Two-parameter noncommutative Gaussian processes

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 225-237).The reality of billion-user networks and multi-terabyte data sets brings forth the need for accurate and computationally tractable descriptions of large random structures, such as random matrices or random graphs. The modern mathematical theory of free probability is increasingly giving rise to analysis tools specifically adapted to such large-dimensional regimes and, more generally, non-commutative probability is emerging as an area of interdisciplinary interest. This thesis develops a new non-commutative probabilistic framework that is both a natural generalization of several existing frameworks (viz. free probability, q-deformed probability) and a setting in which to describe a broader class of random matrix limits. From the practical perspective, this new setting is particularly interesting in its ability to characterize the behavior of large random objects that asymptotically retain a certain degree of commutative structure and therefore fall outside the scope of free probability. The type of commutative structure considered is modeled on the two-parameter families of generalized harmonic oscillators found in physics and the presently introduced framework may be viewed as a two-parameter deformation of classical probability. Specifically, we introduce (1) a generalized Non-commutative Central Limit Theorem giving rise to a two-parameter deformation of the classical Gaussian statistics and (2) a two-parameter continuum of non-commutative probability spaces in which to realize these statistics. The framework that emerges has a remarkably rich combinatorial structure and bears upon a number of well-known mathematical objects, such as a quantum deformation of the Airy function, that had not previously played a prominent role in a probabilistic setting. Finally, the present framework paves the way to new types of asymptotic results, by providing more general asymptotic theorems and revealing new layers of structure in previously known results, notably in the "correlated process version" of Wigner's Semicircle Law.by Natasha Blitvić.Ph.D

Mathematical Aspects of Hopfield Models

Diese Dissertation behandelt zwei Modelle aus der statistischen Mechanik ungeordneter Systeme. Beide sind Varianten des Hopfield-Modells und gehören zur Klasse der Molekularfeldmodelle. Im ersten Teil behandeln wir den Fall mit p-Spin-Wechselwirkungen (p größer als 4 und gerade) und superextensiv vielen Mustern (deren Anzahl M wie die p-te Potenz der Systemgröße N wächst), wobei wir zwei verschiedene Energiefunktionen betrachten. Wir beweisen die Existenz einer kritischen Temperatur, bei welcher der sogenannte Replikaüberlapp von Null auf einen strikt positiven Wert springt. Wir geben obere und untere Schranken für ihren Wert an und zeigen, daß für die eine Wahl der Hamiltonfunktion beide gegen die kritische Temperatur (bis auf einen konstanten Faktor) des Random Energy Model konvergieren, falls p gegen Unendlich strebt. Diese kritische Temperatur fällt mit der kleinsten Temperatur zusammen, für welche die ausgeglühte freie Energie und der Erwartungswert der abgeschreckten freien Energie identisch sind. Der Zusammenhang zwischen diesen beiden Resultaten wird durch eine partielle Integrationsformel geliefert, welche mit Hilfe einer Störungsentwicklung der Boltzmannfaktoren bewiesen wird. Außerdem berechnen wir die Fluktuationen der freien Energie und erhalten, daß sie von der Ordnung Quadratwurzel von N sind. Weiterhin beweisen wir die Existenz einer kritischen Proportionalitätskonstanten für die Anzahl Muster, oberhalb welcher das Minimum der Hamiltonfunktion mit großer Wahrscheinlichkeit nicht in der Nähe eines der Muster angenommen wird. Dies bedeutet, daß, obwohl das Gibbsmaß sich bei kleinen Temperaturen auf einer kleinen Teilmenge des Zustandsraumes konzentriert, dies nicht in der Nähe der Muster geschieht. In einem zweiten Teil beweisen wir obere Schranken für die Norm von gewissen zufälligen Matrizen mit abhängigen Einträgen. Diese Abschätzungen werden im ersten Teil zur Berechnung der Fluktuationen der freien Energie benutzt. Sie werden mit der Chebyshev-Markov-Ungleichung, angewandt auf die Spur von hohen Potenzen der