Distributed Processing of Blind Source Separation

<p>Communication is performed by transmitting signals through a medium. It is common that signals originating from different sources are mixed in the transport medium. The operation of separating source signals without prior information about the sources is referred to as blind source separation (BSS). Blind source separation for wireless sensor networks has recently received attention because of low cost and the easy coverage of large areas. Distributed processing is attractive as it is scalable and consumes low power. Existing distributed BSS algorithms either require a fully connected pattern of connectivity, to ensure the good performance, or require a high computational load at each sensor node, to enhance the scalability. This motivates us to develop distributed BSS algorithms that can be implemented over any arbitrary graph with fully shared computations and with good performance. This thesis presents three studies on distributed algorithms. The first two studies are on existing distributed algorithms that are used in linearly constrained convex optimization problems, which are common in signal processing and machine learning. The studies are aimed at improving the algorithms in terms of computational complexity, communication cost, processors coordination and scalability. This makes them more suitable for implementation on sensor networks, thus forming a basis for the development of distributed BSS algorithms on sensor networks in our third study. In the first study, we consider constrained problems in which the constraint includes a weighted sum of all the decision variables. By formulating a constrained dual problem associated to the original constrained problem, we were able to develop a distributed algorithm that can be run both synchronously and asynchronously on any arbitrary graph with lower communication cost than traditional distributed algorithms. In the second study, we consider constrained problems in which the constraint is separable. By making use of the augmented Lagrangian function and splitting the dual variable (Lagrange multiplier) associated to each partial constraint, we were able to develop a distributed fully asynchronous algorithm with lower computational complexity than traditional distributed algorithms. The simplicity of the algorithm is the consequence of approximating the constraint on the equality of the decoupled dual variables. We also provide a measure of the inaccuracy in such an approximation on the optimal value of the primal objective function. Finally, in the third study, we investigate distributed processing solutions for BSS on sensor networks. We propose two distributed processing schemes for BSS that we refer to as scheme 1 and scheme 2. In scheme 1, each sensor node estimates one specific source signal while in scheme 2, by formulating a consensus optimization problem, each sensor node estimates all source signals in a fully shared computation manner. Our proposed algorithms carry the following features: low computational complexity, low power consumption, low data transmission rate, scalability and excellent performance over arbitrary graphs. Although all of our proposed algorithms share the aforementioned properties, each of them is superior in one or some of the features compared to the others. Comparative experimental results show that among all our proposed distributed BSS algorithms, a variant of scheme 1 performs best when all features are considered. This is achieved by making use of the concept of pairwise mutual information along with adding a sparsity assumption on the parameters of the model that is used in BSS.</p>

Distributed Processing of Blind Source Separation

<p>Communication is performed by transmitting signals through a medium. It is common that signals originating from different sources are mixed in the transport medium. The operation of separating source signals without prior information about the sources is referred to as blind source separation (BSS). Blind source separation for wireless sensor networks has recently received attention because of low cost and the easy coverage of large areas. Distributed processing is attractive as it is scalable and consumes low power. Existing distributed BSS algorithms either require a fully connected pattern of connectivity, to ensure the good performance, or require a high computational load at each sensor node, to enhance the scalability. This motivates us to develop distributed BSS algorithms that can be implemented over any arbitrary graph with fully shared computations and with good performance. This thesis presents three studies on distributed algorithms. The first two studies are on existing distributed algorithms that are used in linearly constrained convex optimization problems, which are common in signal processing and machine learning. The studies are aimed at improving the algorithms in terms of computational complexity, communication cost, processors coordination and scalability. This makes them more suitable for implementation on sensor networks, thus forming a basis for the development of distributed BSS algorithms on sensor networks in our third study. In the first study, we consider constrained problems in which the constraint includes a weighted sum of all the decision variables. By formulating a constrained dual problem associated to the original constrained problem, we were able to develop a distributed algorithm that can be run both synchronously and asynchronously on any arbitrary graph with lower communication cost than traditional distributed algorithms. In the second study, we consider constrained problems in which the constraint is separable. By making use of the augmented Lagrangian function and splitting the dual variable (Lagrange multiplier) associated to each partial constraint, we were able to develop a distributed fully asynchronous algorithm with lower computational complexity than traditional distributed algorithms. The simplicity of the algorithm is the consequence of approximating the constraint on the equality of the decoupled dual variables. We also provide a measure of the inaccuracy in such an approximation on the optimal value of the primal objective function. Finally, in the third study, we investigate distributed processing solutions for BSS on sensor networks. We propose two distributed processing schemes for BSS that we refer to as scheme 1 and scheme 2. In scheme 1, each sensor node estimates one specific source signal while in scheme 2, by formulating a consensus optimization problem, each sensor node estimates all source signals in a fully shared computation manner. Our proposed algorithms carry the following features: low computational complexity, low power consumption, low data transmission rate, scalability and excellent performance over arbitrary graphs. Although all of our proposed algorithms share the aforementioned properties, each of them is superior in one or some of the features compared to the others. Comparative experimental results show that among all our proposed distributed BSS algorithms, a variant of scheme 1 performs best when all features are considered. This is achieved by making use of the concept of pairwise mutual information along with adding a sparsity assumption on the parameters of the model that is used in BSS.</p>

Construction of a class of maximal commutative subalgebras of prime Leavitt path algebras

