Geometry and complexity of O'Hara's algorithm

In this paper we analyze O'Hara's partition bijection. We present three type of results. First, we show that O'Hara's bijection can be viewed geometrically as a certain scissor congruence type result. Second, we obtain a number of new complexity bounds, proving that O'Hara's bijection is efficient in several special cases and mildly exponential in general. Finally, we prove that for identities with finite support, the map of the O'Hara's bijection can be computed in polynomial time, i.e. much more efficiently than by O'Hara's construction.Comment: 20 pages, 4 figure

Symmetric Edit Lenses: A New Foundation for Bidirectional Languages

Lenses are bidirectional transformations between pairs of connected structures capable of translating an edit on one structure into an edit on the other. Most of the extensive existing work on lenses has focused on the special case of asymmetric lenses, where one structures is taken as primary and the other is thought of as a projection or view. Some symmetric variants exist, where each structure contains information not present in the other, but these all lack the basic operation of composition. Additionally, existing accounts do not represent edits carefully, making incremental operation difficult or producing unsatisfactory synchronization candidates. We present a new symmetric formulation which works with descriptions of changes to structures, rather than with the structures themselves. We construct a semantic space of edit lenses between “editable structures”—monoids of edits with a partial monoid action for applying edits—with natural laws governing their behavior. We present generalizations of a number of known constructions on asymmetric lenses and settle some longstanding questions about their properties—in particular, we prove the existence of (symmetric monoidal) tensor products and sums and the non-existence of full categorical products and sums in a category of lenses. Universal algebra shows how to build iterator lenses for structured data such as lists and trees, yielding lenses for operations like mapping, filtering, and concatenation from first principles. More generally, we provide mapping combinators based on the theory of containers. Finally, we present a prototype implementation of the core theory and take a first step in addressing the challenge of translating between user gestures and the internal representation of edits

Algorithmic Transverse Feedback Linearization

The feedback equivalence problem, that there exists a state and feedback transformation between two control systems, has been used to solve a wide range of problems both in linear and nonlinear control theory. Its significance is in asking whether a particular system can be made equivalent to a possibly simpler system for which the control problem is easier to solve. The equivalence can then be utilized to transform a solution to the simpler control problem into one for the original control system. Transverse feedback linearization is one such feedback equivalence problem. It is a feedback equivalence problem first introduced by Banaszuk and Hauser for feedback linearizing the dynamics transverse to an orbit in the state-space. In particular, it asks to find an equivalence between the original nonlinear control-affine system and two subsystems: one that is nonlinear but acts tangent to the orbit, and another that is a controllable, linear system and acts transverse to the orbit. If this controllable, linear subsystem is stabilized, the original system converges upon the orbit. Nielsen and Maggiore generalized this problem to arbitrary smooth manifolds of the state-space, and produced conditions upon which the problem was solvable. Those conditions do not help in finding the specific transformation required to implement the control design, but they did suggest one method to find the required transformation. It relies on the construction of a mathematical object that is difficult to do without system-specific insight. This thesis proposes an algorithm for transverse feedback linearization that computes the required transformation. Inspired by literature that looked at the feedback linearization and dynamic feedback linearization problems, this work suggests turning to the "dual space" and using a tool known as the derived flag. The algorithm proposed is geometric in nature, and gives a different perspective on, not just transverse feedback linearization, but feedback linearization problems more broadly

Combinatorial number theory through diagramming and gesture

Within combinatorial number theory, we study a variety of problems about whole numbers that include enumerative, diagrammatic, or computational elements. We present results motivated by two different areas within combinatorial number theory: the study of partitions and the study of digital representations of integers. We take the perspective that mathematics research is mathematics learning; existing research from mathematics education on mathematics learning and problem solving can be applied to mathematics research. We illustrate this by focusing on the concept of diagramming and gesture as mathematical practice. The mathematics presented is viewed through this lens throughout the document. Joint with H. E. Burson and A. Straub, motivated by recent results working toward classifying $(s, t)$-core partitions into distinct parts, we present results on certain abaci diagrams. We give a recurrence (on $s$) for generating polynomials for $s$-core abaci diagrams with spacing $d$ and maximum position strictly less than $ms-r$ for positive integers $s$, $d$, $m$, and $r$. In the case $r =1$, this implies a recurrence for $(s, ms-1)$-core partitions into $d$-distinct parts, generalizing several recent results. We introduce the sets $Q(b;\{d_1, d_2, \ldots, d_k\})$ to be integers that can be represented as quotients of integers that can be written in base $b$ using only digits from the set $\{d_1, \ldots, d_k\}$. We explore in detail the sets $Q(b;\{d_1, d_2, \ldots, d_k\})$ where $d_1 = 0$ and the remaining digits form proper subsets of the set $\{1, 2, \ldots, b-1\}$ for the cases $b =3$, $b=4$ and $b=5$. We introduce modified multiplication transducers as a computational tool for studying these sets. We conclude with discussion of $Q(b; \{d_1, \ldots d_k\})$ for general $b$ and digit sets including $\{-1, 0, 1\}$. Sections of this dissertation are written for a nontraditional audience (outside of the academic mathematics research community)

