On Weak Higher Dimensional Categories I: Part 1 (
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New Multicategory Boosting Algorithms Based on Multicategory Fisher-Consistent Losses by Hui
The Annals of Applied Statistics 2008, Vol. 2, No. 4, 1290–1306 DOI: 10.1214/08-AOAS198 © Institute of Mathematical Statistics, 2008 NEW MULTICATEGORY BOOSTING ALGORITHMS BASED ON MULTICATEGORY FISHER-CONSISTENT LOSSES BY HUI ZOU1,JI ZHU AND TREVOR HASTIE University of Minnesota , University of Michigan and Stanford University Fisher-consistent loss functions play a fundamental role in the construc- tion of successful binary margin-based classifiers. In this paper we establish the Fisher-consistency condition for multicategory classification problems. Our approach uses the margin vector concept which can be regarded as a multicategory generalization of the binary margin. We characterize a wide class of smooth convex loss functions that are Fisher-consistent for multi- category classification. We then consider using the margin-vector-based loss functions to derive multicategory boosting algorithms. In particular, we de- rive two new multicategory boosting algorithms by using the exponential and logistic regression losses. 1. Introduction. The margin-based classifiers, including the support vector machine (SVM) [Vapnik (1996)] and boosting [Freund and Schapire (1997)], have demonstrated their excellent performances in binary classification problems. Re- cent statistical theory regards binary margin-based classifiers as regularized em- pirical risk minimizers with proper loss functions. Friedman, Hastie and Tibshi- rani (2000) showed that AdaBoost minimizes the novel exponential loss by fitting a forward stage-wise additive model. In the same spirit, Lin (2002) showed that the SVM solves a penalized hinge loss problem and the population minimizer of the hinge loss is exactly the Bayes rule, thus, the SVM directly approximates the Bayes rule without estimating the conditional class probability. -
Generalized Enrichment of Categories
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 168 (2002) 391–406 www.elsevier.com/locate/jpaa Generalized enrichment of categories Tom Leinster Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK Received 1 December 1999; accepted 4 May 2001 Abstract We deÿne the phrase ‘category enriched in an fc-multicategory’ and explore some exam- ples. An fc-multicategory is a very general kind of two-dimensional structure, special cases of which are double categories, bicategories, monoidal categories and ordinary multicategories. Enrichment in an fc-multicategory extends the (more or less well-known) theories of enrichment in a monoidal category, in a bicategory, and in a multicategory. Moreover, fc-multicategories provide a natural setting for the bimodules construction, traditionally performed on suitably co- complete bicategories. Although this paper is elementary and self-contained, we also explain why, from one point of view, fc-multicategories are the natural structures in which to enrich categories. c 2001 Elsevier Science B.V. All rights reserved. MSC: 18D20; 18D05; 18D50; 18D10 A general question in category theory is: given some kind of categorical structure, what might it be enriched in? For instance, suppose we take braided monoidal cat- egories. Then the question asks: what kind of thing must V be if we are to speak sensibly of V-enriched braided monoidal categories? (The usual answer is that V must be a symmetricmonoidal category.) In another paper, [7], I have given an answer to the general question for a certain family of categorical structures (generalized multicategories). -
On the Geometric Realization of Dendroidal Sets
Fabio Trova On the Geometric Realization of Dendroidal Sets Thesis advisor: Prof. Ieke Moerdijk Leiden University MASTER ALGANT University of Padova Et tu ouvriras parfois ta fenˆetre, comme ¸ca,pour le plaisir. Et tes amis seront bien ´etonn´esde te voir rire en regardant le ciel. Alors tu leur diras: “Oui, les ´etoiles,¸came fait toujours rire!” Et ils te croiront fou. Je t’aurai jou´eun bien vilain tour. A Irene, Lorenzo e Paolo a chi ha fatto della propria vita poesia a chi della poesia ha fatto la propria vita Contents Introduction vii Motivations and main contributions........................... vii 1 Category Theory1 1.1 Categories, functors, natural transformations...................1 1.2 Adjoint functors, limits, colimits..........................4 1.3 Monads........................................7 1.4 More on categories of functors............................ 