Statistics on Linear Chord Diagrams

Linear chord diagrams are partitions of $\left[2n\right]$ into $n$ blocks of size two called chords. We refer to a block of the form $\{i,i+1\}$ as a short chord. In this paper, we study the distribution of the number of short chords on the set of linear chord diagrams, as a generalization of the Narayana distribution obtained when restricted to the set of noncrossing linear chord diagrams. We provide a combinatorial proof that this distribution is unimodal and has an expected value of one. We also study the number of pairs $(i,i+1)$ where $i$ is the minimal element of a chord and $i+1$ is the maximal element of a chord. We show that the distribution of this statistic on linear chord diagrams corresponds to the second-order Eulerian triangle and is log-concave.Comment: 10 pages, final revision

Ising Graphical Model

The Ising model is an important model in statistical physics, with over 10,000 papers published on the topic. This model assumes binary variables and only local pairwise interactions between neighbouring nodes. Inference for the general Ising model is NP-hard; this includes tasks such as calculating the partition function, finding a lowest-energy (ground) state and computing marginal probabilities. Past approaches have proceeded by working with classes of tractable Ising models, such as Ising models defined on a planar graph. For such models, the partition function and ground state can be computed exactly in polynomial time by establishing a correspondence with perfect matchings in a related graph. In this thesis we continue this line of research. In particular we simplify previous inference algorithms for the planar Ising model. The key to our construction is the complementary correspondence between graph cuts of the model graph and perfect matchings of its expanded dual. We show that our exact algorithms are effective and efficient on a number of real-world machine learning problems. We also investigate heuristic methods for approximating ground states of non-planar Ising models. We show that in this setting our approximative algorithms are superior than current state-of-the-art methods

An optimizational approach for an algorithmic reassembly of fragmented objects

In Cambodia close to the Thai border, lies the Angkor-style temple of Banteay Chhmar. Like all nearly forgotten temples in remote places, it crumbles under the ages. By today most of it is only a heap of stones. Manually reconstructing these temples is both complex and challenging: The conservation team is confronted with a pile of stones, the original position of which is generally unkown. This reassembly task resembles a large-scale 3D puzzle. Usually, it is resolved by a team of specialists who analyze each stone, using their experience and knowledge of Khmer culture. Possible solutions are tried and retried and the stones are placed in different locations until the correct one is found. The major drawbacks of this technique are: First, since the stones are moved continuously they are further damaged, second, there is a threat to the safety of the workers due to handling very heavy weights, and third because of the high complexity and labour-intensity of the work it takes several months up to several years to solve even a small part of the puzzle. These risks and conditions motivated the development of a virtual approach to reassemble the stones, as computer algorithms are theoretically capable of enumerating all potential solutions in less time, thereby drastically reducing the amount of work required for handling the stones. Furthermore the virtual approach has the potential to reduce the on-site costs of in-situ analysis. The basis for this virtual puzzle algorithm are high-resolution 3D models of more than one hundred stones. The stones can be viewed as polytopes with approximately cuboidal form although some of them contain additional indentations. Exploiting these and related geometric features and using a priori knowledge of the orientation of each stone speeds up the process of matching the stones. The aim of the current thesis is to solve this complex large-scale virtual 3D puzzle. In order to achieve this, a general workflow is developed which involves 1) to simplify the high-resolution models to their most characteristic features, 2) apply an advanced similarity analysis and 3) to match best combinations as well as 4) validate the results. The simplification step is necessary to be able to quickly match potential side-surfaces. It introduces the new concept of a minimal volume box (MVB) designed to closely and storage efficiently resemble Khmer stones.Additionally, this reduced edge-based model is used to segment the high-resolution data according to each side-surface. The second step presents a novel technique allowing to conduct a similarity analysis of virtual temple stones. It is based on several geometric distance functions which determine the relatedness of a potential match and is capable of sorting out unlikely ones. The third step employs graph theoretical methods to combine the similarity values into a correct solution of this large-scale 3D puzzle. The validation demonstrates the high quality and robustness of this newly constructed puzzle workflow. The workflow this thesis presents virtually puzzles digitized stones of fallen straight Khmer temple walls. It is able to virtually and correctly reasemble up to 42 digitized stones requiring a minimum of user-interaction