Topics in Tropical Linear Algebra and Applied Probability

Tropical linear algebra is the study of classical linear algebra problems with arithmetic/ndone over the tropical semiring, namely with addition replaced by max, and multiplication/nreplaced by addition. It allows one to reformulate nonlinear problems into piecewise-linear/nones. This approach has successfully been employed to solve and characterize solutions to/nmany problems in combinatorial optimization, control theory and game theory [5]. Tropical/nspectral theory, the study of tropical eigenvalues and eigenspaces, often plays a central role/nin these applications. We derive the basics of this theory in Chapter 1./nIn Chapter 2 we give a combinatorial description of the cones of linearity of the tropical/neigenvector map. In Chapter 3 we extend this work to cones of linearity of the tropical/neigenspace and polytrope map. Our results contribute to a better understanding of the/npolyhedral foundations of tropical linear algebra./nChapter 4 illustrates the above results in the context of pairwise ranking. Here one as-/nsigns to each pair of candidates a comparison score, and the algorithm produces a cardinal/n(numerically quantified) ranking of candidates. This setup is natural in sport competitions,/nbusiness and decision making. The difficulty lies in the existence of inconsistencies of the/nform A > B > C > A, since pairwise comparisons are performed independently. Tropi-/ncalRank is an algorithm pioneered by Elsner and van den Driessche. Solution sets of this/nranking method are precisely the polytropes studied in Chapter 3. For generic input pair-/nwise comparison matrices, this set contains one unique point that is the tropical eigenvector,/nwhich is then interpreted as the comparison score. In particular, the results in Chapter 3/nprovide a complete classification of all possible solution sets to the optimization problem/nthat TropicalRank solves. This answers open questions from several papers [22, 32] in the/narea./nIn Chapter 4 we also show that TropicalRank belongs to the same parametrized family/nof ranking methods as two other commonly used algorithms, PerronRank and HodgeRank./nFurthermore, we show that HodgeRank and PerronRank asymptotically give the same score/nunder certain random ranking models. Despite their mathematical connections, we can/nconstruct instances in which these three methods produce arbitrarily different rank order.2/nThe last two chapters are topics in applied probability. Chapter 5 studies the exact/nand asymptotic distribution of size-biased permutations of finite sequences with independent/nand identically distributed (i.i.d) terms. The size-biased permutation of a positive summable/nsequence (x1 , x2 , . . .) is the same sequence presented in a random order (x[1], x[2], . . .), where/nx[1] is defined to be xi with probability proportional to its `size' xi ; given that x[1] is xi ,/nthe next term x[2] is defined to be xj for j = i with probability again proportional to its/n`size' xj , and so on. Size-biased permutations model the successive sampling method in/necology and oil discovery, where species (or oil reserves) are discovered in a random order/nproportional to their abundances. In the ranking literature it is known as the Plackett-Luce/nmodel, a parametrized family modeling ranking distributions. Size-biased permutation is/none of the building blocks of combinatorial stochastic processes, an area of probability with/napplications to computer science [78]. Finite i.i.d sequence setup serves as a simple model for/nsuccessive sampling, or ranking with increasing number of items. We study the size-biased/npermutation of such a sequence using two separate methods: Markov chains and induced/norder statistics. By going back and forth between the two approaches, we arrive at more/ngeneral results with simplified proofs, and provide a Poisson coupling argument which leads/nto an explicit formula for the asymptotic distribution of the last few terms in the size-biased/npermutation./nChapter 6 is about the binary Little-Hopfield network. This is an established computa-/ntional model of neural memory storage and retrieval which can have exponential capacity/nrelative to the number of neurons. However, known algorithms have produced networks with/nlinear capacity, and it has been a long-standing open problem whether robust exponential/nstorage is possible. For a network with n neurons, the problem involves a linear program in/nn2 variables and exponentially many constraints. We utilized the action of the symmetric/ngroup on the neuron labels and successfully reduced the problem to a linear program in three/nvariables and three constraints. Thus we explicitly constructed simple networks that answer/nthe question affirmatively, with the best possible asymptotic robustness index. This work/ncalls for further research into Little-Hopfield networks and their applications to theoretical/nneuroscience and computer science.

Advisor: Sturmfels, Bernd