Sets of Complex Unit Vectors with Two Angles and Distance-Regular Graphs

We study {0,\alpha}-sets, which are sets of unit vectors of $\mathbb{C}^m$ in which any two distinct vectors have angle 0 or \alpha. We investigate some distance-regular graphs that provide new constructions of {0,\alpha}-sets using a method by Godsil and Roy. We prove bounds for the sizes of {0,\alpha}-sets of flat vectors, and characterize all the distance-regular graphs that yield {0,\alpha}-sets meeting the bounds at equality.Comment: 15 page

Optimal dual martingales, their analysis and application to new algorithms for Bermudan products

In this paper we introduce and study the concept of optimal and surely optimal dual martingales in the context of dual valuation of Bermudan options, and outline the development of new algorithms in this context. We provide a characterization theorem, a theorem which gives conditions for a martingale to be surely optimal, and a stability theorem concerning martingales which are near to be surely optimal in a sense. Guided by these results we develop a framework of backward algorithms for constructing such a martingale. In turn this martingale may then be utilized for computing an upper bound of the Bermudan product. The methodology is pure dual in the sense that it doesn't require certain input approximations to the Snell envelope. In an It\^o-L\'evy environment we outline a particular regression based backward algorithm which allows for computing dual upper bounds without nested Monte Carlo simulation. Moreover, as a by-product this algorithm also provides approximations to the continuation values of the product, which in turn determine a stopping policy. Hence, we may obtain lower bounds at the same time. In a first numerical study we demonstrate the backward dual regression algorithm in a Wiener environment at well known benchmark examples. It turns out that the method is at least comparable to the one in Belomestny et. al. (2009) regarding accuracy, but regarding computational robustness there are even several advantages.Comment: This paper is an extended version of Schoenmakers and Huang, "Optimal dual martingales and their stability; fast evaluation of Bermudan products via dual backward regression", WIAS Preprint 157

A Characterization of LYM and Rank Logarithmically Concave Partially Ordered Sets and Its Applications

The LYM property of a finite standard graded poset is one of the central notions in Sperner theory. It is known that the product of two finite standard graded posets satisfying the LYM properties may not have the LYM property again. In 1974, Harper proved that if two finite standard graded posets satisfying the LYM properties also satisfy rank logarithmic concavities, then their product also satisfies these two properties. However, Harper's proof is rather non-intuitive. Giving a natural proof of Harper's theorem is one of the goals of this thesis. The main new result of this thesis is a characterization of rank-finite standard graded LYM posets that satisfy rank logarithmic concavities. With this characterization theorem, we are able to give a new, natural proof of Harper's theorem. In fact, we prove a strengthened version of Harper's theorem by weakening the finiteness condition to the rank-finiteness condition. We present some interesting applications of the main characterization theorem. We also give a brief history of Sperner theory, and summarize all the ingredients we need for the main theorem and its applications, including a new equivalent condition for the LYM property that is a key for proving our main theorem

Bipartite Distance-Regular Graphs of Diameter Four

Using a method by Godsil and Roy, bipartite distance-regular graphs of diameter four can be used to construct $\{0,\alpha\}$-sets, a generalization of the widely applied equiangular sets and mutually unbiased bases. In this thesis, we study the properties of these graphs. There are three main themes of the thesis. The first is the connection between bipartite distance-regular graphs of diameter four and their halved graphs, which are necessarily strongly regular. We derive formulae relating the parameters of a graph of diameter four to those of its halved graphs, and use these formulae to derive a necessary condition for the point graph of a partial geometry to be a halved graph. Using this necessary condition, we prove that several important families of strongly regular graphs cannot be halved graphs. The second theme is the algebraic properties of the graphs. We study Krein parameters as the first part of this theme. We show that bipartite-distance regular graphs of diameter four have one ``special" Krein parameter, denoted by \krein. We show that the antipodal bipartite distance-regular graphs of diameter four with \krein=0 are precisely the Hadamard graphs. In general, we show that a bipartite distance-regular graph of diameter four satisfies \krein=0 if and only if it satisfies the so-called $Q$-polynomial property. In relation to halved graphs, we derive simple formulae for computing the Krein parameters of a halved graph in terms of those of the bipartite graph. As the second part of the algebraic theme, we study Terwilliger algebras. We describe all the irreducible modules of the complex space under the Terwilliger algebra of a bipartite distance-regular graph of diameter four, and prove that no irreducible module can contain two linearly independent eigenvectors of the graph with the same eigenvalue. Finally, we study constructions and bounds of $\{0,\alpha\}$-sets as the third theme. We present some distance-regular graphs that provide new constructions of $\{0,\alpha\}$-sets. We prove bounds for the sizes of $\{0,\alpha\}$-sets of flat vectors, and characterize all the distance-regular graphs that yield $\{0,\alpha\}$-sets meeting the bounds at equality. We also study bipartite covers of linear Cayley graphs, and present a geometric condition and a coding theoretic condition for such a cover to produce $\{0,\alpha\}$-sets. Using simple operations on graphs, we show how new $\{0,\alpha\}$-sets can be constructed from old ones

Reporting and Analysing the Environmental Impact of Language Models on the Example of Commonsense Question Answering with External Knowledge

Human-produced emissions are growing at an alarming rate, causing already observable changes in the climate and environment in general. Each year global carbon dioxide emissions hit a new record, and it is reported that 0.5% of total US greenhouse gas emissions are attributed to data centres as of 2021. The release of ChatGPT in late 2022 sparked social interest in Large Language Models (LLMs), the new generation of Language Models with a large number of parameters and trained on massive amounts of data. Currently, numerous companies are releasing products featuring various LLMs, with many more models in development and awaiting release. Deep Learning research is a competitive field, with only models that reach top performance attracting attention and being utilized. Hence, achieving better accuracy and results is often the first priority, while the model's efficiency and the environmental impact of the study are neglected. However, LLMs demand substantial computational resources and are very costly to train, both financially and environmentally. It becomes essential to raise awareness and promote conscious decisions about algorithmic and hardware choices. Providing information on training time, the approximate carbon dioxide emissions and power consumption would assist future studies in making necessary adjustments and determining the compatibility of available computational resources with model requirements. In this study, we infused T5 LLM with external knowledge and fine-tuned the model for Question-Answering task. Furthermore, we calculated and reported the approximate environmental impact for both steps. The findings demonstrate that the smaller models may not always be sustainable options, and increased training does not always imply better performance. The most optimal outcome is achieved by carefully considering both performance and efficiency factors.Comment: Presented at Bonn Sustainable AI 2023 conferenc