143. Constraints on the detectable wave strain from each target lead to constraints on neutron star ellipticity and r-mode amplitude, the strictest of which are ϵ95%=3.1×10-7 and α95%=1.8×10-5 respectively. This analysis is the most comprehensive and sensitive search of continuous gravitational waves from accreting millisecond x-ray pulsars to date