Application of semidefinite programming to maximize the spectral gap produced by node removal

The smallest positive eigenvalue of the Laplacian of a network is called the spectral gap and characterizes various dynamics on networks. We propose mathematical programming methods to maximize the spectral gap of a given network by removing a fixed number of nodes. We formulate relaxed versions of the original problem using semidefinite programming and apply them to example networks.Comment: 1 figure. Short paper presented in CompleNet, Berlin, March 13-15 (2013

Eigenvector Synchronization, Graph Rigidity and the Molecule Problem

The graph realization problem has received a great deal of attention in recent years, due to its importance in applications such as wireless sensor networks and structural biology. In this paper, we extend on previous work and propose the 3D-ASAP algorithm, for the graph realization problem in $\mathbb{R}^3$, given a sparse and noisy set of distance measurements. 3D-ASAP is a divide and conquer, non-incremental and non-iterative algorithm, which integrates local distance information into a global structure determination. Our approach starts with identifying, for every node, a subgraph of its 1-hop neighborhood graph, which can be accurately embedded in its own coordinate system. In the noise-free case, the computed coordinates of the sensors in each patch must agree with their global positioning up to some unknown rigid motion, that is, up to translation, rotation and possibly reflection. In other words, to every patch there corresponds an element of the Euclidean group Euc(3) of rigid transformations in $\mathbb{R}^3$, and the goal is to estimate the group elements that will properly align all the patches in a globally consistent way. Furthermore, 3D-ASAP successfully incorporates information specific to the molecule problem in structural biology, in particular information on known substructures and their orientation. In addition, we also propose 3D-SP-ASAP, a faster version of 3D-ASAP, which uses a spectral partitioning algorithm as a preprocessing step for dividing the initial graph into smaller subgraphs. Our extensive numerical simulations show that 3D-ASAP and 3D-SP-ASAP are very robust to high levels of noise in the measured distances and to sparse connectivity in the measurement graph, and compare favorably to similar state-of-the art localization algorithms.Comment: 49 pages, 8 figure

Optimal Data Collection For Informative Rankings Expose Well-Connected Graphs

Given a graph where vertices represent alternatives and arcs represent pairwise comparison data, the statistical ranking problem is to find a potential function, defined on the vertices, such that the gradient of the potential function agrees with the pairwise comparisons. Our goal in this paper is to develop a method for collecting data for which the least squares estimator for the ranking problem has maximal Fisher information. Our approach, based on experimental design, is to view data collection as a bi-level optimization problem where the inner problem is the ranking problem and the outer problem is to identify data which maximizes the informativeness of the ranking. Under certain assumptions, the data collection problem decouples, reducing to a problem of finding multigraphs with large algebraic connectivity. This reduction of the data collection problem to graph-theoretic questions is one of the primary contributions of this work. As an application, we study the Yahoo! Movie user rating dataset and demonstrate that the addition of a small number of well-chosen pairwise comparisons can significantly increase the Fisher informativeness of the ranking. As another application, we study the 2011-12 NCAA football schedule and propose schedules with the same number of games which are significantly more informative. Using spectral clustering methods to identify highly-connected communities within the division, we argue that the NCAA could improve its notoriously poor rankings by simply scheduling more out-of-conference games.Comment: 31 pages, 10 figures, 3 table

Network connectivity tracking for a team of unmanned aerial vehicles

Algebraic connectivity is the second-smallest eigenvalue of the Laplacian matrix and can be used as a metric for the robustness and efficiency of a network. This connectivity concept applies to teams of multiple unmanned aerial vehicles (UAVs) performing cooperative tasks, such as arriving at a consensus. As a UAV team completes its mission, it often needs to control the network connectivity. The algebraic connectivity can be controlled by altering edge weights through movement of individual UAVs in the team, or by adding and deleting edges. The addition and deletion problem for algebraic connectivity, however, is NP-hard. The contributions of this work are 1) a comparison of four heuristic methods for modifying algebraic connectivity through the addition and deletion of edges, 2) a rule-based algorithm for tracking a connectivity profile through edge weight modification and the addition and deletion of edges, 3) a new, hybrid method for selecting the best edge to add or remove, 4) a distributed method for estimating the eigenvectors of the Laplacian matrix and selecting the best edge to add or remove for connectivity modification and tracking, and 5) an implementation of the distributed connectivity tracking using a consensus controller and double-integrator dynamics

Preserving Measured Structure During Generation and Reduction of Multivariate Point Configurations