10 1.5 Monoidal Categories................................. 13 2 Simplicial Sets 19 2.1 The Simplicial Category ∆ .............................. 19 2.2 The category SSet of Simplicial Sets........................ 21 2.3 Geometric Realization................................ 23 2.4 Classifying Spaces.................................. 28 3 Multicategory Theory 29 3.1 Trees.......................................... 29 3.2 Planar Multicategories................................ 31 3.3 Symmetric multicategories.............................. 34 3.4 (co)completeness of Multicat ............................. 37 3.5 Closed monoidal structure in Multicat ....................... 40 4 Dendroidal Sets 43 4.1 The dendroidal category Ω .............................. 43 4.1.1 Algebraic definition of Ω ........................... 44 4.1.2 Operadic definition of Ω ........................... 45 4.1.3 Equivalence of the definitions........................ 46 4.1.4 Faces and degeneracies............................ 48 4.2 The category dSet of Dendroidal Sets........................ 52 4.3 Nerve of a Multicategory............................... 55 4.4 Closed Monoidal structure on dSet ........................ -
Models of Classical Linear Logic Via Bifibrations of Polycategories
Models of Classical Linear Logic via Bifibrations of Polycategories N. Blanco and N. Zeilberger School of Computer Science University of Birmingham, UK SYCO5, September 2019 N. Blanco and N. Zeilberger ( School of ComputerModels Science of Classical University Linear of Logic Birmingham, via Bifibrations UK ) of PolycategoriesSYCO5, September 2019 1 / 27 Outline 1 Multicategories and Monoidal categories 2 Opfibration of Multicategories 3 Polycategories and Linearly Distributive Categories 4 Bifibration of polycategories N. Blanco and N. Zeilberger ( School of ComputerModels Science of Classical University Linear of Logic Birmingham, via Bifibrations UK ) of PolycategoriesSYCO5, September 2019 2 / 27 Multicategories and Monoidal categories Outline 1 Multicategories and Monoidal categories N. Blanco and N. Zeilberger ( School of ComputerModels Science of Classical University Linear of Logic Birmingham, via Bifibrations UK ) of PolycategoriesSYCO5, September 2019 3 / 27 Multicategories and Monoidal categories Tensor product of vector spaces In linear algebra: universal property C A; B A ⊗ B In category theory as a structure: a monoidal product ⊗ Universal property of tensor product needs many-to-one maps Category with many-to-one maps ) Multicategory N. Blanco and N. Zeilberger ( School of ComputerModels Science of Classical University Linear of Logic Birmingham, via Bifibrations UK ) of PolycategoriesSYCO5, September 2019 3 / 27 Multicategories and Monoidal categories Multicategory1 Definition A multicategory M has: A collection of objects Γ finite list of objects and A objects Set of multimorphisms M(Γ; A) Identities idA : A ! A f :Γ ! A g :Γ ; A; Γ ! B Composition: 1 2 g ◦i f :Γ1; Γ; Γ2 ! B With usual unitality and associativity and: interchange law: (g ◦ f1) ◦ f2 = (g ◦ f2) ◦ f1 where f1 and f2 are composed in two different inputs of g 1Tom Leinster. -
Tom Leinster
Theory and Applications of Categories, Vol. 12, No. 3, 2004, pp. 73–194. OPERADS IN HIGHER-DIMENSIONAL CATEGORY THEORY TOM LEINSTER ABSTRACT. The purpose of this paper is to set up a theory of generalized operads and multicategories and to use it as a language in which to propose a definition of weak n-category. Included is a full explanation of why the proposed definition of n- category is a reasonable one, and of what happens when n ≤ 2. Generalized operads and multicategories play other parts in higher-dimensional algebra too, some of which are outlined here: for instance, they can be used to simplify the opetopic approach to n-categories expounded by Baez, Dolan and others, and are a natural language in which to discuss enrichment of categorical structures. Contents Introduction 73 1 Bicategories 77 2 Operads and multicategories 88 3 More on operads and multicategories 107 4 A definition of weak ω-category 138 A Biased vs. unbiased bicategories 166 B The free multicategory construction 177 C Strict ω-categories 180 D Existence of initial operad-with-contraction 189 References 192 Introduction This paper concerns various aspects of higher-dimensional category theory, and in partic- ular n-categories and generalized operads. We start with a look at bicategories (Section 1). Having reviewed the basics of the classical definition, we define ‘unbiased bicategories’, in which n-fold composites of 1-cells are specified for all natural n (rather than the usual nullary and binary presentation). We go on to show that the theories of (classical) bicategories and of unbiased bicategories are equivalent, in a strong sense. -
