Point Information Gain and Multidimensional Data Analysis

We generalize the Point information gain (PIG) and derived quantities, i.e. Point information entropy (PIE) and Point information entropy density (PIED), for the case of R\'enyi entropy and simulate the behavior of PIG for typical distributions. We also use these methods for the analysis of multidimensional datasets. We demonstrate the main properties of PIE/PIED spectra for the real data on the example of several images, and discuss possible further utilization in other fields of data processing.Comment: 16 pages, 6 figure

On some entropy functionals derived from R\'enyi information divergence

We consider the maximum entropy problems associated with R\'enyi $Q$-entropy, subject to two kinds of constraints on expected values. The constraints considered are a constraint on the standard expectation, and a constraint on the generalized expectation as encountered in nonextensive statistics. The optimum maximum entropy probability distributions, which can exhibit a power-law behaviour, are derived and characterized. The R\'enyi entropy of the optimum distributions can be viewed as a function of the constraint. This defines two families of entropy functionals in the space of possible expected values. General properties of these functionals, including nonnegativity, minimum, convexity, are documented. Their relationships as well as numerical aspects are also discussed. Finally, we work out some specific cases for the reference measure $Q(x)$ and recover in a limit case some well-known entropies

Estimating the number of components of a multicomponent nonstationary signal using the short-term time-frequency Rényi entropy

This article proposes a method for estimating the local number of signals components using the short term Rényi entropy of signals in the time-frequency plane. (Additional details can be found in the comprehensive book on Time-Frequency Signal Analysis and Processing (see http://www.elsevier.com/locate/isbn/0080443354). In addition, the most recent upgrade of the original software package that calculates Time-Frequency Distributions and Instantaneous Frequency estimators can be downloaded from the web site: www.time-frequency.net. This was the first software developed in the field, and it was first released publicly in 1987 at the 1st ISSPA conference held in Brisbane, Australia, and then continuously updated).The time-frequency Rényi entropy provides a measure of complexity of a nonstationary multicomponent signal in the time-frequency plane. When the complexity of a signal corresponds to the number of its components, then this information is measured as the Rényi entropy of the time-frequency distribution (TFD) of the signal. This article presents a solution to the problem of detecting the number of components that are present in short-time interval of the signal TFD, using the short-term Rényi entropy. The method is automatic and it does not require a prior information about the signal. The algorithm is applied on both synthetic and real data, using a quadratic separable kernel TFD. The results confirm that the short-term Rényi entropy can be an effective tool for estimating the local number of components present in the signal. The key aspect of selecting a suitable TFD is also discussed

Quasi-particle spectrum and entanglement generation after a quench in the quantum Potts spin chain

Recently, a non-trivial relation between the quasi-particle spectrum and entanglement entropy production was discovered in non-integrable quenches in the paramagnetic Ising quantum spin chain. Here we study the dynamics of analogous quenches in the quantum Potts spin chain. Tuning the parameters of the system, we observe a sudden increase in the entanglement production rate, which is shown to be related to the appearance of new quasiparticle excitations in the post-quench spectrum. Our results demonstrate the generality of the effect and support its interpretation as the non-equilibrium version of the well-known Gibbs paradox related to mixing entropy which appears in systems with a non-trivial quasi-particle spectrum.Comment: 15 pages, pdflatex, 30 pdf figures. v2: reformatted, 22 pages, typos correcte

Blind image quality assessment through anisotropy

We describe an innovative methodology for determining the quality of digital images. The method is based on measuring the variance of the expected entropy of a given image upon a set of predefined directions. Entropy can be calculated on a local basis by using a spatial/ spatial-frequency distribution as an approximation for a probability density function. The generalized Rényi entropy and the normalized pseudo-Wigner distribution (PWD) have been selected for this purpose. As a consequence, a pixel-by-pixel entropy value can be calculated, and therefore entropy histograms can be generated as well. The variance of the expected entropy is measured as a function of the directionality, and it has been taken as an anisotropy indicator. For this purpose, directional selectivity can be attained by using an oriented 1-D PWD implementation, Our main purpose is to show how such an anisotropy measure can be used as a metric to assess both the fidelity and quality of images. Experimental results show that an index such as this presents some desirable features that resemble those from an ideal image quality function, constituting a suitable quality index for natural images. Namely, in-focus, noise-free natural images have shown a maximum of this metric in comparison with other degraded, blurred, or noisy versions. This result provides a way of identifying in-focus, noise-free images from other degraded versions, allowing an automatic and nonreference classification of images according to their relative quality. It is also shown that the new measure is well correlated with classical reference metrics such as the peak signal-to-noise ratio. © 2007 Optical Society of America.This research has been supported by the following projects: TEC2004-00834, TEC2005-24739-E, TEC2005- 24046-E, and 20045OE184 from the Spanish Ministry of Education and Science and PI040765 from the Spanish Ministry of Health.Peer Reviewe