Learning from samples using coherent lower previsions

Het hoofdonderwerp van dit werk is het afleiden, voorstellen en bestuderen van voorspellende en parametrische gevolgtrekkingsmodellen die gebaseerd zijn op de theorie van coherente onderprevisies. Een belangrijk nevenonderwerp is het vinden en bespreken van extreme onderwaarschijnlijkheden. In het hoofdstuk ‘Modeling uncertainty’ geef ik een inleidend overzicht van de theorie van coherente onderprevisies ─ ook wel theorie van imprecieze waarschijnlijkheden genoemd ─ en de ideeën waarop ze gestoeld is. Deze theorie stelt ons in staat onzekerheid expressiever ─ en voorzichtiger ─ te beschrijven. Dit overzicht is origineel in de zin dat ze meer dan andere inleidingen vertrekt van de intuitieve theorie van coherente verzamelingen van begeerlijke gokken. Ik toon in het hoofdstuk ‘Extreme lower probabilities’ hoe we de meest extreme vormen van onzekerheid kunnen vinden die gemodelleerd kunnen worden met onderwaarschijnlijkheden. Elke andere onzekerheidstoestand beschrijfbaar met onderwaarschijnlijkheden kan geformuleerd worden in termen van deze extreme modellen. Het belang van de door mij bekomen en uitgebreid besproken resultaten in dit domein is voorlopig voornamelijk theoretisch. Het hoofdstuk ‘Inference models’ behandelt leren uit monsters komende uit een eindige, categorische verzameling. De belangrijkste basisveronderstelling die ik maak is dat het bemonsteringsproces omwisselbaar is, waarvoor ik een nieuwe definitie geef in termen van begeerlijke gokken. Mijn onderzoek naar de gevolgen van deze veronderstelling leidt ons naar enkele belangrijke representatiestellingen: onzekerheid over (on)eindige rijen monsters kan gemodelleerd worden in termen van categorie-aantallen (-frequenties). Ik bouw hier op voort om voor twee populaire gevolgtrekkingsmodellen voor categorische data ─ het voorspellende imprecies Dirichlet-multinomiaalmodel en het parametrische imprecies Dirichletmodel ─ een verhelderende afleiding te geven, louter vertrekkende van enkele grondbeginselen; deze modellen pas ik toe op speltheorie en het leren van Markov-ketens. In het laatste hoofdstuk, ‘Inference models for exponential families’, verbreed ik de blik tot niet-categorische exponentiële-familie-bemonsteringsmodellen; voorbeelden zijn normale bemonstering en Poisson-bemonstering. Eerst onderwerp ik de exponentiële families en de aanverwante toegevoegde parametrische en voorspellende previsies aan een grondig onderzoek. Deze aanverwante previsies worden gebruikt in de klassieke Bayesiaanse gevolgtrekkingsmodellen gebaseerd op toegevoegd updaten. Ze dienen als grondslag voor de nieuwe, door mij voorgestelde imprecieze-waarschijnlijkheidsgevolgtrekkingsmodellen. In vergelijking met de klassieke Bayesiaanse aanpak, laat de mijne toe om voorzichtiger te zijn bij de beschrijving van onze kennis over het bemonsteringsmodel; deze voorzichtigheid wordt weerspiegeld door het op deze modellen gebaseerd gedrag (getrokken besluiten, gemaakte voorspellingen, genomen beslissingen). Ik toon ten slotte hoe de voorgestelde gevolgtrekkingsmodellen gebruikt kunnen worden voor classificatie door de naïeve credale classificator.This thesis's main subject is deriving, proposing, and studying predictive and parametric inference models that are based on the theory of coherent lower previsions. One important side subject also appears: obtaining and discussing extreme lower probabilities. In the chapter ‘Modeling uncertainty’, I give an introductory overview of the theory of coherent lower previsions ─ also called the theory of imprecise probabilities ─ and its underlying ideas. This theory allows us to give a more expressive ─ and a more cautious ─ description of uncertainty. This overview is original in the sense that ─ more than other introductions ─ it is based on the intuitive theory of coherent sets of desirable gambles. I show in the chapter ‘Extreme lower probabilities’ how to obtain the most extreme forms of uncertainty that can be modeled using lower probabilities. Every other state of uncertainty describable by lower probabilities can be formulated in terms of these extreme ones. The importance of the results in this area obtained and extensively discussed by me is currently mostly theoretical. The chapter ‘Inference models’ treats learning from samples from a finite, categorical space. My most basic assumption about the sampling process is that it is exchangeable, for which I give a novel definition in terms of desirable gambles. My investigation of the consequences of this assumption leads us to some important representation theorems: uncertainty about (in)finite sample sequences can be modeled entirely in terms of category counts (frequencies). I build on this to give an elucidating derivation from first principles for two popular inference models for categorical data ─ the predictive imprecise Dirichlet-multinomial model and the parametric imprecise Dirichlet model; I apply these models to game theory and learning Markov chains. In the last chapter, ‘Inference models for exponential families’, I enlarge the scope to exponential family sampling models; examples are normal sampling and Poisson sampling. I first thoroughly investigate exponential families and the related conjugate parametric and predictive previsions used in classical Bayesian inference models based on conjugate updating. These previsions serve as a basis for the new imprecise-probabilistic inference models I propose. Compared to the classical Bayesian approach, mine allows to be much more cautious when trying to express what we know about the sampling model; this caution is reflected in behavior (conclusions drawn, predictions made, decisions made) based on these models. Lastly, I show how the proposed inference models can be used for classification with the naive credal classifier

