Sharp Bounds for Optimal Decoding of Low Density Parity Check Codes

Consider communication over a binary-input memoryless output-symmetric channel with low density parity check (LDPC) codes and maximum a posteriori (MAP) decoding. The replica method of spin glass theory allows to conjecture an analytic formula for the average input-output conditional entropy per bit in the infinite block length limit. Montanari proved a lower bound for this entropy, in the case of LDPC ensembles with convex check degree polynomial, which matches the replica formula. Here we extend this lower bound to any irregular LDPC ensemble. The new feature of our work is an analysis of the second derivative of the conditional input-output entropy with respect to noise. A close relation arises between this second derivative and correlation or mutual information of codebits. This allows us to extend the realm of the interpolation method, in particular we show how channel symmetry allows to control the fluctuations of the overlap parameters.Comment: 40 Pages, Submitted to IEEE Transactions on Information Theor

Free energies of Boltzmann Machines: self-averaging, annealed and replica symmetric approximations in the thermodynamic limit

Restricted Boltzmann machines (RBMs) constitute one of the main models for machine statistical inference and they are widely employed in Artificial Intelligence as powerful tools for (deep) learning. However, in contrast with countless remarkable practical successes, their mathematical formalization has been largely elusive: from a statistical-mechanics perspective these systems display the same (random) Gibbs measure of bi-partite spin-glasses, whose rigorous treatment is notoriously difficult. In this work, beyond providing a brief review on RBMs from both the learning and the retrieval perspectives, we aim to contribute to their analytical investigation, by considering two distinct realizations of their weights (i.e., Boolean and Gaussian) and studying the properties of their related free energies. More precisely, focusing on a RBM characterized by digital couplings, we first extend the Pastur-Shcherbina-Tirozzi method (originally developed for the Hopfield model) to prove the self-averaging property for the free energy, over its quenched expectation, in the infinite volume limit, then we explicitly calculate its simplest approximation, namely its annealed bound. Next, focusing on a RBM characterized by analogical weights, we extend Guerra's interpolating scheme to obtain a control of the quenched free-energy under the assumption of replica symmetry: we get self-consistencies for the order parameters (in full agreement with the existing Literature) as well as the critical line for ergodicity breaking that turns out to be the same obtained in AGS theory. As we discuss, this analogy stems from the slow-noise universality. Finally, glancing beyond replica symmetry, we analyze the fluctuations of the overlaps for an estimate of the (slow) noise affecting the retrieval of the signal, and by a stability analysis we recover the Aizenman-Contucci identities typical of glassy systems.Comment: 21 pages, 1 figur

High-Dimensional Random Fields and Random Matrix Theory

Our goal is to discuss in detail the calculation of the mean number of stationary points and minima for random isotropic Gaussian fields on a sphere as well as for stationary Gaussian random fields in a background parabolic confinement. After developing the general formalism based on the high-dimensional Kac-Rice formulae we combine it with the Random Matrix Theory (RMT) techniques to perform analysis of the random energy landscape of $p-$spin spherical spinglasses and a related glass model, both displaying a zero-temperature one-step replica symmetry breaking glass transition as a function of control parameters (e.g. a magnetic field or curvature of the confining potential). A particular emphasis of the presented analysis is on understanding in detail the picture of "topology trivialization" (in the sense of drastic reduction of the number of stationary points) of the landscape which takes place in the vicinity of the zero-temperature glass transition in both models. We will reveal the important role of the GOE "edge scaling" spectral region and the Tracy-Widom distribution of the maximal eigenvalue of GOE matrices for providing an accurate quantitative description of the universal features of the topology trivialization scenario.Comment: 40 pages; 2 figures; In this version the original lecture notes are converted to an article format, new Eqs. (82)-(85) and Appendix about anisotropic fields added, noticed misprints corrected, references updated. references update

Laser scanning vibrometry and modal analysis to characterize a vocal fold replica

