Dissertations / Theses on the topic 'Representations'

This thesis is concerned with the Langlands program; namely the global Langlands correspondence, Langlands functoriality, and a conjecture of Gross. In chapter 1, we cover the most important background material needed for this thesis. This includes material on reductive groups and their root data, the definition of automorphic representations and a general overview of the Langlands program, and Gross' conjecture concerning attaching l-adic Galois representations to automorphic representations on certain reductive groups G over ℚ. In chapter 2, we show that odd-dimensional definite unitary groups satisfy the hypotheses of Gross' conjecture and verify the conjecture in this case using known constructions of automorphic l-adic Galois representations. We do this by verifying a specific case of a generalisation of Gross' conjecture; one should still get l-adic Galois representations if one removes one of his hypotheses but with the cost that their image lies in ^CG(ℚ_l) as opposed to ^LG(ℚ_l). Such Galois representations have been constructed for certain automorphic representations on G, a definite unitary group of arbitrary dimension, and there is a map ^CG(ℚ_l) → ^LG(ℚ_l) precisely when G is odd-dimensional. In chapter 3, which forms the main part of this thesis, we show that G = U_n(B) where B is a rational definite quaternion algebra satisfies the hypotheses of Gross' conjecture. We prove that one can transfer a cuspidal automorphic representation π of G to a π' on Sp_2n (a Jacquet-Langlands type transfer) provided it is Steinberg at some finite place. We also prove this when B is indefinite. One can then transfer π′ to an automorphic representaion of GL_2n+1 using the work of Arthur. Finally, one can attach l-adic Galois representations to these automorphic representations on GL_2n+1, provided we assume π is regular algebraic if B is indefinite, and show that they have orthogonal image.

The goal of the research presented here is to build a natural language processing system for our future natural language applications. We believe that real applications will move our research in Computational Linguistics and Artificial Intelligence forward, and we prefer applications that are attractive to a large number of users. The engine in this system is the wide-coverage grammar Nor- Source, and the research topic is to build a prototype of the natural language processing system. The first part of our research topic is to build a pipeline for our grammar. We present two versions of the pipeline. The first pipeline has unmapped predicates that contain object and event references at the discourse level. Each object and event has an unique identifier in the discourse, and the pipeline performs a simple pronoun resolution. The second pipeline is a pipeline with predicates mapped to a selected domain, and the discourse contains object and event references at the world domain level. The world references are the result of an interpretation with a logic model or a selection of a previous stored situation. The domain ontology predicates are mapped from the underspecified predicates. Both pipelines have a demonstrator and the specified world pipeline’s domain is the classical Box World from Artificial Intelligence. The second part of our research topic is to fill the gap between underspecified predicates and domain specific predicates. The meaning representation produced by NorSource is Minimal Recursion Semantics (MRS), and this representation has underspecified quantifier scope and word senses. We have algorithms for solving the underspecified quantifier scope, but we don’t have algorithms for mapping underspecified predicates to domain specific predicates. The starting point is Vendler’s Aktionsart types. The types have a structure and Moen and Steedman showed that a verb argument can coerce the verb from one Aktionsart type to another. Some verbs have culminating states that are not a part of the surface structure of the sentence. Some verbs have additional structure like sub events and causal relations between sub events. Structure of a verb and coercion of Aktionsart types are outside the scope of an MRS, so we want to incorporate some of these notions into our mapping between underspecified and specified predicates. We use a domain ontology and a mapping algorithm. The ontology contains a collection of concepts with a “is-a” hierarchy, “has” relations, and “use” relation. The ontology also contains templates that use the hierarchy and the relations in order to implement constraints on a predicate and its arguments. The ontology contains complex domain objects that generate possible structures, time points, roles and states. We use the Change Location domain to demonstrate how the mapping algorithm works. We have created a natural language processing system prototype and we have filled the gap between underspecified predicates and domain specific predicates. We can transform our MRSs into expressions in First-Order Logic and reason with them. The tools from the DELPH-IN consortium creates “deep” grammars that offers the meaning representation MRS, and this means that our work can be used by other grammars and languages.

Irreducible representations of a finite group over a field are important because all representations of a group are direct sums of irreducible representations. Maschke tells us that if φ is a representation of the finite group G of order n on the m-dimensional space V over the field K of complex numbers and if U is an invariant subspace of φ, then U has a complementary reducing subspace W . The objective of this thesis is to find all irreducible representations of the dihedral group D2n. The reason we will work with the dihedral group is because it is one of the first and most intuitive non-abelian group we encounter in abstract algebra. I will compute the representations and characters of D2n and my thesis will be an explanation of these computations. When n = 2k + 1 we will show that there are k + 2 irreducible representations of D2n, but when n = 2k we will see that D2n has k + 3 irreducible rep- resentations. To achieve this we will first give some background in group, ring, module, and vector space theory that is used in representation theory. We will then explain what general representation theory is. Finally we will show how we arrived at our conclusion.

The English language has become an international language and is globally used as a lingua franca. Therefore, there has been a shift in English-language education toward teaching English as an interna-tional language (EIL). Teaching from the EIL paradigm means that English is seen as an international language used in communication by people from different linguistic and cultural backgrounds. As the approach to English-language education changes from the traditional native-speaker, target country context, so does the role of culture within English-language teaching. The aim of this thesis is to in-vestigate and analyse cultural representations in two Swedish EFL textbooks used in upper-secondary school to see how they correspond with the EIL paradigm. This is done by focusing on the geograph-ical origin of the cultural content as well as looking at what kinds of culture are represented in the textbooks. A content analysis of the textbooks is conducted, using Kachru’s Concentric Circles of English as the model for the analysis of the geographical origin. Horibe’s model of the three different kinds of culture in EIL is the model used for coding the second part of the analysis. The results of the analysis show that culture of target countries and "Culture as social custom" dominate the cultural content of the textbook. Thus, although there are some indications that the EIL paradigm has influ-enced the textbooks, the traditional approach to culture in language teaching still prevails in the ana-lysed textbooks. Because of the relatively small sample included in the thesis, further studies need to be conducted in order to make conclusions regarding the Swedish context as a whole.

Given the "right" representation, learning is easy. This thesis studies representation learning and meta-learning, with a special focus on sparse representations. Meta-learning is fundamental to machine learning, and it translates to learning to learn itself. The presentation unfolds in two parts. In the first part, we establish learning theoretic results for learning sparse representations. The second part introduces new multi-task and meta-learning paradigms for representation learning. On the sparse representations front, our main pursuits are generalization error bounds to support a supervised dictionary learning model for Lasso-style sparse coding. Such predictive sparse coding algorithms have been applied with much success in the literature; even more common have been applications of unsupervised sparse coding followed by supervised linear hypothesis learning. We present two generalization error bounds for predictive sparse coding, handling the overcomplete setting (more original dimensions than learned features) and the infinite-dimensional setting. Our analysis led to a fundamental stability result for the Lasso that shows the stability of the solution vector to design matrix perturbations. We also introduce and analyze new multi-task models for (unsupervised) sparse coding and predictive sparse coding, allowing for one dictionary per task but with sharing between the tasks' dictionaries. The second part introduces new meta-learning paradigms to realize unprecedented types of learning guarantees for meta-learning. Specifically sought are guarantees on a meta-learner's performance on new tasks encountered in an environment of tasks. Nearly all previous work produced bounds on the expected risk, whereas we produce tail bounds on the risk, thereby providing performance guarantees on the risk for a single new task drawn from the environment. The new paradigms include minimax multi-task learning (minimax MTL) and sample variance penalized meta-learning (SVP-ML). Regarding minimax MTL, we provide a high probability learning guarantee on its performance on individual tasks encountered in the future, the first of its kind. We also present two continua of meta-learning formulations, each interpolating between classical multi-task learning and minimax multi-task learning. The idea of SVP-ML is to minimize the task average of the training tasks' empirical risks plus a penalty on their sample variance. Controlling this sample variance can potentially yield a faster rate of decrease for upper bounds on the expected risk of new tasks, while also yielding high probability guarantees on the meta-learner's average performance over a draw of new test tasks. An algorithm is presented for SVP-ML with feature selection representations, as well as a quite natural convex relaxation of the SVP-ML objective.

