Euclidean distance geometry and applications

Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the input data consists of an incomplete set of distances, and the output is a set of points in Euclidean space that realizes the given distances. We survey some of the theory of Euclidean distance geometry and some of the most important applications: molecular conformation, localization of sensor networks and statics.Comment: 64 pages, 21 figure

Building Information Filtering Networks with Topological Constraints: Algorithms and Applications

We propose a new methodology for learning the structure of sparse networks from data; in doing so we adopt a dual perspective where we consider networks both as weighted graphs and as simplicial complexes. The proposed learning methodology belongs to the family of preferential attachment algorithms, where a network is extended by iteratively adding new vertices. In the conventional preferential attachment algorithm a new vertex is added to the network by adding a single edge to another existing vertex; in our approach a new vertex is added to a set of vertices by adding one or more new simplices to the simplicial complex. We propose the use of a score function to quantify the strength of the association between the new vertex and the attachment points. The methodology performs a greedy optimisation of the total score by selecting, at each step, the new vertex and the attachment points that maximise the gain in the score. Sparsity is enforced by restricting the space of the feasible configurations through the imposition of topological constraints on the candidate networks; the constraint is fulfilled by allowing only topological operations that are invariant with respect to the required property. For instance, if the topological constraint requires the constructed network to be be planar, then only planarity-invariant operations are allowed; if the constraint is that the network must be a clique forest, then only simplicial vertices can be added. At each step of the algorithm, the vertex to be added and the attachment points are those that provide the maximum increase in score while maintaining the topological constraints. As a concrete but general realisation we propose the clique forest as a possible topological structure for the representation of sparse networks, and we allow to specify further constraints such as the allowed range of clique sizes and the saturation of the attachment points. In this thesis we originally introduce the Maximally Filtered Clique Forest (MFCF) algorithm: the MFCF builds a clique forest by repeated application of a suitably invariant operation that we call Clique Expansion operator and adds vertices according to a strategy that greedily maximises the gain in a local score function. The gains produced by the Clique Expansion operator can be validated in a number of ways, including statistical testing, cross-validation or value thresholding. The algorithm does not prescribe a specific form for the gain function, but allows the use of any number of gain functions as long as they are consistent with the Clique Expansion operator. We describe several examples of gain functions suited to different problems. As a specific practical realisation we study the extraction of planar networks with the Triangulated Maximally Filtered Graph (TMFG). The TMFG, in its simplest form, is a specialised version of the MFCF, but it can be made more powerful by allowing the use of specialised planarity invariant operators that are not based on the Clique Expansion operator. We provide applications to two well known applied problems: the Maximum Weight Planar Subgraph Problem (MWPSP) and the Covariance Selection problem. With regards to the Covariance Selection problem we compare our results to the state of the art solution (the Graphical Lasso) and we highlight the benefits of our methodology. Finally, we study the geometry of clique trees as simplicial complexes and note how the statistics based on cliques and separators provides information equivalent to the one that can be achieved by means of homological methods, such as the analysis of Betti numbers, however with our approach being computationally more efficient and intuitively simpler. Finally, we use the geometric tools developed to provide a possible methodology for inferring the size of a dataset generated by a factor model. As an example we show that our tools provide a solution for inferring the size of a dataset generated by a factor model

Receding-horizon switched linear system design: a semidefinite programming approach with distributed computation

This dissertation presents a framework for analysis and controller synthesis problems for switched linear systems. These are multi-modal systems whose parameters vary within a finite set according to the state of a discrete time automaton; the switching signal may be unconstrained or may be drawn from a language of admissible switching signals. This model of system dynamics and discrete logic has many applications in a number of engineering contexts. A receding-horizon type approach is taken by designing controllers with access to a finite-length preview of future modes and finite memory of past modes; the length of both preview and memory are taken as design choices. The results developed here take the form of nested sequences of SDP feasibility problems. These conditions are exact in that the feasibility of any element of the sequence is sufficient to construct a suitable controller, while the existence of a suitable controller necessitates the feasibility of some element of the sequence. Considered first is the problem of controller synthesis for the stabilization of switched systems. These developments serve both as a control problem of interest and a demonstration of the methods used to solve subsequent switched control problems. Exact conditions for the existence of a controller are developed, along with converse results which rule out levels of closed-loop stability based on the infeasibility of individual SDP problems. This permits the achievable closed-loop performance level to be approximated to arbitrary accuracy. Examined next are two different performance problems: one of disturbance attenuation and one of windowed variance. For each problem, controller synthesis conditions are presented exactly in the form of SDP feasibility problems which may be optimized to determine levels of performance. In both cases, the performance level may be taken as uniform or allowed to vary based on the switching path encountered. The controller synthesis conditions presented here can grow both large and computationally intensive, but they share a common structural sparsity which may be exploited. The last part of this dissertation examines this structure and presents a distributed approach to solving such problems. This maintains the tractability of these results even at large scales, expanding the scope of systems to which these methods can be applied

Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System

Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics