6 research outputs found
R\'enyi Entropy Power Inequalities via Normal Transport and Rotation
Following a recent proof of Shannon's entropy power inequality (EPI), a
comprehensive framework for deriving various EPIs for the R\'enyi entropy is
presented that uses transport arguments from normal densities and a change of
variable by rotation. Simple arguments are given to recover the previously
known R\'enyi EPIs and derive new ones, by unifying a multiplicative form with
constant c and a modification with exponent {\alpha} of previous works. In
particular, for log-concave densities, we obtain a simple transportation proof
of a sharp varentropy bound.Comment: 17 page. Entropy Journal, to appea
The Statistical Foundations of Entropy
In the last two decades, the understanding of complex dynamical systems underwent important conceptual shifts. The catalyst was the infusion of new ideas from the theory of critical phenomena (scaling laws, renormalization group, etc.), (multi)fractals and trees, random matrix theory, network theory, and non-Shannonian information theory. The usual Boltzmann–Gibbs statistics were proven to be grossly inadequate in this context. While successful in describing stationary systems characterized by ergodicity or metric transitivity, Boltzmann–Gibbs statistics fail to reproduce the complex statistical behavior of many real-world systems in biology, astrophysics, geology, and the economic and social sciences.The aim of this Special Issue was to extend the state of the art by original contributions that could contribute to an ongoing discussion on the statistical foundations of entropy, with a particular emphasis on non-conventional entropies that go significantly beyond Boltzmann, Gibbs, and Shannon paradigms. The accepted contributions addressed various aspects including information theoretic, thermodynamic and quantum aspects of complex systems and found several important applications of generalized entropies in various systems
MANIFOLD REPRESENTATIONS OF MUSICAL SIGNALS AND GENERATIVE SPACES
Tra i diversi campi di ricerca nell\u2019ambito dell\u2019informatica musicale, la sintesi e la generazione di segnali audio incarna la pluridisciplinalita\u300 di questo settore, nutrendo insieme le pratiche scientifiche e musicale dalla sua creazione. Inerente all\u2019informatica dalla sua creazione, la generazione audio ha ispirato numerosi approcci, evolvendo colle pratiche musicale e gli progressi tecnologici e scientifici. Inoltre, alcuni processi di sintesi permettono anche il processo inverso, denominato analisi, in modo che i parametri di sintesi possono anche essere parzialmente o totalmente estratti dai suoni, dando una rappresentazione alternativa ai segnali analizzati. Per di piu\u300, la recente ascesa dei algoritmi di l\u2019apprendimento automatico ha vivamente interrogato il settore della ricerca scientifica, fornendo potenti data-centered metodi che sollevavano diversi epistemologici interrogativi, nonostante i sui efficacia. Particolarmente, un tipo di metodi di apprendimento automatico, denominati modelli generativi, si concentrano sulla generazione di contenuto originale usando le caratteristiche che hanno estratti dei dati analizzati. In tal caso, questi modelli non hanno soltanto interrogato i precedenti metodi di generazione, ma anche sul modo di integrare questi algoritmi nelle pratiche artistiche. Mentre questi metodi sono progressivamente introdotti nel settore del trattamento delle immagini, la loro applicazione per la sintesi di segnali audio e ancora molto marginale. In questo lavoro, il nostro obiettivo e di proporre un nuovo metodo di audio sintesi basato su questi nuovi tipi di generativi modelli, rafforazti dalle nuove avanzati dell\u2019apprendimento automatico. Al primo posto, facciamo una revisione dei approcci esistenti nei settori dei sistemi generativi e di sintesi sonore, focalizzando sul posto di nostro lavoro rispetto a questi disciplini e che cosa possiamo aspettare di questa collazione. In seguito, studiamo in maniera piu\u300 precisa i modelli generativi, e come possiamo utilizzare questi recenti avanzati per l\u2019apprendimento di complesse distribuzione di suoni, in un modo che sia flessibile e nel flusso creativo del utente. Quindi proponiamo un processo di inferenza / generazione, il quale rifletta i processi di analisi/sintesi che sono molto usati nel settore del trattamento del segnale audio, usando modelli latenti, che sono basati sull\u2019utilizzazione di un spazio continuato di alto livello, che usiamo per controllare la generazione. Studiamo dapprima i risultati preliminari ottenuti con informazione spettrale estratte da diversi tipi di dati, che valutiamo qualitativamente e quantitativamente. Successiva- mente, studiamo come fare per rendere questi metodi piu\u300 adattati ai segnali audio, fronteggiando tre diversi aspetti. Primo, proponiamo due diversi metodi di