Matrizen, bewiesen. Das zentrale Resultat dazu ist eine Darstellung der Spur von diesen hohen Potenzen als Wege auf gewissen bipartiten Graphen. Dies transformiert das Berechnen des Erwartungswertes der Spur auf das kombinatorische Problem, die maximale Anzahl kreisförmiger Teilgraphen eines gegebenen Eulergraphen zu bestimmen. Die Resultate zeigen, dass die Abhängigkeit zwischen den Einträgen eine wichtige Rolle spielt und nicht vernachlässigt werden kann. Im letzten Teil schließlich betrachten wir ein Hopfield-Modell mit zwei Gauß'schen Mustern. Wir zeigen, da$szlig; überabzählbar viele extremale Gibbszustände existieren, welche durch den Einheitskreis indiziert werden. Diese Symmetrie wird zufällig gebrochen im Sinne, daß der Metazustand von einem Kontinuum von Paaren von extremalen Gibbsmaßen getragen wird, welche durch eine globale Spinsymmetrie verknüpft sind. Wir beweisen diese Resultate mit Hilfe der zufälligen Ratenfunktion des induzierten Maßes auf den Überlapparametern. Insbesondere zeigen wir, daß die Symmetriebrechung durch die Fluktuationen der Ratenfunktion auf den (entarteten) Minima ihrer Erwartung erzwungen wird. Diese Fluktuationen werden durch einen zufälligen Prozeß auf dem Einheitskreis beschrieben, dessen globale Minima die Menge (schlussendlich ein Paar) der extremalen Zustände auswählen.This thesis is concerned with two models from equilibrium statistical mechanics of disordered systems. Both of them are variants of the Hopfield model, and belong to the class of mean-field models. In the first part, we treat the case of p-spin interactions (with p larger than 4 and even) and super-extensively many patterns (their number M scaling as the (p-1)th power of the system size N). We consider two choices of the Hamiltonians. We find that there exists a critical temperature, at which the replica overlap changes from 0 to a strictly positive value. We give upper and lower bounds for its value, and show that for one choice of the Hamiltonian, both of them converge as p tends to infinity to the critical temperature (up to a constant factor) of the random energy model. This critical temperature coincides with the minimum temperature for which annealed free energy and mean of the quenched free energy are equal. The relation between the two results is furnished by an integration by parts formula that is proved by perturbative expansion of the Boltzmann factors. We also calculate the fluctuations of the free energy which are shown to be of the order of the square root of the system size N. Furthermore, we find that there exists a critical scaling constant for the number of patterns above which with large probability the minimum of the Hamiltonian is not realized in the vicinity of any of the patterns. This means that while there is a condensation for low temperatures, the Gibbs measure does not concentrate around the patterns. In a second part of the thesis, we prove upper bounds on the norm of certain random matrices with dependent entries. These estimates are used in Part I to prove the fluctuations of the free energy. They are proved by the Chebyshev-Markov inequality, applied to the trace of large powers of the matrices. The key ingredient is a representation of the trace of these large powers in terms of walks on an appropriate bipartite graph. This reduces the calculation of the expectation of the trace to the combinatorial problem of counting the maximum number of sub-circuits of a given circuit. The results show that the dependence between the entries cannot be neglected. Finally, in the last part, we consider a two pattern Hopfield model with Gaussian patterns. We show that there are uncountably many pure states indexed by the circle. This symmetry is randomly broken in the sense that the metastate is supported on a continuum of pairs of pure states that are related by a global (spin-flip) symmetry. We prove these results by studying the random rate function of the induced measure on the overlap parameters. In particular, the breaking of the symmetry is shown to be due to the fluctuations of this rate function at the (degenerate) minima of its expectation. These fluctuations are described by a random process on the circle whose global minima determine the chosen set (eventually a pair) of states

