The Dispersion of the Gauss-Markov Source

The Gauss-Markov source produces U_i = aU_(i–1) + Z_i for i ≥ 1, where U_0 = 0, |a| 0, and we show that the dispersion has a reverse waterfilling representation. This is the first finite blocklength result for lossy compression of sources with memory. We prove that the finite blocklength rate-distortion function R(n; d; ε) approaches the rate-distortion function R(d) as R(n; d; ε) = R(d)+ √ V(d)/n Q–1(ε)+o(1√n), where V (d) is the dispersion, ε ε 2 (0; 1) is the excess-distortion probability, and Q^(-1) is the inverse Q-function. We give a reverse waterfilling integral representation for the dispersion V (d), which parallels that of the rate-distortion functions for Gaussian processes. Remarkably, for all 0 < d ≥ σ^2 (1+|σ|)^2, R(n; d; ε) of the Gauss-Markov source coincides with that of Z_i, the i.i.d. Gaussian noise driving the process, up to the second-order term. Among novel technical tools developed in this paper is a sharp approximation of the eigenvalues of the covariance matrix of n samples of the Gauss-Markov source, and a construction of a typical set using the maximum likelihood estimate of the parameter a based on n observations

The Dispersion of the Gauss-Markov Source

The Gauss-Markov source produces U_i=aU_(i-1)+ Z_i for i ≥ 1, where U_0 = 0, |a| 0, and we show that the dispersion has a reverse waterfilling representation. This is the first finite blocklength result for lossy compression of sources with memory. We prove that the finite blocklength rate-distortion function R(n, d, ε) approaches the rate-distortion function R(d) as R(n, d, ε) = R(d)+√{[V(d)/n]}Q^(-1)(ε)+o([1/(√n)]), where V(d) is the dispersion, ε ∈ (0,1) is the excess-distortion probability, and Q^(-1) is the inverse of the Q-function. We give a reverse waterfilling integral representation for the dispersion V (d), which parallels that of the rate-distortion functions for Gaussian processes. Remarkably, for all 0 <; d ≤ σ2/(1+|a|)^2 ,R(n, d, c) of the Gauss-Markov source coincides with that of Zi, the i.i.d. Gaussian noise driving the process, up to the second-order term. Among novel technical tools developed in this paper is a sharp approximation of the eigenvalues of the covariance matrix of n samples of the Gauss-Markov source, and a construction of a typical set using the maximum likelihood estimate of the parameter a based on n observations

The Dispersion of the Gauss-Markov Source

The Gauss-Markov source produces U_i = aU_(i–1) + Z_i for i ≥ 1, where U_0 = 0, |a| 0, and we show that the dispersion has a reverse waterfilling representation. This is the first finite blocklength result for lossy compression of sources with memory. We prove that the finite blocklength rate-distortion function R(n; d; ε) approaches the rate-distortion function R(d) as R(n; d; ε) = R(d)+ √ V(d)/n Q–1(ε)+o(1√n), where V (d) is the dispersion, ε ε 2 (0; 1) is the excess-distortion probability, and Q^(-1) is the inverse Q-function. We give a reverse waterfilling integral representation for the dispersion V (d), which parallels that of the rate-distortion functions for Gaussian processes. Remarkably, for all 0 < d ≥ σ^2 (1+|σ|)^2, R(n; d; ε) of the Gauss-Markov source coincides with that of Z_i, the i.i.d. Gaussian noise driving the process, up to the second-order term. Among novel technical tools developed in this paper is a sharp approximation of the eigenvalues of the covariance matrix of n samples of the Gauss-Markov source, and a construction of a typical set using the maximum likelihood estimate of the parameter a based on n observations

Analysis and correction of the helium speech effect by autoregressive signal processing

SIGLELD:D48902/84 / BLDSC - British Library Document Supply CentreGBUnited Kingdo

Computation of the one-dimensional unwrapped phase

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2006.Includes bibliographical references (p. 101-102). "Cepstrum bibliography" (p. 67-100).In this thesis, the computation of the unwrapped phase of the discrete-time Fourier transform (DTFT) of a one-dimensional finite-length signal is explored. The phase of the DTFT is not unique, and may contain integer multiple of 27r discontinuities. The unwrapped phase is the instance of the phase function chosen to ensure continuity. This thesis presents existing algorithms for computing the unwrapped phase, discussing their weaknesses and strengths. Then two composite algorithms are proposed that use the existing ones, combining their strengths while avoiding their weaknesses. The core of the proposed methods is based on recent advances in polynomial factoring. The proposed methods are implemented and compared to the existing ones.by Zahi Nadim Karam.S.M