Theory and research results about the structure of social representations have been built usually from the study of isolated representations. The studies aiming at identifying structural relations involving two or more representations are more recent. In the literature, different terms have been employed to refer to inter-related social representation sets, managed or not by a superior ideological stance; in those cases, we refer to representation families, systems or networks. In this context, there are coordination relations, in which associations can be identified at the same level of a social thinking architecture. Traditionally, the studies of representations in conjunction – presenting element intersections at the level of their cores or object labels – have been privileged. The present research aims at contributing to the knowledge about disjoint representations. When, at an inter-representation level, there are two or more representation structures linked by relations between cognems or between object-label words, we propose the name of representational system to the macrostructure that is formed, with relations formalized by the basic cognitive schemes model. The main research aim was to identify relations among elements of different social representations, submitting to test the existence of connection points among representations at the same level of the social thinking architecture. It is a model that conceives inter-representation relations at the level of cognems. All the studies concerned the investigation of a representational system including the social representation on aging. Most of the studies were conducted with university undergraduate samples from Padua, Italy – and Studies 1-A and 1-B also had a Brazilian sample of undergraduates and an Italian sample of elderly people. The first studies had the purpose of characterizing the representation system. After a preliminary qualitative study conducted with interviews with people from different age groups, which has allowed to identify the cognems of the social representation on aging, Studies 1-A and 1-B were comparative investigations that characterized the structural status of the elements of that representation and also structural differences linked to age groups and cultural context. Study 2 was a survey marked by paired evaluations of proximity between objects linked to the aging theme which allowed the identification of three social representations highly connected to aging, by means prototypical and similarity analyses: death, health and family. Study 3 had a mixed nature, with characteristics of both a quasi-experiment and a survey. It provided the characterization of the structures of the chosen social representations in the system and also assessed the intensity of relations between social objects in that system. Additionally, participants evaluated pairs of elements of the representation on aging and the other representations and indicated if they perceived a connection or not, enabling the identification of possible connection points. After the identification of those possible relationships, the focus shifted to testing the plausibility of a model for connection points including a bridge relation between those elements. Study 4 consisted of context manipulations of emphasis or relativization of a peripheral element of the social representation on health that was connected to elements from the social representation on aging. It was observed that a context change relative to a peripheral element of a representation interferes on the activation of schemes relative to a second representation, regardless of their structural status; it is a first empirical evidence of the validity of a theoretical conception of coordination relations involving disjoint representations of a same system by means of connection points. Finally, Studies 5-A and 5-B aimed at verifying if denying or confirming the information of cognems from connection points would be associated with activation differences of semantic and evaluative relations (bridge relations) with elements connected to them, from the representation on aging. Differences were identified in the sense that compatibility with the content of the manipulated cognem was associated with higher valences of at least one of the hyperconnectors. As a conclusion, evidence was found of relations among social representations at the level of their elements, and the conception of a theoretical model of inter-representation connection points presented promising results. The model contributes to the understanding of mechanisms of association of social representations in disjunction and also opens possibilities for application especially concerning social representation dynamics, which might also take representation systems into account.
Le teorie e i risultati di ricerca sulla struttura delle rappresentazioni sociali sono stati costruiti, in generale, a partire dallo studio di rappresentazioni isolate. Gli studi volti a identificare le relazioni strutturali che coinvolgono due o più rappresentazioni sono più recenti. In letteratura, termini diversi sono stati impiegati per riferirsi a insiemi interconnessi di rappresentazioni sociali, gestiti o non da un’istanza ideologica superiore; in questi casi, ci riferiamo a famiglie, sistemi o reti di rappresentazioni. In questo contesto, vi sono relazioni di coordinamento, in cui le associazioni possono essere identificate allo stesso livello dell’architettura del pensiero sociale. Tradizionalmente, sono stati privilegiati gli studi di rappresentazioni in congiunzione, che presentano intersezioni di elementi al livello dei loro nuclei o etichette di oggetto. La presente ricerca si propone di contribuire alla conoscenza di rappresentazioni disgiunte. Quando, a livello inter-rappresentazione, ci sono due o più strutture legate da relazioni tra cognemi o tra etichette di oggetti, proponiamo il nome di sistema rappresentazionale per la macrostruttura che ne risulta, con relazioni formalizzate dal modello di schemi cognitivi di base. L'obiettivo principale della ricerca è consistito nell’identificazione di relazioni tra elementi di rappresentazioni sociali diverse, sottoponendo a prova l'esistenza di punti di connessione tra rappresentazioni allo stesso livello dell’architettura del pensiero sociale. Si tratta di un modello che concepisce le relazioni interrappresentazione al livello dei cognemi. Tutti i cinque studi condotti hanno coinvolto un sistema di rappresentazioni, tenendo la rappresentazione sociale dell’invecchiamento come punto di riferimento. La maggior parte degli studi è stata svolta con campioni di convenienza di studenti universitari di Padova, Italia. Gli Studi 1-A e 1-B hanno avuto anche un campione brasiliano di studenti e un campione italiano di anziani. I primi studi avevano lo scopo di caratterizzare il sistema rappresentazionale. Dopo un’indagine preliminare qualitativa condotta con interviste a persone di diverse fasce di età, la quale ha permesso di identificare i cognemi della rappresentazione sociale sull’invecchiamento, gli Studi 1-A e 1-B sono stati realizzati tramite ricerche comparative che hanno caratterizzato lo statuto strutturale degli elementi di quella rappresentazione e anche delle differenze strutturali legate a gruppi di età e contesti culturali diversi. Lo Studio 2 è consistito in una ricerca caratterizzata da valutazioni appaiate di prossimità tra oggetti legati al tema dell'invecchiamento, la quale ha permesso l'identificazione di tre rappresentazioni sociali altamente connesse all'invecchiamento attraverso analisi prototipiche e di similitudine: morte, salute e famiglia. Lo Studio 3 ha avuto un carattere misto, con caratteristiche sia di quasi-esperimento sia di survey. Ha fornito la caratterizzazione delle strutture delle rappresentazioni sociali scelte dal sistema ed anche permesso di valutare l'intensità delle relazioni tra oggetti sociali in quel sistema. Inoltre, i partecipanti hanno valutato le coppie di elementi della rappresentazione sull’invecchiamento e sugli altri tre oggetti e hanno indicato se percepivano una relazione o meno tra di loro, permettendo l’individuazione di possibili punti di connessione. Dopo l'individuazione delle relazioni possibili, l'attenzione è stata rivolta a verificare la plausibilità di un modello di punti di connessione tra elementi con una relazione ponte. Lo Studio 4 è consistito nella manipolazione di un elemento periferico della rappresentazione sociale della salute (tramite sua enfasi o relativizzazione), collegata ad elementi della rappresentazione sociale dell’ invecchiamento. E' stato osservato che un cambiamento di contesto rispetto ad un elemento periferico di una rappresentazione interferisce sull’attivazione di schemi relativi ad una seconda rappresentazione del sistema, indipendentemente dal suo statuto strutturale. Si tratta di una prima evidenza empirica della validità di una concezione teorica che sottolinea le relazioni di coordinamento fra rappresentazioni disgiunte in uno stesso sistema per mezzo di punti di connessione. Infine, gli Studi 5-A e 5-B avevano lo scopo di verificare se il negare o confermare le informazioni di cognemi in punti di connessione fosse associato a differenze di attivazione di relazioni semantiche e valutative (relazioni ponte) con elementi ad essi connessi, della rappresentazione dell’invecchiamento. Sono state individuate delle differenze, nel senso che una compatibilità con il contenuto del cognema manipolato è stata associata con valenze più alte di almeno uno degli iperconnettori. In conclusione, si sono trovate evidenze empiriche riguardo relazioni tra rappresentazioni sociali a livello dei loro elementi, e ha trovato sostegno, con risultati promettenti, la concezione di un modello teorico di punti di connessione tra rappresentazioni. Il modello conntribuisce alla comprensione dei meccanismi di associazione di rappresentazioni sociali in disgiunzione e apre anche la possibilità di applicazioni soprattutto per quanto riguarda le dinamiche delle rappresentazioni sociali, attraverso interventi mirati sui sistemi di rappresentazione presi in esame.