Reinforced Multicategory Support Vector Machines
Supplementary materials for this article are available online. PleaseclicktheJCGSlinkathttp://pubs.amstat.org. Reinforced Multicategory Support Vector Machines Yufeng L IU and Ming YUAN Support vector machines are one of the most popular machine learning methods for classification. Despite its great success, the SVM was originally designed for binary classification. Extensions to the multicategory case are important for general classifica- tion problems. In this article, we propose a new class of multicategory hinge loss func- tions, namely reinforced hinge loss functions. Both theoretical and numerical properties of the reinforced multicategory SVMs (MSVMs) are explored. The results indicate that the proposed reinforced MSVMs (RMSVMs) give competitive and stable performance when compared with existing approaches. R implementation of the proposed methods is also available online as supplemental materials. Key Words: Fisher consistency; Multicategory classification; Regularization; SVM. 1. INTRODUCTION Classification is a very important statistical task for information extraction from data. Among numerous classification techniques, the Support Vector Machine (SVM) is one of the most well-known large-margin classifiers and has achieved great success in many ap- plications (Boser, Guyon, and Vapnik 1992; Cortes and Vapnik 1995). The basic concept behind the binary SVM is to find a separating hyperplane with maximum separation be- tween the two classes. Because of its flexibility in estimating the decision boundary using kernel learning as well as its ability in handling high-dimensional data, the SVM has be- come a very popular classifier and has been widely applied in many different fields. More details about the SVM can be found, for example, in the works of Cristianini and Shawe- Taylor (2000), Hastie, Tibshirani, and Friedman (2001), Schölkopf and Smola (2002). -
Locally Cartesian Closed Categories, Coalgebras, and Containers
U.U.D.M. Project Report 2013:5 Locally cartesian closed categories, coalgebras, and containers Tilo Wiklund Examensarbete i matematik, 15 hp Handledare: Erik Palmgren, Stockholms universitet Examinator: Vera Koponen Mars 2013 Department of Mathematics Uppsala University Contents 1 Algebras and Coalgebras 1 1.1 Morphisms .................................... 2 1.2 Initial and terminal structures ........................... 4 1.3 Functoriality .................................... 6 1.4 (Co)recursion ................................... 7 1.5 Building final coalgebras ............................. 9 2 Bundles 13 2.1 Sums and products ................................ 14 2.2 Exponentials, fibre-wise ............................. 18 2.3 Bundles, fibre-wise ................................ 19 2.4 Lifting functors .................................. 21 2.5 A choice theorem ................................. 22 3 Enriching bundles 25 3.1 Enriched categories ................................ 26 3.2 Underlying categories ............................... 29 3.3 Enriched functors ................................. 31 3.4 Convenient strengths ............................... 33 3.5 Natural transformations .............................. 41 4 Containers 45 4.1 Container functors ................................ 45 4.2 Natural transformations .............................. 47 4.3 Strengths, revisited ................................ 50 4.4 Using shapes ................................... 53 4.5 Final remarks ................................... 56 i Introduction -
Equivalence of Multicategory SVM and Simplex Cone SVM: Fast Computations and Statistical Theory