Diffeomorphic Transformations for Time Series Analysis: An Efficient Approach to Nonlinear Warping

The proliferation and ubiquity of temporal data across many disciplines has sparked interest for similarity, classification and clustering methods specifically designed to handle time series data. A core issue when dealing with time series is determining their pairwise similarity, i.e., the degree to which a given time series resembles another. Traditional distance measures such as the Euclidean are not well-suited due to the time-dependent nature of the data. Elastic metrics such as dynamic time warping (DTW) offer a promising approach, but are limited by their computational complexity, non-differentiability and sensitivity to noise and outliers. This thesis proposes novel elastic alignment methods that use parametric \& diffeomorphic warping transformations as a means of overcoming the shortcomings of DTW-based metrics. The proposed method is differentiable \& invertible, well-suited for deep learning architectures, robust to noise and outliers, computationally efficient, and is expressive and flexible enough to capture complex patterns. Furthermore, a closed-form solution was developed for the gradient of these diffeomorphic transformations, which allows an efficient search in the parameter space, leading to better solutions at convergence. Leveraging the benefits of these closed-form diffeomorphic transformations, this thesis proposes a suite of advancements that include: (a) an enhanced temporal transformer network for time series alignment and averaging, (b) a deep-learning based time series classification model to simultaneously align and classify signals with high accuracy, (c) an incremental time series clustering algorithm that is warping-invariant, scalable and can operate under limited computational and time resources, and finally, (d) a normalizing flow model that enhances the flexibility of affine transformations in coupling and autoregressive layers.Comment: PhD Thesis, defended at the University of Navarra on July 17, 2023. 277 pages, 8 chapters, 1 appendi

Quelques Aspects des Réseaux Multi-Cellules Multi-Utilisateurs MIMO : Délai, Conception d'Emetteur-Récepteur, Sélection d'Utilisateurs et Topologie

In order to meet ever-growing needs for capacity in wireless networks, transmission techniques and the system models used to study their performances have rapidly evolved. From single-user single-antenna point-to-point communications to modern multi-cell multi-antenna cellular networks there have been large advances in technology. Along the way, several assumptions are made in order to have either more realistic models, but also to allow simpler analysis. We analyze three aspects of actual networks and try to benefit from them when possible or conversely, to mitigate their negative impact. This sometimes corrects overly optimistic results, for instance when delay in the channel state information (CSI) acquisition is no longer neglected. However, this sometimes also corrects overly pessimistic results, for instance when in a broadcast channel (BC) the number of users is no longer limited to be equal to the number of transmit antennas or when partial connectivity is taken into account in cellular networks.We first focus on the delay in the CSI acquisition because it greatly impairs the channel multiplexing gain if nothing is done to use the dead time during which the transmitters are not transmitting and do not yet have the CSI. We review and propose different schemes to use this dead time to improve the multiplexing gain in both the BC and the interference channel (IC). We evaluate the more relevant net multiplexing gain, taking into account the training and feedback overheads. Results are surprising because potential schemes to fight delay reveal to be burdened by impractical overheads in the BC. In the IC, an optimal scheme is proposed. It allows avoiding any loss of multiplexing gain even for significant feedback delay. Concerning the number of users, we propose a new criterion for the greedy user selection in a BC to benefit of the multi-user diversity, and two interference alignment schemes for the IC to benefit of having multiple users in each cell. Finally, partially connected cellular networks are considered and schemes to benefit from said partial connectivity to increase the multiplexing gain are proposed.Afin de répondre au besoin sans cesse croissant de capacité dans les réseaux sans fil, les techniques de transmission, et les modèles utilisés pour les étudier, ont évolués rapidement. De simples communications point à point avec une seuleantenne nous sommes passé aux réseaux cellulaires de nos jours: de multiples cellules et de multiples antennes à l’émission et à la réception. Progressivement, plusieurs hypothèses ont été faites, soit afin d’avoir des modèles réalistes, mais aussi parfois pour permettre une analyse plus simple. Nous examinons et analysons l’impact de trois aspects des réseaux réels. Cela revient parfois à corriger des résultats trop optimistes, par exemple lorsque le délai dans l’acquisition des coefficients des canaux n’est plus négligé. Cela revient parfois à corriger des résultats trop pessimistes, par exemple, lorsque dans un canal de diffusion (BC) le nombre d’utilisateurs n’est plus limité au nombre d’antennes d’émission ou lorsque la connectivité partielle est prise en compte dans les réseaux cellulaires. Plus précisément, dans cette thèse, nous nous concentrons sur le délai dans l’acquisition des coefficients des canaux par l’émetteur puisque sa prise en comptedétériore grandement le gain de multiplexage du canal si rien n’est fait pour utiliser efficacement le temps mort au cours duquel les émetteurs ne transmettent pas et n’ont pas encore la connaissance du canal. Nous examinons et proposons des schémas de transmission pour utiliser efficacement ce temps mort afin d’améliorer le gain de multiplexage. Nous évaluons le gain de multiplexage net, plus pertinent, en tenant compte le temps passé à envoyer symboles d’apprentissage et à les renvoyer aux transmetteurs. Les résultats sont surprenant puisque les schémas contre le retard de connaissance de canal se révèle être impraticables à cause du cout du partage de la connaissance des canaux. Dans les réseaux multi-cellulaires, un schéma de transmission optimal est proposé et permet de n’avoir aucune perte de gain de multiplexage même en cas de retard important dans la connaissance de canal. En ce qui concerne le nombre d’utilisateurs, nous proposons un nouveau critère pour la sélection des utilisateurs de les configurations à une seule cellule afin de bénéficier de la diversité multi-utilisateurs, et nous proposons deux schémas d’alignement d’interférence pour systèmes multi-cellulaires afin de bénéficier du fait qu’il y a généralement plusieurs utilisateurs dans chaque cellule. Enfin, les réseaux cellulaires partiellement connectés sont étudiés et des schémas bénéficiant de la connectivité partielle pour augmenter le gain de multiplexage sont proposés