Every positive integer has a unique radix 2 representation which uses the digits {0,1}. However, if we allow digits other than 0 and 1, say {0,1,-1}, then a positive integer has many representations. Of these redundant representations, it is possible to choose one that has few nonzero digits. It is well known that using representations of integers with few nonzero digits allows certain algebraic operations to be done more quickly. This thesis is concerned with various representations of integers that are related to efficient implementations of algebraic operations in cryptographic algorithms. The topics covered here include:

• The width-w nonadjacent form (w-NAF). We prove that the w-NAF of an integer has a minimal number of nonzero digits; that is, no other representation of an integer, which uses the w-NAF digits, can have fewer nonzero digits than its w-NAF.
• A left-to-right analogue of the w-NAF. We introduce a new family of radix 2 representations which use the same digits as the w-NAF, but have the property that they can be computed by sliding a window from left to right across the binary representation of an integer. We show these new representations have a minimal number of nonzero digits.
• Joint representations. Solinas introduced a {0,1,-1}-radix 2 representation for pairs of integers called the joint sparse form. We consider generalizations of the joint sparse form which represent r≥2 integers and use digits other than {0,1,-1}. We show how to construct a {0,1,2,3}-joint representation that has a minimal number of nonzero columns.
• Nonadjacent digit sets. It is well known that if x equals 3 or -1 then every nonnegative integer has a unique {0,1,x}-nonadjacent form; that is, a {0,1,x}-radix 2 representation with the property that, of any two consecutive digits, at most one is nonzero. We investigate what other values of x have this property.

Mathematics
Ph.D.
Here we investigate a variety of ways to represent polytopes and related objects. We define a class of posets, which includes all abstract polytopes, giving a unique representative among posets having a particular labeled flag graph and characterize the labeled flag graphs of abstract polytopes. We show that determining the realizability of an abstract polytope is equivalent to solving a low rank matrix completion problem. For any given polytope, we provide a new construction for the known result that there is a combinatorial polytope with a specified ridge that is always projectively equivalent to the given polytope, and we show how this makes a naturally arising subclass of intractable problems tractable. We give necessary and sufficient conditions for realizing a polytope's interval poset, which is the polytopal analog of a poset's Hasse diagram. We then provide a counter example to the general realizablity of a polytope's interval poset.
Temple University--Theses

Artificial intelligence systems are seeking to learn better representations. One of the most desirable properties in these representations is disentanglement. Disentangled representations show merits of interpretability and generalizability. Through these representations, the world around us can be decomposed into explanatory factors of variation, and can thus be more easily understood by not only machines but humans. Disentanglement is akin to the reverse engineering process of a video game, where based on exploring the beautiful open world we need to figure out what underlying controllable factors that actually render/generate these fantastic dynamics. This thesis mainly discusses the problem of how such "reverse engineering" can be achieved using deep learning techniques in the computer vision domain. Although there have been plenty of works tackling this challenging problem, this thesis shows that an important ingredient that is highly effective but largely neglected by existing works is the modeling of visual variation. We show from various perspectives that by integrating the modeling of visual variation in generative models, we can achieve superior unsupervised disentanglement performance that has never been seen before. Specifically, this thesis will cover various novel methods based on technical insights such as variation consistency, variation predictability, perceptual simplicity, spatial constriction, Lie group decomposition, and contrastive nature in semantic changes. Besides the proposed methods, this thesis also touches on topics such as variational autoencoders, generative adversarial networks, latent space examination, unsupervised disentanglement metrics, and neural network architectures. We hope the observations, analysis, and methods presented in this thesis can inspire and contribute to future works in disentanglement learning and related machine learning fields.

This research is based on Middle School Mathematics Project (MSMP) funded by the Interagency Educational Research Initiative through a grant to the American Association for the Advancement of Science. Both teacher’s instructional representations and students’ written representations were coded and analyzed to investigate the nature and structure of the representations in teaching fractions, decimals and percents in middle school classrooms in three school districts in Texas. The study further explored the relationship between both the quality and quantity of instructional representations and students’ written representations, and the relationship between students’ written representations and their achievements. This dissertation used a mixed approach utilizing both quantitative and qualitative methods. The data was collected in the first two years of a five-year study. A total of 14 sixth grade mathematics teachers from three school districts in Texas were selected from the MSMP project. Before the actual videotaping procedure, a professional development focusing on multiple representations was held for the teachers. Both pretests and posttests were used to examine the relationship between the structure of students’ written representations and their achievements. The results showed that the both the quantity and quality of teachers’ instructional representations varied a lot. Symbolic representations were the predominant representations in classroom teaching. Structures of instructional representations converge to content sub-constructs rather than format sub-constructs. Here subconstructs include part-whole, measure, quotient, multiplication by one and cross product. Instead, format sub-constructs include real world, manipulatives, pictures, spoken symbolic representations and written symbolic representations, however, connections between these sub-constructs were not statistically significant. Within the three content sub-constructs (part-whole, quotient, and multiplication by one) that revealed by students’ written representations, quotient and multiplication by one significantly predicated the students’ posttest scores. It was also found that, among the three quality criteria (accuracy, comprehensibility and connections) of instructional representations, the comprehensibility score significantly predicated students’ achievement in the posttests.