Separability between signal and noise components using the distribution of scaled Hankel matrix eigenvalues with application in biomedical signals.

Biomedical signals are records from human and animal bodies. These records are considered as nonlinear time series, which hold important information about the physiological activities of organisms, and embrace many subjects of interest. However, biomedical signals are often corrupted by artifacts and noise, which require separation or signal extraction before any statistical evaluation. Another challenge in analysing biomedical signals is that their data is often non-stationary, particularly when there is an abnormal event observed within the signal, such as epileptic seizure, and can also present chaotic behaviour. The literature suggests that distinguishing chaos from noise continues to remain a highly contentious issue in the modern age, as it has been historically. This is because chaos and noise share common properties, which in turn make them indistinguishable. We seek to provide a viable solution to this problem by presenting a novel approach for the separability between signal and noise components and the differentiation of noise from chaos. Several methods have been used for the analysis of and discrimination between different categories of biomedical signals, but many of these are based on restrictive assumptions of the normality, stationarity and linearity of the observed data. Therefore, an improved technique which is robust in its analysis of non-stationary time series is of paramount importance in accurate diagnosis of human diseases. The SSA (Singular Spectrum Analysis) technique does not depend on these assumptions, which could be very helpful for analysing and modelling biomedical data. Therefore, the main aim of the thesis is to provide a novel approach for developing the SSA technique, and then apply it to the analysis of biomedical signals. SSA is a reliable technique for separating an arbitrary signal from a noisy time series (signal+noise). It is based upon two main selections: window length, L; and the number of eigenvalues, r. These values play an important role in the reconstruction and forecasting stages. However, the main issue in extracting signals using the SSA procedure lies in identifying the optimal values of L and r required for signal reconstruction. The aim of this thesis is to develop theoretical and methodological aspects of the SSA technique, to present a novel approach to distinguishing between deterministic and stochastic processes, and to present an algorithm for identifying the eigenvalues corresponding to the noise component, and thereby choosing the optimal value of r relating to the desired signal for separability between signal and noise. The algorithm used is considered as an enhanced version of the SSA method, which decomposes a noisy signal into the sum of a signal and noise. Although the main focus of this thesis is on the selection of the optimal value of r, we also provide some results and recommendations to the choice of L for separability. Several criteria are introduced which characterise this separability. The proposed approach is based on the distribution of the eigenvalues of a scaled Hankel matrix, and on dynamical systems, embedding theorem, matrix algebra and statistical theory. The research demonstrates that the proposed approach can be considered as an alternative and promising technique for choosing the optimal values of r and L in SSA, especially for biomedical signals and genetic time series. For the theoretical development of the approach, we present new theoretical results on the eigenvalues of a scaled Hankel matrix, provide some properties of the eigenvalues, and show the effect of the window length and the rank of the Hankel matrix on the eigenvalues. The new theoretical results are examined using simulated and real time series. Furthermore, the effect of window length on the distribution of the largest and smallest eigenvalues of the scaled Hankel matrix is also considered for the white noise process. The results indicate that the distribution of the largest eigenvalue for the white noise process has a positive skewed distribution for different series lengths and different values of window length, whereas the distribution of the smallest eigenvalue has a different pattern with L; the distribution changes from left to right when L increases. These results, together with other results obtained by the different criteria introduced and used in this research, are very promising for the identification of the signal subspace. For the practical aspect and empirical results, various biomedical signals and genetics time series are used. First, to achieve the objectives of the thesis, a comprehensive study has been made on the distribution, pattern; and behaviour of scaled Furthermore, the normal distribution with different parameters is considered and the effect of scale and shape parameters are evaluated. The correlation between eigenvalues is also assessed, using parametric and non-parametric association criteria. In addition, the distribution of eigenvalues for synthetic time series generated from some well known low dimensional chaotic systems are analysed in-depth. The results yield several important properties with broad application, enabling the distinction between chaos and noise in time series analysis. At this stage, the main result of the simulation study is that the findings related to the series generated from normal distribution with mean zero (white noise process) are totally different from those obtained for other series considered in this research, which makes a novel contribution to the area of signal processing and noise reduction. Second, the proposed approach and its criteria are applied to a number of simulated and real data with different levels of noise and structures. Our results are compared with those obtained by common and well known criteria in order to evaluate, enhance and confirm the accuracy of the approach and its criteria. The results indicate that the proposed approach has the potential to split the eigenvalues into two groups; the first corresponding to the signal and the second to the noise component. In addition, based on the results, the optimal value of L that one needs for the reconstruction of a noise free signal from a noisy series should be the median of the series length. The results confirm that the performance of the proposed approach can improve the quality of the reconstruction step for signal extraction. Finally, the thesis seeks to explore the applicability of the proposed approach for discriminating between normal and epileptic seizure electroencephalography (EEG) signals, and filtering the signal segments to make them free from noise. Various criteria based on the largest eigenvalue are also presented and used as features to distinguish between normal and epileptic EEG segments. These features can be considered as useful information to classify brain signals. In addition, the approach is applied to the removal of nonspecific noise from Drosophila segmentation genes. Our findings indicate that when extracting signal from different genes, for optimised signal and noise separation, a different number of eigenvalues need to be chosen for each gene

Teaching/Learning Physics: Integrating Research into Practice

The GIREP-MPTL International conference on Teaching/Learning Physics: Integrating Research into Practice [GIREP-MPTL 2014] was held from 7 to 12 July 2014 at the University of Palermo, Italy. The conference has been organised by the Groupe International de Recherche sur l’Enseignement de la Physique [GIREP] and the Multimedia in Physics Teaching and Learning [MPTL] group and it has been sponsored by the International Commission on Physics Education [ICPE] – Commission 14 of the International Union for Pure and Applied Physics [IUPAP], the European Physical Society – Physics Education Division [EPS-PED], the Latin American Physics Education Network [LAPEN] and the Società Italiana di Fisica [SIF]. The theme of the conference, Teaching/Learning Physics: Integrating Research into Practice, underlines aspects of great relevance in contemporary science education. In fact, during the last few years, evidence based Physics Education Research provided results concerning the ways and strategies to improve student conceptual understanding, interest in Physics, epistemological awareness and insights for the construction of a scientific citizenship. However, Physics teaching practice seems resistant to adopting adapting these findings to their own situation and new research based curricula find difficulty in affirming and spread, both at school and university levels. The conference offered an opportunity for in-depth discussions of this apparently wide-spread tension in order to find ways to do better. The purpose of the GIREP-MPTL 2014 was to bring together people working in physics education research and in physics education at schools from all over the world to allow them to share research results and exchange their experience. About 300 teachers, educators, and researchers, from all continents and 45 countries have attended the Conference contributing with 177 oral presentations, 15 workshops, 11 symposia, and around 60 poster presentations, together with 11 keynote addresses (general talks). After the conference, 147 papers have been submitted for the GIREP-MPTL 2014 International Conference proceedings. Each paper has been reviewed by at least two reviewers, from countries that are different to those of the authors and on the basis of criteria described on the Conference web site. Papers were subsequently revised by authors according to reviewers’ comments and the accepted papers are reported in this book, divided in 8 Sections on the basis of the keywords suggested by authors. The other book section (actually, the first one) contains the papers that six of the keynote talkers sent for publication in this Proceedings Book. We would like to thank all the authors that contributed with their papers to the realization of this book and all the referees that with their criticism helped authors to improve the quality of the papers