Topics in Tropical Linear Algebra and Applied Probability

Tropical linear algebra is the study of classical linear algebra problems with arithmeticdone over the tropical semiring, namely with addition replaced by max, and multiplicationreplaced by addition. It allows one to reformulate nonlinear problems into piecewise-linearones. This approach has successfully been employed to solve and characterize solutions tomany problems in combinatorial optimization, control theory and game theory [5]. Tropicalspectral theory, the study of tropical eigenvalues and eigenspaces, often plays a central rolein these applications. We derive the basics of this theory in Chapter 1.In Chapter 2 we give a combinatorial description of the cones of linearity of the tropicaleigenvector map. In Chapter 3 we extend this work to cones of linearity of the tropicaleigenspace and polytrope map. Our results contribute to a better understanding of thepolyhedral foundations of tropical linear algebra.Chapter 4 illustrates the above results in the context of pairwise ranking. Here one as-signs to each pair of candidates a comparison score, and the algorithm produces a cardinal(numerically quantified) ranking of candidates. This setup is natural in sport competitions,business and decision making. The difficulty lies in the existence of inconsistencies of theform A > B > C > A, since pairwise comparisons are performed independently. Tropi-calRank is an algorithm pioneered by Elsner and van den Driessche. Solution sets of thisranking method are precisely the polytropes studied in Chapter 3. For generic input pair-wise comparison matrices, this set contains one unique point that is the tropical eigenvector,which is then interpreted as the comparison score. In particular, the results in Chapter 3provide a complete classification of all possible solution sets to the optimization problemthat TropicalRank solves. This answers open questions from several papers [22, 32] in thearea.In Chapter 4 we also show that TropicalRank belongs to the same parametrized familyof ranking methods as two other commonly used algorithms, PerronRank and HodgeRank.Furthermore, we show that HodgeRank and PerronRank asymptotically give the same scoreunder certain random ranking models. Despite their mathematical connections, we canconstruct instances in which these three methods produce arbitrarily different rank order.2The last two chapters are topics in applied probability. Chapter 5 studies the exactand asymptotic distribution of size-biased permutations of finite sequences with independentand identically distributed (i.i.d) terms. The size-biased permutation of a positive summablesequence (x1 , x2 , . . .) is the same sequence presented in a random order (x[1], x[2], . . .), wherex[1] is defined to be xi with probability proportional to its `size' xi ; given that x[1] is xi ,the next term x[2] is defined to be xj for j = i with probability again proportional to its`size' xj , and so on. Size-biased permutations model the successive sampling method inecology and oil discovery, where species (or oil reserves) are discovered in a random orderproportional to their abundances. In the ranking literature it is known as the Plackett-Lucemodel, a parametrized family modeling ranking distributions. Size-biased permutation isone of the building blocks of combinatorial stochastic processes, an area of probability withapplications to computer science [78]. Finite i.i. d sequence setup serves as a simple model forsuccessive sampling, or ranking with increasing number of items. We study the size-biasedpermutation of such a sequence using two separate methods: Markov chains and inducedorder statistics. By going back and forth between the two approaches, we arrive at moregeneral results with simplified proofs, and provide a Poisson coupling argument which leadsto an explicit formula for the asymptotic distribution of the last few terms in the size-biasedpermutation.Chapter 6 is about the binary Little-Hopfield network. This is an established computa-tional model of neural memory storage and retrieval which can have exponential capacityrelative to the number of neurons. However, known algorithms have produced networks withlinear capacity, and it has been a long-standing open problem whether robust exponentialstorage is possible. For a network with n neurons, the problem involves a linear program inn2 variables and exponentially many constraints. We utilized the action of the symmetricgroup on the neuron labels and successfully reduced the problem to a linear program in threevariables and three constraints. Thus we explicitly constructed simple networks that answerthe question affirmatively, with the best possible asymptotic robustness index. This workcalls for further research into Little-Hopfield networks and their applications to theoreticalneuroscience and computer science.