The topic of LGBT rights has been increasingly discussed and debated over recent years. More and more scholars show their interests in the field of LGBT representations in media. However, not many studies involved LGBT representations in social media. This paper explores LGBT representations on Facebook by analysing posts on an open page and in a private group, including both representations of the self as the identity of sexual minorities, content that is displayed on Facebook and the similarities and differences in two different settings (a private group and an open page). This study aims to fill the gap of self- representations of sexual minorities in non-anonymous online settings. Main concepts in this study include communication, gender, and sexuality, media representation, power, identity, stereotypes, self-representations, and discourse. A content analysis was conducted on posts from an LGBT open page and a private group with CDA (Critical Discourse Analysis) being used as the method. Results suggest that both the open page and private group express values of non- discrimination, equality, and respect by presenting links, images and with an extra form of representation - personal statements in the group. Compared to the neutral environment on the open page, more personal emotions are involved in the private group and some of them tend to be negative. However, some images can go extreme, they overemphasise same-sex marriages but neglect other ones.

In this thesis, a number of novel watermarking techniques for different 3D representations are presented. A novel watermarking method is proposed for the mono-view video, which might be interpreted as the basic implicit representation of 3D scenes. The proposed method solves the common flickering problem in the existing video watermarking schemes by means of adjusting the watermark strength with respect to temporal contrast thresholds of human visual system (HVS), which define the maximum invisible distortions in the temporal direction. The experimental results indicate that the proposed method gives better results in both objective and subjective measures, compared to some recognized methods in the literature. The watermarking techniques for the geometry and image based representations of 3D scenes, denoted as 3D watermarking, are examined and classified into three groups, as 3D-3D, 3D-2D and 2D-2D watermarking, in which the pair of symbols identifies whether the watermark is embedded-detected in a 3D model or a 2D projection of it. A detailed literature survey on 3D-3D watermarking is presented that mainly focuses on protection of the intellectual property rights of the 3D geometrical representations. This analysis points out the specific problems in 3D-3D geometry watermarking , such as the lack of a unique 3D scene representation, standardization for the coding schemes and benchmarking tools on 3D geometry watermarking. For 2D-2D watermarking category, the copyright problem for the emerging free-view televisions (FTV) is introduced. The proposed watermarking method for this original problem embeds watermarks into each view of the multi-view video by utilizing the spatial sensitivity of HVS. The hidden signal in a selected virtual view is detected by computing the normalized correlation between the selected view and a generated pattern, namely rendered watermark, which is obtained by applying the same rendering operations which has occurred on the selected view to the original watermark. An algorithm for the estimation of the virtual camera position and rotation is also developed based on the projective planar relations between image planes. The simulation results show the applicability of the method to the FTV systems. Finally, the thesis also presents a novel 3D-2D watermarking method, in which a watermark is embedded into 3-D representation of the object and detected from a 2-D projection (image) of the same model. A novel solution based on projective invariants is proposed which modifies the cross ratio of the five coplanar points on the 3D model according to the watermark bit and extracts the embedded bit from the 2D projections of the model by computing the cross-ratio. After presenting the applicability of the algorithm via simulations, the future directions for this novel problem for 3D watermarking are addressed.

The Bloch representation of an n-qubit channel provides a way to represent quantum channels as certain affine transformations on [special characters omitted]. In higher dimensions (n > 1), the correspondence between quantum channels and their Bloch representations is not well-understood. Partly motivated by the ability to simplify the calculation of information theoretic quantities of a qubit channel using the Bloch representation, in this thesis we investigate the correspondence between a channel and its Bloch representation with an emphasis on unital n-qubit channels, in which case the Bloch representation is linear.

The thesis is divided into three main sections. First we focus our attention on qubit channels. For certain sets of quantum channels, we establish the surprising existence of a special isomorphism into the set of classical channels. We classify the sets of qubit channels with this property and show that information theoretic quantities are preserved by such classical representations. In a natural progression, we prove some well-known facts about SO(3), the proofs of which are either nonexistent or difficult to find in the literature. Some of this work is based on [12, 13].

In the next section, we consider the multi-qubit channels and show that every finite group can be realized as a subgroup of the quantum channels; this approach allows for the construction of a quantum representation for the free affine monoid over any finite group and gives a classical representation for it. We extend some fundamental results from [26, 28] to the multi-qubit case, including that the set of diagonal Bloch matrices is equal to the free affine monoid over the involution group [special characters omitted]. Some of this work appeared in [10].

Lastly, we study the extreme points for the set of n-qubit channels. There are two types of extreme points: invertible and non-invertible; invertible channels are non-singular maps for which the inverse is also a channel. We briefly study the non-invertible extreme points and then parameterize and analyze the invertiblen-qubit Bloch matrices, which form a compact connected Lie group. We calculate the Lie algebra and give an explicit generating set for the invertible Bloch matrices and a maximal torus.

Languages differ in their sound patterns, but these differences are, to a large extent, systematic. One goal of Universal Grammar (Chomsky 1957, 1965) is to account for the systematic patterns which are attested across languages. Toward this end, Universal Grammar is considered to contain a set of phonological primitives such as features, and some restrictions on their combination. However, in rule-based phonology, it is assumed that rules are part of the grammar of an individual language. By their very nature, rules describe operations. As such, they are not well-suited to express restrictions on the ways in which segments may combine when no overt operation is involved. To account for such restrictions, Chomsky & Halle (Sound Pattern of English (SPE): 1968) supplemented rules with Morpheme Structure Constraints (MSCs) which define the possible morpheme shapes that a particular language allows (see also Halle 1959). Thus, in SPE, both MSCs and rules played a role in accounting for the phonological patterns observed in languages.

Open coordination systems are a means of performing distributed computing where the processes in the system are not known to each other until the system is run. In an ideal world these processes could be of any language and running on any hardware or operating system. Traditionally systems have limited processes either to a single language or so that they can only communicate using simple types.

Topological embeddings of complete graphs and complete multipartite graphs give rise to combinatorial designs when the faces of the embeddings are triangles. In this case, the blocks of the design correspond to the triangular faces of the embedding. These designs include Steiner, twofold and Mendelsohn triple systems, as well as Latin squares. We look at construction methods, structural properties and other problems concerning these cases. In addition, we look at graph representations by Steiner triple systems and by combinatorial embecldings. This is closely related to finding independent sets in triple systems. We examine which graphs can be represented in Steiner triple systems and combinatorial embeddings of small orders and give several bounds including a bound on the order of Steiner triple systems that are guaranteed to represent all graphs of a given maximum degree. Finally, we provide an enumer- ation of graphs of up to six edges representable by Steiner triple systems.