regolarizzazione di questo generativo spazio che sono specificamente sviluppati per l\u2019audio : una strategia basata sulla traduzione segnali / simboli, e una basata su vincoli percettivi. Poi, proponiamo diversi metodi per fronteggiare il aspetto temporale dei segnali audio, basati sull\u2019estrazione di rappresentazioni multiscala e sulla predizione, che permettono ai generativi spazi ottenuti di anche modellare l\u2019aspetto dinamico di questi segnali. Per finire, cambiamo il nostro approccio scientifico per un punto di visto piu\u301 ispirato dall\u2019idea di ricerca e creazione. Primo, descriviamo l\u2019architettura e il design della nostra libreria open-source, vsacids, sviluppata per permettere a esperti o non-esperti musicisti di provare questi nuovi metodi di sintesi. Poi, proponiamo una prima utilizzazione del nostro modello con la creazione di una performance in real- time, chiamata \ue6go, basata insieme sulla nostra libreria vsacids e sull\u2019uso di une agente di esplorazione, imparando con rinforzo nel corso della composizione. Finalmente, tramo dal lavoro presentato alcuni conclusioni sui diversi modi di migliorare e rinforzare il metodo di sintesi proposto, nonche\u301 eventuale applicazione artistiche.Among the diverse research fields within computer music, synthesis and generation of audio signals epitomize the cross-disciplinarity of this domain, jointly nourishing both scientific and artistic practices since its creation. Inherent in computer music since its genesis, audio generation has inspired numerous approaches, evolving both with musical practices and scientific/technical advances. Moreover, some syn- thesis processes also naturally handle the reverse process, named analysis, such that synthesis parameters can also be partially or totally extracted from actual sounds, and providing an alternative representation of the analyzed audio signals. On top of that, the recent rise of machine learning algorithms earnestly questioned the field of scientific research, bringing powerful data-centred methods that raised several epistemological questions amongst researchers, in spite of their efficiency. Especially, a family of machine learning methods, called generative models, are focused on the generation of original content using features extracted from an existing dataset. In that case, such methods not only questioned previous approaches in generation, but also the way of integrating this methods into existing creative processes. While these new generative frameworks are progressively introduced in the domain of image generation, the application of such generative techniques in audio synthesis is still marginal. In this work, we aim to propose a new audio analysis-synthesis framework based on these modern generative models, enhanced by recent advances in machine learning. We first review existing approaches, both in sound synthesis and in generative machine learning, and focus on how our work inserts itself in both practices and what can be expected from their collation. Subsequently, we focus a little more on generative models, and how modern advances in the domain can be exploited to allow us learning complex sound distributions, while being sufficiently flexible to be integrated in the creative flow of the user. We then propose an inference / generation process, mirroring analysis/synthesis paradigms that are natural in the audio processing domain, using latent models that are based on a continuous higher-level space, that we use to control the generation. We first provide preliminary results of our method applied on spectral information, extracted from several datasets, and evaluate both qualitatively and quantitatively the obtained results. Subsequently, we study how to make these methods more suitable for learning audio data, tackling successively three different aspects. First, we propose two different latent regularization strategies specifically designed for audio, based on and signal / symbol translation and perceptual constraints. Then, we propose different methods to address the inner temporality of musical signals, based on the extraction of multi-scale representations and on prediction, that allow the obtained generative spaces that also model the dynamics of the signal. As a last chapter, we swap our scientific approach to a more research & creation-oriented point of view: first, we describe the architecture and the design of our open-source library, vsacids, aiming to be used by expert and non-expert music makers as an integrated creation tool. Then, we propose an first musical use of our system by the creation of a real-time performance, called aego, based jointly on our framework vsacids and an explorative agent using reinforcement learning to be trained during the performance. Finally, we draw some conclusions on the different manners to improve and reinforce the proposed generation method, as well as