UC Berkeley

Tropical linear algebra is the study of classical linear algebra problems with arithmetic

done over the tropical semiring, namely with addition replaced by max, and multiplication

replaced by addition. It allows one to reformulate nonlinear problems into piecewise-linear

ones. This approach has successfully been employed to solve and characterize solutions to

many problems in combinatorial optimization, control theory and game theory [5]. Tropical

spectral theory, the study of tropical eigenvalues and eigenspaces, often plays a central role

in 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 tropical

eigenvector map. In Chapter 3 we extend this work to cones of linearity of the tropical

eigenspace and polytrope map. Our results contribute to a better understanding of the

polyhedral 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 the

form 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 this

ranking 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 3

provide a complete classification of all possible solution sets to the optimization problem

that TropicalRank solves. This answers open questions from several papers [22, 32] in the

area.

In Chapter 4 we also show that TropicalRank belongs to the same parametrized family

of ranking methods as two other commonly used algorithms, PerronRank and HodgeRank.

Furthermore, we show that HodgeRank and PerronRank asymptotically give the same score

under certain random ranking models. Despite their mathematical connections, we can

construct instances in which these three methods produce arbitrarily different rank order.2

The last two chapters are topics in applied probability. Chapter 5 studies the exact

and asymptotic distribution of size-biased permutations of finite sequences with independent

and identically distributed (i.i.d) terms. The size-biased permutation of a positive summable

sequence (x1 , x2 , . . .) is the same sequence presented in a random order (x[1], x[2], . . .), where

x[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 in

ecology and oil discovery, where species (or oil reserves) are discovered in a random order

proportional to their abundances. In the ranking literature it is known as the Plackett-Luce

model, a parametrized family modeling ranking distributions. Size-biased permutation is

one of the building blocks of combinatorial stochastic processes, an area of probability with

applications to computer science [78]. Finite i.i.d sequence setup serves as a simple model for

successive sampling, or ranking with increasing number of items. We study the size-biased

permutation of such a sequence using two separate methods: Markov chains and induced

order statistics. By going back and forth between the two approaches, we arrive at more

general results with simplified proofs, and provide a Poisson coupling argument which leads

to an explicit formula for the asymptotic distribution of the last few terms in the size-biased

permutation.

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 capacity

relative to the number of neurons. However, known algorithms have produced networks with

linear capacity, and it has been a long-standing open problem whether robust exponential

storage is possible. For a network with n neurons, the problem involves a linear program in

n2 variables and exponentially many constraints. We utilized the action of the symmetric

group on the neuron labels and successfully reduced the problem to a linear program in three

variables and three constraints. Thus we explicitly constructed simple networks that answer

the question affirmatively, with the best possible asymptotic robustness index. This work

calls for further research into Little-Hopfield networks and their applications to theoretical

neuroscience and computer science.