The theme of this thesis is the study of moduli stacks of representations of an associative algebra, with an eye toward continuous representations of profinite groups such as Galois groups. The central object of study is the geometry of the map \(\bar{\psi}\) from the moduli stack of representations to the moduli scheme of pseudorepresentations. The first chapter culminates in showing that \(\bar{\psi}\) is very close to an adequate moduli space of Alper. In particular, \(\bar{\psi}\) is universally closed. The second chapter refines the results of the first chapter. In particular, certain projective subschemes of the fibers of \(\bar{\psi}\) are identified, generalizing a suggestion of Kisin. The third chapter applies the results of the first two chapters to moduli groupoids of continuous representations and pseudorepresentations of profinite algebras. In this context, the moduli formal scheme of pseudorepresentations is semi-local, with each component Spf \(B_\bar{D}\) being the moduli of deformations of a given finite field-valued pseudorepresentation \(\bar{D}\). Under a finiteness condition, it is shown that \(\bar{\psi}\) is not only formally finite type over Spf \(B_\bar{D}\), but arises as the completion of a finite type algebraic stack over Spec \(B_\bar{D}\). Finally, the fourth chapter extends Kisin's construction of loci of coefficient spaces for p-adic local Galois representations cut out by conditions from p-adic Hodge theory. The result is extended from the case that the coefficient ring is a complete Noetherian local ring to the more general case that the coefficient space is a Noetherian formal scheme.
Mathematics

Thesis: Sc. D., Massachusetts Institute of Technology, Department of Mathematics, 2018.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 133-136).
This thesis studies the alternating sum of cohomology groups of a Springer fiber (in characteristic 0), called a total Springer representation, as a representation of both the Weyl group and the stabilizer of the corresponding nilpotent element. For classical types, we present explicit formulas for the decomposition of total Springer representations into irreducible ones of the corresponding Weyl group using Kostka-Foulkes polynomials. Also, the character value at any element contained in the maximal parabolic subgroup(s) of type A is explicitly given in terms of Green polynomials. As a result, closed formulas for the Euler characteristic of Springer fibers are deduced. Our proof relies on analysis of geometry of Springer fibers and combinatorics of symmetric functions. Moreover, we provide formulas for the character value of a total Springer representation at any element in the stabilizer of the corresponding nilpotent element. For exceptional types, the character values of total Springer representations are completely known. Here, we only describe the decomposition of such representations into irreducible ones of stabilizers of corresponding nilpotent elements.
by Dongkwan Kim.
Sc. D.

Thesis: M. Eng., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2012.
Cataloged from PDF version of thesis.
Includes bibliographical references (page 54).
I discuss the lessons learned during the design and implementation of three knowledge representations systems for sensemaking. The focus is on the tension that exists between a knowledge representation's role as a surrogate for the world and its role as a facilitator of computational reasoning. Each system accepts natural language inputs and implements a bidirectional model of sensemaking. One system emphasizes inference while the other two systems emphasize their role as a representation for the world. I discuss the differences between these systems and what gives rise to these differences.
by Hong-Linh Q. Nguyen.
M. Eng.

When understanding narrative events we must keep track of a number of (often) changing dimensions, such as who is the focus of the narrative, what they are doing, and where they are doing it; amongst which, this ‘where’ dimension is thought to be crucial in forming even the simplest representation of events. Thus, in order to understand how we represent events, research has often focused upon how we represent space. While research from the spatial cognition and language comprehension literature has shown that we can form a mental representation of space that is rich in detail, maintaining categorical (i.e. room-by-room) as well as Euclidean distance (i.e. absolute distance), this research has primarily used tasks which may bias towards one representation over another. Thus, the research presented in this thesis set out to explore which representation of space is maintained in the absence of an overt task, and the implications that this representation has for the organisation and accessibility for information maintained in a mental representation of events (or, an ‘event model’). Using eye-tracking methods and ‘look-and-listen’ tasks in the visual world paradigm throughout to explore how manipulations of space influence accessibility for objects – in terms of fixations on objects depicted in a visual display during language comprehension – the experiments laid out in the present thesis found that comprehenders spontaneously form a categorical (but not Euclidean) representation of space, constructing and maintaining separate ‘event models’ based around spatial units (e.g. rooms) in which information (e.g. objects) is maintained. While Experiment 1 found no influence of space on accessibility for the target on mention using a concurrent viewing and listening paradigm, by replacing the visual scene with a blank-screen prior to the onset of the narrative, Experiment 2 established that, during narrative comprehension, following the movement of a protagonist from one location to another, potential targets located in the initial location were less accessible prior to mention for the target. Experiment 3 explored the locus of this effect and found that, on mention, targets were less accessible following a spatial shift, regardless of whether the objects were in the same or a separate location to the protagonist. Experiments 4-6 built upon Experiments 1-3 to further explore how a spatial shift made by an object (carried by a protagonist) can modulate the structure of events, asking how access for a target is resolved when associated with two event models (vs. one event model). Here, visual scenes depicted two rooms, containing several objects, separated by a boundary. Narratives described an object moving from one location to another with a protagonist, necessitating movement across the boundary (or not). The most reliable finding was that both representations of the moved object were less accessible on mention when associated with two event models (vs. one event model), suggesting that competition occurs between event models prior to access for a target if the target is represented across more than one event model. Experiment 7 aimed to address how event structure can increase, rather than reduce accessibility for a target. Here, discourses separating a target into a different event model (vs. the same event model) to that of a semantically-related competitor increased accessibility for the target on mention and reduced accessibility for the competitor, suggesting that information in an accessed (foregrounded) event model is more accessible than information in other (backgrounded) event models. Together, the findings of the conducted experiments support the notion that comprehenders form and use event models in discourse comprehension when necessary during passive listening tasks, and that the form and structure of these event models modulates accessibility for information when doing so. The experiments established here have further supported and built upon general models of event cognition, establishing how and when event models, and the information maintained within them, are accessed.