possible further creative applications.A\u300 travers les diffe\u301rents domaines de recherche de la musique computationnelle, l\u2019analysie et la ge\u301ne\u301ration de signaux audio sont l\u2019exemple parfait de la trans-disciplinarite\u301 de ce domaine, nourrissant simultane\u301ment les pratiques scientifiques et artistiques depuis leur cre\u301ation. Inte\u301gre\u301e a\u300 la musique computationnelle depuis sa cre\u301ation, la synthe\u300se sonore a inspire\u301 de nombreuses approches musicales et scientifiques, e\u301voluant de pair avec les pratiques musicales et les avance\u301es technologiques et scientifiques de son temps. De plus, certaines me\u301thodes de synthe\u300se sonore permettent aussi le processus inverse, appele\u301 analyse, de sorte que les parame\u300tres de synthe\u300se d\u2019un certain ge\u301ne\u301rateur peuvent e\u302tre en partie ou entie\u300rement obtenus a\u300 partir de sons donne\u301s, pouvant ainsi e\u302tre conside\u301re\u301s comme une repre\u301sentation alternative des signaux analyse\u301s. Paralle\u300lement, l\u2019inte\u301re\u302t croissant souleve\u301 par les algorithmes d\u2019apprentissage automatique a vivement questionne\u301 le monde scientifique, apportant de puissantes me\u301thodes d\u2019analyse de donne\u301es suscitant de nombreux questionnements e\u301piste\u301mologiques chez les chercheurs, en de\u301pit de leur effectivite\u301 pratique. En particulier, une famille de me\u301thodes d\u2019apprentissage automatique, nomme\u301e mode\u300les ge\u301ne\u301ratifs, s\u2019inte\u301ressent a\u300 la ge\u301ne\u301ration de contenus originaux a\u300 partir de caracte\u301ristiques extraites directement des donne\u301es analyse\u301es. Ces me\u301thodes n\u2019interrogent pas seulement les approches pre\u301ce\u301dentes, mais aussi sur l\u2019inte\u301gration de ces nouvelles me\u301thodes dans les processus cre\u301atifs existants. Pourtant, alors que ces nouveaux processus ge\u301ne\u301ratifs sont progressivement inte\u301gre\u301s dans le domaine la ge\u301ne\u301ration d\u2019image, l\u2019application de ces techniques en synthe\u300se audio reste marginale. Dans cette the\u300se, nous proposons une nouvelle me\u301thode d\u2019analyse-synthe\u300se base\u301s sur ces derniers mode\u300les ge\u301ne\u301ratifs, depuis renforce\u301s par les avance\u301es modernes dans le domaine de l\u2019apprentissage automatique. Dans un premier temps, nous examinerons les approches existantes dans le domaine des syste\u300mes ge\u301ne\u301ratifs, sur comment notre travail peut s\u2019inse\u301rer dans les pratiques de synthe\u300se sonore existantes, et que peut-on espe\u301rer de l\u2019hybridation de ces deux approches. Ensuite, nous nous focaliserons plus pre\u301cise\u301ment sur comment les re\u301centes avance\u301es accomplies dans ce domaine dans ce domaine peuvent e\u302tre exploite\u301es pour l\u2019apprentissage de distributions sonores complexes, tout en e\u301tant suffisamment flexibles pour e\u302tre inte\u301gre\u301es dans le processus cre\u301atif de l\u2019utilisateur. Nous proposons donc un processus d\u2019infe\u301rence / g\ue9n\ue9ration, refle\u301tant les paradigmes d\u2019analyse-synthe\u300se existant dans le domaine de ge\u301ne\u301ration audio, base\u301 sur l\u2019usage de mode\u300les latents continus que l\u2019on peut utiliser pour contro\u302ler la ge\u301ne\u301ration. Pour ce faire, nous e\u301tudierons de\u301ja\u300 les re\u301sultats pre\u301liminaires obtenus par cette me\u301thode sur l\u2019apprentissage de distributions spectrales, prises d\u2019ensembles de donne\u301es diversifie\u301s, en adoptant une approche a\u300 la fois quantitative et qualitative. Ensuite, nous proposerons d\u2019ame\u301liorer ces me\u301thodes de manie\u300re spe\u301cifique a\u300 l\u2019audio sur trois aspects distincts. D\u2019abord, nous proposons deux strate\u301gies de re\u301gularisation diffe\u301rentes pour l\u2019analyse de signaux audio : une base\u301e sur la traduction signal/ symbole, ainsi qu\u2019une autre base\u301e sur des contraintes perceptives. Nous passerons par la suite a\u300 la dimension temporelle de ces signaux audio, proposant de nouvelles me\u301thodes base\u301es sur l\u2019extraction de repre\u301sentations temporelles multi-e\u301chelle et sur une ta\u302che supple\u301mentaire de pre\u301diction, permettant la mode\u301lisation de caracte\u301ristiques dynamiques par les espaces ge\u301ne\u301ratifs obtenus. En dernier lieu, nous passerons d\u2019une approche scientifique a\u300 une approche plus oriente\u301e vers un point de vue recherche & cre\u301ation. Premie\u300rement, nous pre\u301senterons notre librairie open-source, vsacids, visant a\u300 e\u302tre employe\u301e par des cre\u301ateurs experts et non-experts comme un outil inte\u301gre\u301. Ensuite, nous proposons une premie\u300re utilisation musicale de notre syste\u300me par la cre\u301ation d\u2019une performance temps re\u301el, nomme\u301e \ue6go, base\u301e a\u300 la fois sur notre librarie et sur un agent d\u2019exploration appris dynamiquement par renforcement au cours de la performance. Enfin, nous tirons les conclusions du travail accompli jusqu\u2019a\u300 maintenant, concernant les possibles ame\u301liorations et de\u301veloppements de la me\u301thode de synthe\u300se propose\u301e, ainsi que sur de possibles applications cre\u301atives