Equivalence of Multicategory SVM and Simplex Cone SVM: Fast Computations and Statistical Theory Guillaume A. Pouliot 1 Abstract attempts have been made at extending the methodology to The multicategory SVM (MSVM) of Lee et al. accommodate classification with K>2 categories (Sun et (2004) is a natural generalization of the classical, al., 2017; Dogan et al., 2016; Lopez et al., 2016; Kumar binary support vector machines (SVM). However, et al., 2017, survey available in Ma and Guo, 2014). Lee, its use has been limited by computational diffi- Lin and Wahba (2004) propose what is arguably the natural culties. The simplex-cone SVM (SCSVM) of multicategory generalization of binary SVM. For instance, Mroueh et al. (2012) is a computationally ef- their multicategory SVM (MSVM) is Fisher consistent (i.e., ficient multicategory classifier, but its use has the classification rule it produces converges to the Bayes been limited by a seemingly opaque interpreta- rule), which is a key property and motivation for the use tion. We show that MSVM and SCSVM are in standard SVM. Furthermore, it encompasses standard SVM fact exactly equivalent, and provide a bijection as a special case. between their tuning parameters. MSVM may However, the method has not been widely used in applica- then be entertained as both a natural and com- tion, nor has it been studied from a statistical perspective, putationally efficient multicategory extension of the way SVM has been. Amongst the machine learning SVM. We further provide a Donsker theorem for community, MSVM has not gathered popularity commen- finite-dimensional kernel MSVM and partially surate to that of SVM. -
In Praise of Quaternions
IN PRAISE OF QUATERNIONS JOACHIM LAMBEK With an appendix on the algebra of biquaternions Michael Barr Abstract. This is a survey of some of the applications of quaternions to physics in the 20th century. In the first half century, an elegant presentation of Maxwell's equations and special relativity was achieved with the help of biquaternions, that is, quaternions with complex coefficients. However, a quaternionic derivation of Dirac's celebrated equation of the electron depended on the observation that all 4 × 4 real matrices can be generated by quaternions and their duals. On examine quelques applications des quaternions `ala physique du vingti`emesi`ecle.Le premier moiti´edu si`ecleavait vu une pr´esentation ´el´egantes des equations de Maxwell et de la relativit´e specialle par les quaternions avec des coefficients complexes. Cependant, une d´erivation de l'´equationc´el`ebrede Dirac d´ependait sur l'observation que toutes les matrices 4 × 4 r´eelles peuvent ^etre gener´eespar les representations reguli`eresdes quaternions. 1. Prologue. This is an expository article attempting to acquaint algebraically inclined readers with some basic notions of modern physics, making use of Hamilton's quaternions rather than the more sophisticated spinor calculus. While quaternions play almost no r^olein mainstream physics, they afford a quick entry into the world of special relativity and allow one to formulate the Maxwell-Lorentz theory of electro-magnetism and the Dirac equation of the electron with a minimum of mathematical prerequisites. Marginally, quaternions even give us a glimpse of the Feynman diagrams appearing in the standard model. As everyone knows, quaternions were invented (discovered?) by William Rowan Hamilton. -
Cyclic Multicategories, Multivariable Adjunctions and Mates
J. K-Theory 13 (2014), 337–396 ©2013 ISOPP doi:10.1017/is013012007jkt250 Cyclic multicategories, multivariable adjunctions and mates by EUGENIA CHENG,NICK GURSKI AND EMILY RIEHL Abstract A multivariable adjunction is the generalisation of the notion of a 2-variable adjunction, the classical example being the hom/tensor/cotensor trio of functors, to n C 1 functors of n variables. In the presence of multivariable adjunctions, natural transformations between certain composites built from multivariable functors have “dual” forms. We refer to corresponding natural transformations as multivariable or parametrised mates, generalising the mates correspondence for ordinary adjunctions, which enables one to pass between natural transformations involving left adjoints to those involving right adjoints. A central problem is how to express the naturality (or functoriality) of the parametrised mates, giving a precise characterization of the dualities so-encoded. We present the notion of “cyclic double multicategory” as a structure in which to organise multivariable adjunctions