Les avancées récentes en apprentissage profond et en traitement d'image présentent l'opportunité d'unifier ces deux champs de recherche complémentaires pour une meilleure résolution du problème de classification d'images dans des catégories sémantiques. L'apprentissage profond apporte au traitement d'image le pouvoir de représentation nécessaire à l'amélioration des performances des méthodes de classification d'images. Cette thèse propose de nouvelles méthodes d'apprentissage de représentations visuelles profondes pour la résolution de cette tache. L'apprentissage profond a été abordé sous deux angles. D'abord nous nous sommes intéressés à l'apprentissage non supervisé de représentations latentes ayant certaines propriétés à partir de données en entrée. Il s'agit ici d'intégrer une connaissance à priori, à travers un terme de régularisation, dans l'apprentissage d'une machine de Boltzmann restreinte (RBM). Nous proposons plusieurs formes de régularisation qui induisent différentes propriétés telles que la parcimonie, la sélectivité et l'organisation en structure topographique. Le second aspect consiste au passage graduel de l'apprentissage non supervisé à l'apprentissage supervisé de réseaux profonds. Ce but est réalisé par l'introduction sous forme de supervision, d'une information relative à la catégorie sémantique. Deux nouvelles méthodes sont proposées. Le premier est basé sur une régularisation top-down de réseaux de croyance profonds à base de RBMs. Le second optimise un cout intégrant un critre de reconstruction et un critre de supervision pour l'entrainement d'autoencodeurs profonds. Les méthodes proposées ont été appliquées au problme de classification d'images. Nous avons adopté le modèle sac-de-mots comme modèle de base parce qu'il offre d'importantes possibilités grâce à l'utilisation de descripteurs locaux robustes et de pooling par pyramides spatiales qui prennent en compte l'information spatiale de l'image. L'apprentissage profonds avec agrÉgation spatiale est utilisé pour apprendre un dictionnaire hiÉrarchique pour l'encodage de reprÉsentations visuelles de niveau intermÉdiaire. Cette mÉthode donne des rÉsultats trs compétitifs en classification de scènes et d'images. Les dictionnaires visuels appris contiennent diverses informations non-redondantes ayant une structure spatiale cohérente. L'inférence est aussi très rapide. Nous avons par la suite optimisé l'étape de pooling sur la base du codage produit par le dictionnaire hiérarchique précédemment appris en introduisant introduit une nouvelle paramétrisation dérivable de l'opération de pooling qui permet un apprentissage par descente de gradient utilisant l'algorithme de rétro-propagation. Ceci est la premire tentative d'unification de l'apprentissage profond et du modèle de sac de mots. Bien que cette fusion puisse sembler évidente, l'union de plusieurs aspects de l'apprentissage profond de représentations visuelles demeure une tache complexe à bien des égards et requiert encore un effort de recherche important
Recent advancements in the areas of deep learning and visual information processing have presented an opportunity to unite both fields. These complementary fields combine to tackle the problem of classifying images into their semantic categories. Deep learning brings learning and representational capabilities to a visual processing model that is adapted for image classification. This thesis addresses problems that lead to the proposal of learning deep visual representations for image classification. The problem of deep learning is tackled on two fronts. The first aspect is the problem of unsupervised learning of latent representations from input data. The main focus is the integration of prior knowledge into the learning of restricted Boltzmann machines (RBM) through regularization. Regularizers are proposed to induce sparsity, selectivity and topographic organization in the coding to improve discrimination and invariance. The second direction introduces the notion of gradually transiting from unsupervised layer-wise learning to supervised deep learning. This is done through the integration of bottom-up information with top-down signals. Two novel implementations supporting this notion are explored. The first method uses top-down regularization to train a deep network of RBMs. The second method combines predictive and reconstructive loss functions to optimize a stack of encoder-decoder networks. The proposed deep learning techniques are applied to tackle the image classification problem. The bag-of-words model is adopted due to its strengths in image modeling through the use of local image descriptors and spatial pooling schemes. Deep learning with spatial aggregation is used to learn a hierarchical visual dictionary for encoding the image descriptors into mid-level representations. This method achieves leading image classification performances for object and scene images. The learned dictionaries are diverse and non-redundant. The speed of inference is also high. From this, a further optimization is performed for the subsequent pooling step. This is done by introducing a differentiable pooling parameterization and applying the error backpropagation algorithm. This thesis represents one of the first attempts to synthesize deep learning and the bag-of-words model. This union results in many challenging research problems, leaving much room for further study in this area

Ce travail s'inscrit dans le cadre de la classification des representations p-adiques du groupe de galois absolu d'un corps local par des modules (verifiant certaines proprietes, notamment l'action d'un operateur de frobenius), realise par j. -m. Fontaine dans le grothendieck festschrift. Une representation est dite surconvergente lorsque son module associe provient d'un module sur un anneau de series bornees et convergentes en dehors d'un voisinage de 0. Dans le cas de caracteristique mixte, on montre que les representations p-adiques surconvergentes forment une sous-categorie tannakienne, stable par extension et changement de base, contenant les representations abeliennes. On montre ensuite que l'on peut retrouver le module des periodes d'une representation p-adique v surconvergente a partir de son module. Dans le cas de caracteristique p, on etablit, en exhibant un contre-exemple, que les conditions de ramification ne suffisent pas a assurer la surconvergence d'une representation. Dans une derniere partie, on retrouve et precise des resultats de bloch, kato et perrin-riou sur l'exponentielle de bloch et kato relatif a la representation v = q#p(r)

Bibliography: leaves 80-82.
In this dissertation I have sought to examine some contemporary representations of Elizabeth I. Beginning with an outline of how the queen struggled to construct her feminine power within and beyond dominant patriarchal discourse, I go on to explore closely three modern filmic treatments of the queen: The Virgin Queen (1955), Mary, Queen of Scots (1971) and Elizabeth (1998). These films are discussed in terms of their engagement with Elizabeth's iconography and mythic biography, and in terms of the anxieties reproduced as a consequence of their grounding in particular historical eras.

This thesis consists of four papers about the intersection between semigroup theory, category theory and representation theory. We say that a representation of a semigroup by a matrix semigroup is effective if it is injective and define the effective dimension of a semigroup S as the minimal n such that S has an effective representation by square matrices of size n. A multisemigroup is a generalization of a semigroup where the multiplication is set-valued, but still associative. A 2-category consists of objects, 1-morphisms and 2-morphisms. A finitary 2-category has finite dimensional vector spaces as objects and linear maps as morphisms. This setting permits the notion of indecomposable 1-morphisms, which turn out to form a multisemigroup. Paper I computes the effective dimension Hecke-Kiselman monoids of type A. Hecke-Kiselman monoids are defined by generators and relations, where the generators are vertices and the relations depend on arrows in a given quiver. Paper II computes the effective dimension of path semigroups and truncated path semigroups. A path semigroup is defined as the set of all paths in a quiver, with concatenation as multiplication. It is said to be truncated if we introduce the relation that all paths of length N are zero. Paper III defines the notion of a multisemigroup with multiplicities and discusses how it better captures the structure of a 2-category, compared to a multisemigroup (without multiplicities). Paper IV gives an example of a family of 2-categories in which the multisemigroup with multiplicities is not a semigroup, but where the multiplicities are either 0 or 1. We describe these multisemigroups combinatorially.

Representation theory is concerned with the ways of writing elements of abstract algebraic structures as linear transformations of vector spaces. Typical structures amenable to representation theory are groups, associative algebras and Lie algebras. In this thesis we study linear representations of finite groups. The study focuses on character theory and how character theory can be used to extract information from a group. Prior to that, concepts needed to treat character theory, and some of their ramifications, are investigated. The study is based on existing literature, with excessive use of examples to illuminate important aspects. An example treating a class of p-groups is also discussed.

Body representation disorders have a key role in the characterization of obesity. So far, the literature consistently pointed to a negative attitudinal body image. Conversely, after reviewing the pertinent literature, it emerges that more incoherent results have been reported for the self-perceived body size. Chapter 2 tries to clarify this issue by adopting a more innovative theoretical framework (i.e., the implicit/explicit model; Longo, 2015). For the first time, we probed the implicit representation underlying position sense in obesity, reporting a similar representation to healthy weight participants. Importantly, this result shows that not all components of body representation are affected by obesity. Chapter 3 addresses another aspect of body representation that has been neglected in obesity, namely bodily self-consciousness. The Rubber Hand Illusion has been traditionally used to investigate the mechanisms underlying body awareness. Our results show that individuals with obesity have comparable subjective experience of the illusion, while the effect of the illusion on self-location is reduced. This dissociation can be interpreted as the result of a preserved visuo-tactile integration and an altered visuo-proprioceptive integration in obesity. However, in Chapter 4 we reported that individuals with obesity have a reduced temporal resolution of visuo-tactile integration, meaning that they integrated stimuli over an extended range of asynchronies than healthy weight participants. In fact, this evidence predicts that in the RHI individuals with obesity might perceive more synchronously the asynchronous stimulation, showing a greater effect of the illusion also in this condition. Nevertheless, we failed to show this pattern of results in our study with an interval of asynchronous stimulation of 1000 ms (usually adopted in the RHI paradigm). We hypothesized that smaller time-lags, which are inside the temporal binding window of individuals with obesity and outside the temporal binding widow of healthy weight participants, might not be perceived by individuals with obesity but detected by healthy weight individuals. Accordingly, a dissimilar susceptibility to the illusion should be observed. Chapter 5 investigates this issue by adopting a modified version of the RHI that enables a parametrical modulation of the timing of the stimulation. However, we could not replicate the RHI even in healthy weight participants. The possible methodological reasons for this failure are discussed. Overall, this work tries to fill some gaps in the previous literature about body representation in obesity. Moreover, our findings provide an important clue about the possible cognitive mechanisms involved in body representation disorders in obesity. However, many questions still need an answer: due to the complexity of the domain a comprehensive knowledge of the topic might be challenging. A deep understanding of obesity is fundamental to develop multidisciplinary and efficacious rehabilitative protocols. Indeed, better treatments would significantly ameliorate individuals’ well-being but also contribute to reduce the huge health costs related to obesity comorbidities.