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Functional inequalities in quantum information theory
Functional inequalities constitute a very powerful toolkit in studying various problems arising in classical information theory, statistics and many-body systems. Extensions of these tools to the noncommutative setting have been introduced in the beginning of the 90's in order to study the asymptotic properties of certain quantum Markovian evolutions. In this thesis, we study various extensions and problems arising from the specific noncommutative nature of such processes.
The first logarithmic Sobolev inequality to be proved, due to Gross, was for the Ornstein Uhlenbeck semigroup, that is the Brownian motion with friction on the real line. The generalization of this result to the quantum Ornstein Uhlenbeck semigroup was found very recently by Carlen and Maas, and de Palma and Huber by means of different techniques. The latter proof consists of a quantum generalization of the so-called entropy power inequality. Here, we consider another possible version of the entropy power inequality and use it to derive asymptotic properties of the frictionless quantum Brownian motion.
The proof of Carlen and Maas discussed in the previous paragraph relies on their new quantum extension of the classical notion of displacement convexity. This is classically known to imply most of the usual functional inequalities such as the modified logarithmic Sobolev inequality and Poincaré's inequality. Here, we further study the framework introduced by Carlen and Maas. In particular, we show how displacement convexity implies quantum functional and transportation cost inequalities. The latter are then used to derive certain concentration inequalities of quantum states in the spirit of Bobkov and Goetze. These concentration inequalities are used in order to derive finite sample size bounds for the task of quantum parameter estimation.
The main advantage of classical logarithmic Sobolev inequalities over other methods resides in their tensorization property: the strong log-Sobolev constant of the product of independent Markovian evolutions is equal to the maximum over the set of strong log-Sobolev constants of the individual evolutions. However, this property is strongly believed to fail in the non-commutative case, due to the non-multiplicativity of noncommutative Lp to Lq norms. In this thesis, we show tensorization of the logarithmic Sobolev constants for the simplest quantum Markov semigroup, namely the generalized depolarizing semigroup. Moreover, we consider a new general method to overcome the issue of tensorization for general primitive quantum Markov semigroups by looking at their contractivity properties under the completely bounded Lp to Lq norms. This method was first investigated in the restricted case of unital semigroups by Beigi and King.
Noncommutative functional inequalities considered in the present literature only deal with primitive quantum Markovian semigroups which model memoryless irreversible dynamics converging to a specific faithful state. However, quantum Markov semigroups can in general display a much richer behavior referred to as decoherence: In particular, under some mild conditions, any such semigroup is known to converge to an algebra of observables which effectively evolve unitarily. Here, we introduce the concept of a decoherence-free logarithmic Sobolev inequality, and the related notion of hypercontractivity of the associated evolution, to study the decoherence rate of non-primitive quantum Markov semigroups. Moreover, we utilize the transference method recently introduced by Gao, Junge and LaRacuente, in order to find decoherence times associated to a class of decoherent Markovian evolutions of great importance in the field of quantum error protection, namely collective decoherence semigroups.
Finally, we develop the notion of quantum reverse hypercontractivity, first introduced by Cubitt, Kastoryano, Montanaro and Temme in the unital case, and apply it in conjunction with the tensorization of the modified logarithmic Sobolev inequality for the generalized depolarizing semigroup in order to find strong converse rates in quantum hypothesis testing and for the classical capacity of classical-quantum channels. Moreover, the transference method also allows us to find strong converse bounds on the various capacities of quantum Markovian evolutions