Synchro-Transient-Extracting Transform for the Analysis of Signals with Both Harmonic and Impulsive Components

Time-frequency analysis (TFA) techniques play an increasingly important role in the field of machine fault diagnosis attributing to their superiority in dealing with nonstationary signals. Synchroextracting transform (SET) and transient-extracting transform (TET) are two newly emerging techniques that can produce energy concentrated representation for nonstationary signals. However, SET and TET are only suitable for processing harmonic signals and impulsive signals, respectively. This poses a challenge for each of these two techniques when a signal contains both harmonic and impulsive components. In this paper, we propose a new TFA technique to solve this problem. The technique aims to combine the advantages of SET and TET to generate energy concentrated representations for both harmonic and impulsive components of the signal. Furthermore, we theoretically demonstrate that the proposed technique retains the signal reconstruction capability. The effectiveness of the proposed technique is verified using numerical and real-world signals

Detecting the number of components in a non-stationary signal using the Rényi entropy of its time-frequency distributions

A time-frequency distribution provides many advantages in the analysis of multicomponent non-stationary signals. The simultaneous signal representation with respect to the time and frequency axis defines the signal amplitude, frequency, bandwidth, and the number of components at each time moment. The Rényi entropy, applied to a time-frequency distribution, is shown to be a valuable indicator of the signal complexity. The aim of this paper is to determine which of the treated time-frequency distributions (TFDs) (namely, the Wigner-Ville distribution, the Choi-Williams distribution, and the spectrogram) has the best properties for estimation of the number of components when there is no prior knowledge of the signal. The optimal Rényi entropy parameter α is determined for each TFD. Accordingly, the effects of different time durations, bandwidths and amplitudes of the signal components on the Rényi entropy have been analysed. The concept of a class, when the Rényi entropy is applied to TFDs, is also introduced

Phases and phase transitions in non-equilibrium quantum matter

This thesis focuses on two recent examples of non-equilibrium quantum phase transitions. In the first part we discuss discrete time crystals (DTCs), which are defined by the fact that they spontaneously break discrete time-translation symmetry. In early realizations of DTCs, many-body localization (MBL) played a crucial role in preventing the periodic drive from heating the system to infinite temperature, which would preclude any possibility of symmetry-breaking. This thesis explores the possibility that dissipation may play an equivalent role, allowing for the possibility of time-translation symmetry-breaking without MBL. We describe the results of an experiment exploring DTC order in a doped semiconductor system with significant dissipation, and a potential description of the interplay of driving, dissipation and interactions using a central spin model. In the second part we discuss measurement-induced phase transitions, where the steady-state entanglement can undergo a phase transition as a function of the measurement rate. First we explore the role of the underlying unitary dynamics in the nature of the phase transition. Previous work has revealed an apparent dichotomy between interacting and non-interacting systems, where interacting systems have a phase transition from volume-law to area-law entanglement at a finite critical measurement rate p, whereas the volume-law for non-interacting systems is destroyed at any p > 0. We study this transition for MBL systems, and find an interpolation between these extremes depending on the measurement basis. We discuss the relevance of the emergent integrability characteristic of MBL and how this intersects with the measurements. Next we study the critical properties of this transition in random 1+1D and 2+1D Clifford circuits, aiming to explore connections with percolation. We utilize a graph-state based simulation algorithm, which provides access to geometric properties of entanglement. We find bulk exponents close to percolation, but possible differences in surface behaviour