Body representation disorders have a key role in the characterization of obesity. So far, the literature consistently pointed to a negative attitudinal body image. Conversely, after reviewing the pertinent literature, it emerges that more incoherent results have been reported for the self-perceived body size. Chapter 2 tries to clarify this issue by adopting a more innovative theoretical framework (i.e., the implicit/explicit model; Longo, 2015). For the first time, we probed the implicit representation underlying position sense in obesity, reporting a similar representation to healthy weight participants. Importantly, this result shows that not all components of body representation are affected by obesity. Chapter 3 addresses another aspect of body representation that has been neglected in obesity, namely bodily self-consciousness. The Rubber Hand Illusion has been traditionally used to investigate the mechanisms underlying body awareness. Our results show that individuals with obesity have comparable subjective experience of the illusion, while the effect of the illusion on self-location is reduced. This dissociation can be interpreted as the result of a preserved visuo-tactile integration and an altered visuo-proprioceptive integration in obesity. However, in Chapter 4 we reported that individuals with obesity have a reduced temporal resolution of visuo-tactile integration, meaning that they integrated stimuli over an extended range of asynchronies than healthy weight participants. In fact, this evidence predicts that in the RHI individuals with obesity might perceive more synchronously the asynchronous stimulation, showing a greater effect of the illusion also in this condition. Nevertheless, we failed to show this pattern of results in our study with an interval of asynchronous stimulation of 1000 ms (usually adopted in the RHI paradigm). We hypothesized that smaller time-lags, which are inside the temporal binding window of individuals with obesity and outside the temporal binding widow of healthy weight participants, might not be perceived by individuals with obesity but detected by healthy weight individuals. Accordingly, a dissimilar susceptibility to the illusion should be observed. Chapter 5 investigates this issue by adopting a modified version of the RHI that enables a parametrical modulation of the timing of the stimulation. However, we could not replicate the RHI even in healthy weight participants. The possible methodological reasons for this failure are discussed. Overall, this work tries to fill some gaps in the previous literature about body representation in obesity. Moreover, our findings provide an important clue about the possible cognitive mechanisms involved in body representation disorders in obesity. However, many questions still need an answer: due to the complexity of the domain a comprehensive knowledge of the topic might be challenging. A deep understanding of obesity is fundamental to develop multidisciplinary and efficacious rehabilitative protocols. Indeed, better treatments would significantly ameliorate individuals’ well-being but also contribute to reduce the huge health costs related to obesity comorbidities.

La tesi si pone l'obiettivo di analizzare la tragedia "Othello" di William Shakespeare, comparandola con alcuni adattamenti. Partendo dall'analisi dell'opera shakespeariana e analizzandone soprattutto l'aspetto linguistico, la tesi si sposterà sull'adattamento lirico di Giuseppe Verdi, sottolineandone la nuova vena "romantica". Procederà poi con il film di Orson Welles, cercando di dare una possibile lettura del film-capolavoro del regista statunitense. Concluderà, infine, con l'adattamento moderno di Carmelo Bene per cerare di dimostrare una sorta di vicinanza tra il drammaturgo inglese e il teatro moderno.

A growing amount of evidence supports the hypothesis that humans are able, from the earliest age, to process numerical information in the absence of language. This work addresses the question of the nature of the internal representation for processing numerosities from three perspective: developmental, adults' skilled performance, and the peculiar case of synaesthesia. In our studies with children we addressed the development of the mental representation for numbers. In the first experiment we showed that, before formal teaching, preschoolers possess multiple numerical representations that follow a specific developmental trend. Indeed, they first rely on an intuitive representation where numbers are distributed logarithmically and progressively, with numerical practice and increasing knowledge, they shift to a formal and linear representation. Moreover, preschool children can exhibit both types of representations according to the familiarity with the context. In the second study, we tested the hypothesis that non-numerical sequences may also rely on a similar representation and follow the same developmental pattern. By studying children from the last year of kindergarten to 3rd grade we observed that numerical and non-numerical sequences have different mental representations. Indeed, only the numerical sequence shows the classical effects that support the hypothesis of a logarithmic representation. Moreover, we observed that children start to learn linearity in the numerical domain and then generalize the principle to all ordinal sequences. In our third study we investigated adults numerical representation of symbolic and non symbolic material. The aim was to test if the basic ability of discriminating between numerosities could explain higher level processes such as approximate calculation and symbolic number comparison. Indeed, if the preverbal approximate system of the numerical representation forms the basis of more complex numerical and mathematical knowledge, it should influence performance in other numerical tasks. Moreover, the crossing of symbolic and non-symbolic format of the stimuli for the approximate calculation task allowed us to qualify previous findings about the operational momentum effect in approximate arithmetic (i.e., the tendency to overestimate additions and underestimate subtractions). Indeed, we observed that the effect may be explained by the tendency to underestimate numerosities and that this bias is proportional to the set size. In the last experiment we investigated the relation between colour and numerical representation in NM, a number-colour synaesthete. Results showed that, in spite of not reporting colours for numerosities, our synaesthete was subject to interference effects. From these results we suggest a new model that accounts for the implicit and explicit synaesthetic effects by suggesting the existence of primary and secondary synaesthetic connections ("pseudo-synaesthesia"). Our results and model questions previous work on bi-directional effects and the operational definition of synaesthesia.