Duality, Derivative-Based Training Methods and Hyperparameter Optimization for Support Vector Machines

In this thesis we consider the application of Fenchel's duality theory and gradient-based methods for the training and hyperparameter optimization of Support Vector Machines. We show that the dualization of convex training problems is possible theoretically in a rather general formulation. For training problems following a special structure (for instance, standard training problems) we find that the resulting optimality conditions can be interpreted concretely. This approach immediately leads to the well-known notion of support vectors and a formulation of the Representer Theorem. The proposed theory is applied to several examples such that dual formulations of training problems and associated optimality conditions can be derived straightforwardly. Furthermore, we consider different formulations of the primal training problem which are equivalent under certain conditions. We also argue that the relation of the corresponding solutions to the solution of the dual training problem is not always intuitive. Based on the previous findings, we consider the application of customized optimization methods to the primal and dual training problems. A particular realization of Newton's method is derived which could be used to solve the primal training problem accurately. Moreover, we introduce a general convergence framework covering different types of decomposition methods for the solution of the dual training problem. In doing so, we are able to generalize well-known convergence results for the SMO method. Additionally, a discussion of the complexity of the SMO method and a motivation for a shrinking strategy reducing the computational effort is provided. In a last theoretical part, we consider the problem of hyperparameter optimization. We argue that this problem can be handled efficiently by means of gradient-based methods if the training problems are formulated appropriately. Finally, we evaluate the theoretical results concerning the training and hyperparameter optimization approaches practically by means of several example training problems

Performance modelling and the representation of large scale distributed system functions

This thesis presents a resource based approach to model generation for performance characterization and correctness checking of large scale telecommunications networks. A notion called the timed automaton is proposed and then developed to encapsulate behaviours of networking equipment, system control policies and non-deterministic user behaviours. The states of pooled network resources and the behaviours of resource consumers are represented as continually varying geometric patterns; these patterns form part of the data operated upon by the timed automata. Such a representation technique allows for great flexibility regarding the level of abstraction that can be chosen in the modelling of telecommunications systems. None the less, the notion of system functions is proposed to serve as a constraining framework for specifying bounded behaviours and features of telecommunications systems. Operational concepts are developed for the timed automata; these concepts are based on limit preserving relations. Relations over system states represent the evolution of system properties observable at various locations within the network under study. The declarative nature of such permutative state relations provides a direct framework for generating highly expressive models suitable for carrying out optimization experiments. The usefulness of the developed procedure is demonstrated by tackling a large scale case study, in particular the problem of congestion avoidance in networks; it is shown that there can be global coupling among local behaviours within a telecommunications network. The uncovering of such a phenomenon through a function oriented simulation is a contribution to the area of network modelling. The direct and faithful way of deriving performance metrics for loss in networks from resource utilization patterns is also a new contribution to the work area

Contributions on secretary problems, independent sets of rectangles and related problems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 187-198).We study three problems arising from different areas of combinatorial optimization. We first study the matroid secretary problem, which is a generalization proposed by Babaioff, Immorlica and Kleinberg of the classical secretary problem. In this problem, the elements of a given matroid are revealed one by one. When an element is revealed, we learn information about its weight and decide to accept it or not, while keeping the accepted set independent in the matroid. The goal is to maximize the expected weight of our solution. We study different variants for this problem depending on how the elements are presented and on how the weights are assigned to the elements. Our main result is the first constant competitive algorithm for the random-assignment random-order model. In this model, a list of hidden nonnegative weights is randomly assigned to the elements of the matroid, which are later presented to us in uniform random order, independent of the assignment. The second problem studied is the jump number problem. Consider a linear extension L of a poset P. A jump is a pair of consecutive elements in L that are not comparable in P. Finding a linear extension minimizing the number of jumps is NP-hard even for chordal bipartite posets. For the class of posets having two directional orthogonal ray comparability graphs, we show that this problem is equivalent to finding a maximum independent set of a well-behaved family of rectangles. Using this, we devise combinatorial and LP-based algorithms for the jump number problem, extending the class of bipartite posets for which this problem is polynomially solvable and improving on the running time of existing algorithms for certain subclasses. The last problem studied is the one of finding nonempty minimizers of a symmetric submodular function over any family of sets closed under inclusion. We give an efficient O(ns)-time algorithm for this task, based on Queyranne's pendant pair technique for minimizing unconstrained symmetric submodular functions. We extend this algorithm to report all inclusion-wise nonempty minimal minimizers under hereditary constraints of slightly more general functions.by José Antonio Soto.Ph.D