Considering prime Leavitt path algebras L K ⁢ ( E ) {L_{K}(E)} , with E being an arbitrary graph with at least two vertices, and K being any field, we construct a class of maximal commutative subalgebras of L K ⁢ ( E ) {L_{K}(E)} such that, for every algebra A from this class, A has zero intersection with the commutative core ℳ K ⁢ ( E ) {\mathcal{M}_{K}(E)} of L K ⁢ ( E ) {L_{K}(E)} defined and studied in [C. Gil Canto and A. Nasr-Isfahani, The commutative core of a Leavitt path algebra, J. Algebra 511 2018, 227–248]. We also give a new proof of the maximality, as a commutative subalgebra, of the commutative core ℳ R ⁢ ( E ) {\mathcal{M}_{R}(E)} of an arbitrary Leavitt path algebra L R ⁢ ( E ) {L_{R}(E)} , where E is an arbitrary graph and R is a commutative unital ring.

Associative spectra of graph algebras II

A necessary and sufficient condition is presented for a graph algebra to satisfy a bracketing identity. The associative spectrum of an arbitrary graph algebra is shown to be either constant or exponentially growing.

A note on the regular ideals of Leavitt path algebras

We show that, for an arbitrary graph, a regular ideal of the associated Leavitt path algebra is also graded. As a consequence, for a row-finite graph, we obtain that the quotient of the associated Leavitt path by a regular ideal is again a Leavitt path algebra and that Condition (L) is preserved by quotients by regular ideals. Furthermore, we describe the vertex set of a regular ideal and make a comparison between the theory of regular ideals in Leavitt path algebras and in graph C*-algebras.

Statistical analysis of random walks on network

This paper describes an investigation of analytical formulas for parameters in random walks. Random walks are used to model situations in which an object moves in a sequence of steps in randomly chosen directions. Given a graph and a starting point, we select a neighbor of it at random, and move to this neighbor; then we select a neighbor of this point at random, and move to it etc. It is a fundamental dynamic process that arise in many models in mathematics, physics, informatics and can be used to model random processes inherent to many important applications. Different aspects of the theory of random walks on graphs are surveyed. In particular, estimates on the important parameters of hitting time, commute time, cover time are discussed in various works. In some papers, authors have derived an analytical expression for the distribution of the cover time for a random walk over an arbitrary graph that was tested for small values of n. However, this work will show the simplified analytical expressions for distribution of hitting time, commute time, cover time for bigger values of n. Moreover, this work will present the probability mass function and the cumulative distribution function for hitting time, commute time.

On Hamilton-Connectivity and Detour Index of Certain Families of Convex Polytopes

A convex polytope is the convex hull of a finite set of points in the Euclidean space ℝ n . By preserving the adjacency-incidence relation between vertices of a polytope, its structural graph is constructed. A graph is called Hamilton-connected if there exists at least one Hamiltonian path between any of its two vertices. The detour index is defined to be the sum of the lengths of longest distances, i.e., detours between vertices in a graph. Hamiltonian and Hamilton-connected graphs have diverse applications in computer science and electrical engineering, whereas the detour index has important applications in chemistry. Checking whether a graph is Hamilton-connected and computing the detour index of an arbitrary graph are both NP-complete problems. In this paper, we study these problems simultaneously for certain families of convex polytopes. We construct two infinite families of Hamilton-connected convex polytopes. Hamilton-connectivity is shown by constructing Hamiltonian paths between any pair of vertices. We then use the Hamilton-connectivity to compute the detour index of these families. A family of non-Hamilton-connected convex polytopes has also been constructed to show that not all convex polytope families are Hamilton-connected.

An Inverse QSAR Method Based on a Two-Layered Model and Integer Programming

A novel framework for inverse quantitative structure–activity relationships (inverse QSAR) has recently been proposed and developed using both artificial neural networks and mixed integer linear programming. However, classes of chemical graphs treated by the framework are limited. In order to deal with an arbitrary graph in the framework, we introduce a new model, called a two-layered model, and develop a corresponding method. In this model, each chemical graph is regarded as two parts: the exterior and the interior. The exterior consists of maximal acyclic induced subgraphs with bounded height, the interior is the connected subgraph obtained by ignoring the exterior, and the feature vector consists of the frequency of adjacent atom pairs in the interior and the frequency of chemical acyclic graphs in the exterior. Our method is more flexible than the existing method in the sense that any type of graphs can be inferred. We compared the proposed method with an existing method using several data sets obtained from PubChem database. The new method could infer more general chemical graphs with up to 50 non-hydrogen atoms. The proposed inverse QSAR method can be applied to the inference of more general chemical graphs than before.

Minimum Linear Arrangement of the Cartesian Product of Optimal Order Graph and Path

The minimum linear arrangement of an arbitrary graph is the embedding of the vertices of the graph onto the line topology in such a way that the sum of the distances between adjacent vertices in the graph is minimized. This minimization can be attained by finding an optimal ordering of the vertex set of the graph and labeling the vertices of the line in that order. In this paper, we compute the minimum linear arrangement of the Cartesian product of certain sequentially optimal order graphs which include interconnection networks such as hypercube, folded hypercube, complete Josephus cube and locally twisted cube with path and the edge faulty counterpart of sequentially optimal order graphs.