Gaussian latent tree model constraints for linguistics and other applications

The relationships between languages are often modelled as phylogenetic trees whereby there is a single shared ancestral language at the root and contemporary languages appear as leaves. These can be thought of as directed acyclic graphs with hidden variables, specifically Bayesian networks. However, from a statistical perspective there is often no formal assessment of the suitability of these latent tree models. A lot of the work that seeks to address this has focused on discrete variable models. However, when observations are instead considered as functional data, the high dimensional approximations are often better considered in a Gaussian context. The high dimensional data is often inefficiently stored and so the first challenge is to project this data to a low dimension while retaining the information of interest. One approach is to use the newly developed tool named separable-canonical variate analysis to form a basis. Extending the techniques for assessing latent tree model compatibility to beyond discrete variables, the complete set of Gaussian tree constraints are derived for the first time. This set comprises equations and inequality statements in terms of correlations of observed variables. These statements must in theory be adhered to for a Gaussian latent tree model to be appropriate for a given data set. Using the separable-canonical variate analysis basis to obtain a truncated representation, the suitability of a phylogenetic tree can then be plainly assessed. However, in practice it is desirable to allow for some sampling error and as such probabilistic tools are developed alongside the theoretical derivation of Gaussian tree constraints. The proposed methodology is implemented in an in-depth study of a real linguistic data set to assess the phylogenies of five Romance languages. This application is distinctive as the data set consists of acoustic recordings, these are treated as functional data, and moreover these are then being used to compare languages in a phylogenetic context. As a consequence a wide range of theory and tools are called upon from the multivariate and functional domains, and the powerful new separable-canonical function analysis and separable-canonical variate analysis are used. Utilising the newly derived Gaussian tree constraints for hidden variable models provides a first insight into features of spoken languages that appear to be tree-compatible

Complexity in Economic and Social Systems

There is no term that better describes the essential features of human society than complexity. On various levels, from the decision-making processes of individuals, through to the interactions between individuals leading to the spontaneous formation of groups and social hierarchies, up to the collective, herding processes that reshape whole societies, all these features share the property of irreducibility, i.e., they require a holistic, multi-level approach formed by researchers from different disciplines. This Special Issue aims to collect research studies that, by exploiting the latest advances in physics, economics, complex networks, and data science, make a step towards understanding these economic and social systems. The majority of submissions are devoted to financial market analysis and modeling, including the stock and cryptocurrency markets in the COVID-19 pandemic, systemic risk quantification and control, wealth condensation, the innovation-related performance of companies, and more. Looking more at societies, there are papers that deal with regional development, land speculation, and the-fake news-fighting strategies, the issues which are of central interest in contemporary society. On top of this, one of the contributions proposes a new, improved complexity measure