International audienceVocal folds are composed of elastic, soft, multilayer material, and are set to various vibration regimes during phonation, while speaking or singing. To explore such vibration phenomena, a vocal folds replica has been built, allowing to control physical parameters (subglottal pressure, vocal folds stiffness, and glottal aperture) in order to understand their respective contribution. Vocal folds are imitated by latex tubes filled with water under variable pressure. The present study aims at presenting mechanical measurements performed on a single vocal fold replica by means of a shaker provided with an accelerometer in conjunction with a laser vibrometer. This vibration measurement protocol yields a series of frequency response functions over a specific area of the vocal fold. Modal analysis is then performed using an algorithm based on the least square complex exponentials (LSCE) method, which has been developed for single input-multiple output (SIMO) systems. Results are further compared with those from the rational fraction polynomial (RFP) method. Although results are in fair accordance, the observed discrepancies are quantified and discussed

Experimental validation and physical modelling of vocal folds pathologies

Voiced sounds involve self-sustained vocal folds oscillations due to the interaction between the airflow and the vocal folds. Common vocal folds pathologies like polyps and anatomical asymmetry degrade the mechanical vocal fold properties and consequently disturb the normal oscillation pattern resulting in an abnormal sound production. Treatment of voice abnormalities would benefit from an improved understanding between the pathology and the resulting oscillation pattern which motivates physical vocal folds modelling. The current study applies a theoretical vocal folds model to vocal folds pathologies. The theoretical vocal folds model is validated using an experimental set-up simulating the human phonatory apparatus. It consists in a pressure reservoir, a self-oscillating latex replica of the vocal folds and an acoustical resonator. The effects of pathologies are simulated by modifying the replica's geometry, elasticity, and homogeneity under controlled experimental conditions. In general, we observed a close match between measurements and theoretical predictions, which is all the more surprising considering the crudeness of the theoretical mode

Design and Implementation of a Distributed Middleware for Parallel Execution of Legacy Enterprise Applications

A typical enterprise uses a local area network of computers to perform its business. During the off-working hours, the computational capacities of these networked computers are underused or unused. In order to utilize this computational capacity an application has to be recoded to exploit concurrency inherent in a computation which is clearly not possible for legacy applications without any source code. This thesis presents the design an implementation of a distributed middleware which can automatically execute a legacy application on multiple networked computers by parallelizing it. This middleware runs multiple copies of the binary executable code in parallel on different hosts in the network. It wraps up the binary executable code of the legacy application in order to capture the kernel level data access system calls and perform them distributively over multiple computers in a safe and conflict free manner. The middleware also incorporates a dynamic scheduling technique to execute the target application in minimum time by scavenging the available CPU cycles of the hosts in the network. This dynamic scheduling also supports the CPU availability of the hosts to change over time and properly reschedule the replicas performing the computation to minimize the execution time. A prototype implementation of this middleware has been developed as a proof of concept of the design. This implementation has been evaluated with a few typical case studies and the test results confirm that the middleware works as expected

A Taxonomy of Data Grids for Distributed Data Sharing, Management and Processing

Data Grids have been adopted as the platform for scientific communities that need to share, access, transport, process and manage large data collections distributed worldwide. They combine high-end computing technologies with high-performance networking and wide-area storage management techniques. In this paper, we discuss the key concepts behind Data Grids and compare them with other data sharing and distribution paradigms such as content delivery networks, peer-to-peer networks and distributed databases. We then provide comprehensive taxonomies that cover various aspects of architecture, data transportation, data replication and resource allocation and scheduling. Finally, we map the proposed taxonomy to various Data Grid systems not only to validate the taxonomy but also to identify areas for future exploration. Through this taxonomy, we aim to categorise existing systems to better understand their goals and their methodology. This would help evaluate their applicability for solving similar problems. This taxonomy also provides a "gap analysis" of this area through which researchers can potentially identify new issues for investigation. Finally, we hope that the proposed taxonomy and mapping also helps to provide an easy way for new practitioners to understand this complex area of research.Comment: 46 pages, 16 figures, Technical Repor