and mates. While the standard mates correspondence is described using an isomorphism of double categories, the multivariable version requires the framework of “double multicategories”. Moreover, we show that the analogous isomorphisms of double multicate- gories give a cyclic action on the multimaps, yielding the notion of “cyclic double multicategory”. The work is motivated by and applied to Riehl’s approach to algebraic monoidal model categories. Key Words: Adjunctions, mates, multicategories. Mathematics Subject Classification 2010: 18A40, 18D99. Contents 1 Mates 343 2 Multivariable adjunctions 350 2.1 Definition of multivariable adjunctions . ............. 351 2.2 Amotivatingexample........................ 358 2.3 Composition ............................. 359 2.4 Multivariablemates.......................... 365 338 E. CHENG,N.GURSKI &E.RIEHL 3 Cyclic double multicategories 374 3.1 Plainmulticategories........................ -
Operator Categories, Multicategories, and Homotopy Coherent Algebra
OPERATOR CATEGORIES, MULTICATEGORIES, AND HOMOTOPY COHERENT ALGEBRA by Clark Barwick What's all this then?| Here are revised notes for my ill-fated talk in the Norwegian Topology Symposium in Bergen in June, 2007. As a result of poor planning, a tedious narrative structure, and a basic lack of familiarity with computer-based presentations, I did not manage to explore nearly as much material as I had intended. This was, for me, a genuine loss, as I was excited to see what interesting applications \real" topologists might find for my little pet project. As penance for my failings as a speaker, I've spent the last three months or so revising and expanding the slides from the talk; here is the result. I have included more details and examples throughout, and I have interspersed a fair number of friendly, informal comments and attempts at humor in order to cut down the inevitable turgidity of the prose. The detailed proofs of the results announced here will be compiled elsewhere | mostly as threads developed in [3] and [4] |, and for the most part, I feel satisfied with sketching the proofs of the more surprising results here. As the completion and revision of the book have taken far longer than expected, this short note can be seen as a kind of progress report on my work so far in these areas. I never did finish writing up these notes, and they haven't been edited in a year. I'm making them available publicly mostly to express the ideas. Some introductory remarks. | The aim of the theory of operator categories is to provide a new set of \sharper tools" (to borrow an expression from Hilbert) for discussing the interaction between algebraic structures and homotopy theory. -
From Rules of Grammar to Laws of Nature
From rules of grammar to laws of nature Joachim Lambek McGill University, Montreal December 3, 2013 Abstract We look at questions debated by pre-Socratic philosophers, emphasize their linguistic motivation and point out attempted answers by modern physics. We pursue a math- ematical treatment of the syntax and morphology of natural languages, investigate the foundations of mathematics and speculate about a quaternionic approach to special rela- tivistic quantum mechanics. 0 Introduction Many questions about the nature of the world are concealed in our language, even if we are not always aware of the hidden linguistic assumptions. I believe that most, if not all philosophi- cal investigations are motivated by reflections on the structure of language. For example, the intense debate by pre-Socratic philosophers whether matter is continuous or discrete is antici- pated by grammatical distinctions, still revealed by those between mass nouns and count nouns in English. Modern physics seems to have resolved this question in favour of the latter, although there still is some disagreement about the nature of space and time, as already discussed by the Greek philosopher Zeno. As far as we know, language was invented between fifty and a hundred thousand years ago by a small group of humanoid primates in Africa, allowing them to spread over the whole globe and grow to several billions in number. It is believed by some experts that all present day languages have a common origin. Modern English is a member of the Germanic branch of the Indo-European family of lan- guages, which stretch all the way from Iceland to Ceylon and are still recognizably related.