Ma thèse étudie le problème de la reconnaissance des objets et des textures. Dans ce cadre, il est nécessaire de construire des représentations de signaux avec des propriétés d'invariance et de stabilité qui ne sont pas satisfaites par des approches linéaires. Les opérateurs de Scattering itèrent des décompositions en ondelettes et rectifications avec des modules complexes. Ces opérateurs définissent une transformée non-linéaire avec des propriétés remarquables ; en particulier, elle est localement invariante par translation et Lipschitz continue par rapport à l'action des difféomorphismes. De plus, les opérateurs de Scattering définissent une représentation des processus stationnaires qui capture les moments d'ordre supérieur, et qui peut être estimée avec faible variance à partir d'un petit nombre de réalisations. Dans cette thèse, nous obtenons des nouvelles propriétés mathématiques de la représentation en scattering, et nous montrons leur efficacité pour la reconnaissance des objets et textures. Grâce à sa continuité Lipschitz par rapport à l'action des difféomorphismes, la transformée en scattering est capable de linéariser les petites déformations. Cette propriété peut être exploitée en pratique avec un classificateur génératif affine, qui nous permet d'obtenir l'état de l'art sur la reconnaissance des chiffres manuscrites. Nous étudions ensuite les représentations en Scattering des textures dans le cadre des images et du son. Nous montrons leur capacité à discriminer des phénomènes non-gaussiens avec des estimateurs à faible variance, ce qui nous permet d'obtenir de l'état de l'art pour la reconnaissance des textures. Finalement, nous nous intéressons aux propriétés du Scattering pour l'analyse multifractale. Nous introduisons une renormalisation des coéfficients en Scattering qui permet d'identifier de façon efficace plusieurs paramètres multifractales; en particulier, nous obtenons une nouvelle caractérisation de l'intermittence à partir des coefficients de Scattering ré-normalisés, qui peuvent s'estimer de façon consistante
This thesis addresses the problem of pattern and texture recognition from a mathematical perspective. These high level tasks require signal representations enjoying specific invariance, stability and consistency properties, which are not satisfied by linear representations. Scattering operators cascade wavelet decompositions and complex modulus, followed by a lowpass filtering. They define a non-linear representation which is locally translation invariant and Lipschitz continuous to the action of diffeomorphisms. They also define a texture representation capturing high order moments and which can be consistently estimated from few realizations. The thesis derives new mathematical properties of scattering representations and demonstrates its efficiency on pattern and texture recognition tasks. Thanks to its Lipschitz continuity to the action of diffeomorphisms, small deformations of the signal are linearized, which can be exploited in applications with a generative affine classifier yielding state-of-the-art results on handwritten digit classification. Expected scattering representations are applied on image and auditory texture datasets, showing their capacity to capture high order moments information with consistent estimators. Scattering representations are particularly efficient for the estimation and characterization of fractal parameters. A renormalization of scattering coefficients is introduced, giving a new insight on fractal description, with the ability in particular to characterize multifractal intermittency using consistent estimators

Image representation is a challenging task. In particular, in order to obtain better performances in different image processing applications such as video surveillance, autonomous driving, crime scene detection and automatic inspection, effective and efficient image representation is a fundamental need. The performance of these applications usually depends on how accurately images are classified into their corresponding groups or how precisely relevant images are retrieved from a database based on a query. Accuracy in image classification and precision in image retrieval depend on the effectiveness of image representation. Existing image representation methods have some limitations. For example, spatial pyramid matching, which is a popular method incorporating spatial information in image-level representation, has not been fully studied to date. In addition, the strengths of pyramid match kernel and spatial pyramid matching are not combined for better image matching. Kernel descriptors based on gradient, colour and shape overcome the limitations of histogram-based descriptors, but suffer from information loss, noise effects and high computational complexity. Furthermore, the combined performance of kernel descriptors has limitations related to computational complexity, higher dimensionality and lower effectiveness. Moreover, the potential of a global texture descriptor which is based on human visual perception has not been fully explored to date. Therefore, in this research project, kernel-based effective and efficient image representation methods are proposed to address the above limitations. An enhancement is made to spatial pyramid matching in terms of improved rotation invariance. This is done by investigating different partitioning schemes suitable to achieve rotation-invariant image representation and the proposal of a weight function for appropriate level contribution in image matching. In addition, the strengths of pyramid match kernel and spatial pyramid are combined to enhance matching accuracy between images. The existing kernel descriptors are modified and improved to achieve greater effectiveness, minimum noise effects, less dimensionality and lower computational complexity. A novel fusion approach is also proposed to combine the information related to all pixel attributes, before the descriptor extraction stage. Existing kernel descriptors are based only on gradient, colour and shape information. In this research project, a texture-based kernel descriptor is proposed by modifying an existing popular global texture descriptor. Finally, all the contributions are evaluated in an integrated system. The performances of the proposed methods are qualitatively and quantitatively evaluated on two to four different publicly available image databases. The experimental results show that the proposed methods are more effective and efficient in image representation than existing benchmark methods.
Doctor of Philosophy

Skeletal representations are important shape descriptors which encode topological and geometrical properties of shapes and reduce their dimension. Skeletons are used in several fields of science and attract the attention of many researchers. In the biocad field, the analysis of structural properties such as porosity of biomaterials requires the previous computation of a skeleton. As the size of three-dimensional images become larger, efficient and robust algorithms that extract simple skeletal structures are required. The most popular and prominent skeletal representation is the medial axis, defined as the shape points which have at least two closest points on the shape boundary. Unfortunately, the medial axis is highly sensitive to noise and perturbations of the shape boundary. That is, a small change of the shape boundary may involve a considerable change of its medial axis. Moreover, the exact computation of the medial axis is only possible for a few classes of shapes. For example, the medial axis of polyhedra is composed of non planar surfaces, and its accurate and robust computation is difficult. These problems led to the emergence of approximate medial axis representations. There exists two main approximation methods: the shape is approximated with another shape class or the Euclidean metric is approximated with another metric. The main contribution of this thesis is the combination of a specific shape and metric simplification. The input shape is approximated with an orthogonal shape, which are polygons or polyhedra enclosed by axis-aligned edges or faces, respectively. In the same vein, the Euclidean metric is replaced by the L infinity or Chebyshev metric. Despite the simpler structure of orthogonal shapes, there are few works on skeletal representations applied to orthogonal shapes. Much of the efforts have been devoted to binary images and volumes, which are a subset of orthogonal shapes. Two new skeletal representations based on this paradigm are introduced: the cube skeleton and the scale cube skeleton. The cube skeleton is shown to be composed of straight line segments or planar faces and to be homotopical equivalent to the input shape. The scale cube skeleton is based upon the cube skeleton, and introduces a family of skeletons that are more stable to shape noise and perturbations. In addition, the necessary algorithms to compute the cube skeleton of polygons and polyhedra and the scale cube skeleton of polygons are presented. Several experimental results confirm the efficiency, robustness and practical use of all the presented methods.

In this thesis an algorithm for producing saliency maps as well as an algorithm for detecting salient regions based on the saliency map was developed. The saliency values are computed as center-surround differences and a local descriptor called the region p-channel is used to represent center and surround respectively. An integral image representation called the integral p-channel is used to speed up extraction of the local descriptor for any given image region. The center-surround difference is calculated as either histogram or p-channel dissimilarities.

Ground truth was collected using human subjects and the algorithm’s ability to detect salient regions was evaluated against this ground truth. The algorithm was also compared to another saliency algorithm.

Two different center-surround interpretations are tested, as well as several p-channel and histogram dissimilarity measures. The results show that for all tested settings the best performing dissimilarity measure is the so called diffusion distance. The performance comparison showed that the algorithm developed in this thesis outperforms the algorithm against which it was compared, both with respect to region detection and saliency ranking of regions. It can be concluded that the algorithm shows promising results and further investigation of the algorithm is recommended. A list of suggested approaches for further research is provided.