Inherent in any multivariate data is structure, which describes the general shape and distribution of the underlying point configuration. While there are potentially many types of structure that could be of interest, consider restricting interest to two general types: geometric structure, the general shape of a point configuration, and probabilistic structure, the general distribution of points within the configuration. The ability to quantify geometric structure is an important step in many common statistical analyses. For instance, general neighbourhood structure is captured using a k-nearest neighbour graph in dimension reduction techniques such as isomap and locally-linear embedding. Neighbourhood graphs are also used in sensor network localization, which has applications in fields such as environmental habitat monitoring and wildlife monitoring. Another geometric graph, the convex hull, is also used in wildlife monitoring as a rough estimate of an animal's home range. The identification of areas of high and low density is one example of measuring the probability structure of a configuration, which can be done using a wide variety of methods. One such method is using kernel density estimation, which can be viewed as a weighted sum of nearby points. Kernel density estimation has widely varying applications, including in regression analysis, and is used in general to assess certain features of the data (modality, skewness, etc.). Related to the idea of measuring structure is the concept of "Cognostics", which has been formalized as scatterplot diagnostics (or scagnostics). Scagnostics provides a framework through which interesting structure can be measured in a configuration. The central idea is to numerically summarize the structure of a large number of two-dimensional point configurations via measures calculated on geometric graphs. This allows the interesting views to be quickly identified, and ultimately examined visually, while the views deemed to be uninteresting are simply discarded. While a good starting point, several issues in the current framework need to be addressed. For instance, while each measure is designed to be in [0,1], there are some that, when measured over tens of thousands of configurations, fail to achieve this range. In addition, there is a lot of structure that could be considered interesting that is not captured by the current framework. These issues, among others, will be addressed and rectified so that the current scagnostic framework can continue to be built upon. With tools to measure structure, attention is turned to making use of the structural information contained in the configuration. Consider the problem of preserving measured structure during the task of data aggregation, more commonly known as binning. Existing methods of data aggregation tend to exist on two ends of the structure retention spectrum. Through experimentation, methods such as equal width and hexagonal binning will be shown to tend to retain the shape of the configuration, at the expense of the density, while methods such as equal frequency and random sampling tend to retain relative density at the expense of overall shape. Tree-based binning, a general binning framework inspired by classification and regression trees, is proposed to bridge the gap between these sets of specialist algorithms. GapBin, a specially designed tree-based binning algorithm, will be shown through experimentation to provide a trade-off in low dimensional space between geometric structure retention and probabilistic structure retention. In higher dimensions, it will be shown to be the superior algorithm in terms of structure retention among those considered. Next, the general problem of constructing a configuration with a given underlying structure is considered. For example, the minimal spanning tree is known to carry important clustering information. Of interest then, is the generation of configurations with a given minimal spanning tree structure. The problem of generating a configuration with a known minimal spanning tree is equivalent to completing a Euclidean distance matrix where the only known entries are those in the minimal spanning tree. For this problem, there are several solutions, including those of Alfakih et. al., Fang & O'Leary, and Trosset. None of these algorithms, however, are designed to retain the structure of the minimal spanning tree. In addition, the sparsity of the Euclidean distance matrix containing only the minimal spanning tree results in completions that are not accurate as compared to the known completion. This leads to issues in the point configurations of the resulting completions. To resolve these, two new algorithms are proposed which are designed to retain the structure of the minimal spanning tree, leading to more accurate completions of these sparse matrices. To complement the algorithms presented, implementation of these algorithms in the statistical programming language R will also be discussed. In particular, the R package treebinr for tree-based binning, and edmcr for Euclidean distance matrix completions will be presented

Of keyboards and beyond - optimization in human-computer interaction

In this thesis, we present optimization frameworks in the area of Human-Computer Interaction. At first, we discuss keyboard layout problems with a special focus on a project we participated in, which aimed at designing the new French keyboard standard. The special nature of this national-scale project and its optimization ingredients are discussed in detail; we specifically highlight our algorithmic contribution to this project. Exploiting the special structure of this design problem, we propose an optimization framework that was efficiently computes keyboard layouts and provides very good optimality guarantees in form of tight lower bounds. The optimized layout that we showed to be nearly optimal was the basis of the new French keyboard standard recently published in the National Assembly in Paris. Moreover, we propose a relaxation for the quadratic assignment problem (a generalization of keyboard layouts) that is based on semidefinite programming. In a branch-and-bound framework, this relaxation achieves competitive results compared to commonly used linear programming relaxations for this problem. Finally, we introduce a modeling language for mixed integer programs that especially focuses on the challenges and features that appear in participatory optimization problems similar to the French keyboard design process.Diese Arbeit behandelt Ansätze zu Optimierungsproblemen im Bereich Human-Computer Interaction. Zuerst diskutieren wir Tastaturbelegungsprobleme mit einem besonderen Fokus auf einem Projekt, an dem wir teilgenommen haben: die Erstellung eines neuen Standards für die französische Tastatur. Wir gehen auf die besondere Struktur dieses Problems und unseren algorithmischen Beitrag ein: ein Algorithmus, der mit Optimierungsmethoden die Struktur dieses speziellen Problems ausnutzt. Mithilfe dieses Algorithmus konnten wir effizient Tastaturbelegungen berechnen und die Qualität dieser Belegungen effektiv (in Form von unteren Schranken) nachweisen. Das finale optimierte Layout, welches mit unserer Methode bewiesenermaßen nahezu optimal ist, diente als Grundlage für den kürzlich in der französischen Nationalversammlung veröffentlichten neuen französischen Tastaturstandard. Darüberhinaus beschreiben wir eine Relaxierung für das quadratische Zuweisungsproblem (eine Verallgemeinerung des Tastaturbelegungsproblems), die auf semidefinieter Programmierung basiert. Wir zeigen, dass unser Algorithmus im Vergleich zu üblich genutzten linearen Relaxierung gut abschneidet. Abschließend definieren und diskutieren wir eine Modellierungssprache für gemischt integrale Programme. Diese Sprache ist speziell auf die besonderen Herausforderungen abgestimmt, die bei interaktiven Optimierungsproblemen auftreten, welche einen ähnlichen Charakter haben wie der Prozess des Designs der französischen Tastatur

Combinatorial Optimization

This report summarizes the meeting on Combinatorial Optimization where new and promising developments in the field were discussed. Th

Leveraging DAGs for Asset Allocation

openThe aim of this thesis is to develop optimization techniques for financial portfolios, in order to exploit information regarding causal relationships between considered financial variables, described through directed acyclic graphs (DAGs) encoding the causal structure underlying the data. More precisely we consider: (i) a budget B; (ii) a set of N_A investible financial assets; (iii) a set of N_F non-investible financial factors, causally determining the evolution of the returns for the considered financial assets. The objective of this thesis is to investigate the utility of causal information in asset allocation tasks by theorizing and testing different models for portfolio optimization. These models should be more resistant to sudden shocks to the market structure and should loose less in terms of performance when the system is subjected to an unpredictable shock. This work is divided into three main chapters: In chapter 2 some theoretical background material about portfolio optimization, DAGs and causality is introduced; in chapter 3 are introduced portfolio optimization models based on Markovitz's framework and afterwards are theorized and explained a series of different methods for asset allocation based on graph clustering techniques; In chapter 4 all the models presented in the previous chapter are tested against a randomly sampled dataset based on the causal structure of the system, both in the static and intervened cases (where a sudden event changed the causal structure), in order to assess their performances before and after a shock occurred and the results obtained are discussed. Results showed the utility of causal information and causal graph structure in asset allocation tasks and causal models proposed proved to be more stable than benchmark ones in case of soft and hard interventions on the system. Future research can be conducted to improve further the methods proposed and to better exploit causal information and graph clustering techniques.The aim of this thesis is to develop optimization techniques for financial portfolios, in order to exploit information regarding causal relationships between considered financial variables, described through directed acyclic graphs (DAGs) encoding the causal structure underlying the data. More precisely we consider: (i) a budget B; (ii) a set of N_A investible financial assets; (iii) a set of N_F non-investible financial factors, causally determining the evolution of the returns for the considered financial assets. The objective of this thesis is to investigate the utility of causal information in asset allocation tasks by theorizing and testing different models for portfolio optimization. These models should be more resistant to sudden shocks to the market structure and should loose less in terms of performance when the system is subjected to an unpredictable shock. This work is divided into three main chapters: In chapter 2 some theoretical background material about portfolio optimization, DAGs and causality is introduced; in chapter 3 are introduced portfolio optimization models based on Markovitz's framework and afterwards are theorized and explained a series of different methods for asset allocation based on graph clustering techniques; In chapter 4 all the models presented in the previous chapter are tested against a randomly sampled dataset based on the causal structure of the system, both in the static and intervened cases (where a sudden event changed the causal structure), in order to assess their performances before and after a shock occurred and the results obtained are discussed. Results showed the utility of causal information and causal graph structure in asset allocation tasks and causal models proposed proved to be more stable than benchmark ones in case of soft and hard interventions on the system. Future research can be conducted to improve further the methods proposed and to better exploit causal information and graph clustering techniques