Upper and lower bounds on the learning curve for Gaussian processes

In this paper we introduce and illustrate non-trivial upper and lower bounds on the learning curves for one-dimensional Gaussian Processes. The analysis is carried out emphasising the effects induced on the bounds by the smoothness of the random process described by the Modified Bessel and the Squared Exponential covariance functions. We present an explanation of the early, linearly-decreasing behavior of the learning curves and the bounds as well as a study of the asymptotic behavior of the curves. The effects of the noise level and the lengthscale on the tightness of the bounds are also discussed.

General bounds on Bayes errors for regression with Gaussian processes

Based on a simple convexity lemma, we develop bounds for different types of Bayesian prediction errors for regression with Gaussian processes. The basic bounds are formulated for a fixed training set. Simpler expressions are obtained for sampling from an input distribution which equals the weight function of the covariance kernel, yielding asymptotically tight results. The results are compared with numerical experiments.

Nonlinear inverse synthesis and eigenvalue division multiplexing in optical fiber channels

We scrutinize the concept of integrable nonlinear communication channels, resurrecting and extending the idea of eigenvalue communications in a novel context of nonsoliton coherent optical communications. Using the integrable nonlinear Schrödinger equation as a channel model, we introduce a new approach - the nonlinear inverse synthesis method - for digital signal processing based on encoding the information directly onto the nonlinear signal spectrum. The latter evolves trivially and linearly along the transmission line, thus, providing an effective eigenvalue division multiplexing with no nonlinear channel cross talk. The general approach is illustrated with a coherent optical orthogonal frequency division multiplexing transmission format. We show how the strategy based upon the inverse scattering transform method can be geared for the creation of new efficient coding and modulation standards for the nonlinear channel. © Published by the American Physical Society.

Soliton's eigenvalue based analysis on the generation mechanism of rogue wave phenomenon in optical fibers exhibiting weak third order dispersion

One of the extraordinary aspects of nonlinear wave evolution which has been observed as the spontaneous occurrence of astonishing and statistically extraordinary amplitude wave is called rogue wave. We show that the eigenvalues of the associated equation of nonlinear Schrödinger equation are almost constant in the vicinity of rogue wave and we validate that optical rogue waves are formed by the collision between quasi-solitons in anomalous dispersion fiber exhibiting weak third order dispersion.

Analysis of rogue wave phenomenon in optical fiber based on soliton's eigenvalue

The impact of third-order dispersion (TOD) on optical rogue wave phenomenon is investigated numerically. We validate the TOD coefficient by utilizing the eigenvalue of the associated equation of the nonlinear Schrödinger equation (NLSE). © 2014 OSA.

Eigenvalue communications in nonlinear fiber channels

Continuous progress in optical communication technology and corresponding increasing data rates in core fiber communication systems are stimulated by the evergrowing capacity demand due to constantly emerging new bandwidth-hungry services like cloud computing, ultra-high-definition video streams, etc. This demand is pushing the required capacity of optical communication lines close to the theoretical limit of a standard single-mode fiber, which is imposed by Kerr nonlinearity [1–4]. In recent years, there have been extensive efforts in mitigating the detrimental impact of fiber nonlinearity on signal transmission, through various compensation techniques. However, there are still many challenges in applying these methods, because a majority of technologies utilized in the inherently nonlinear fiber communication systems had been originally developed for linear communication channels. Thereby, the application of ”linear techniques” in a fiber communication systems is inevitably limited by the nonlinear properties of the fiber medium. The quest for the optimal design of a nonlinear transmission channels, development of nonlinear communication technqiues and the usage of nonlinearity in a“constructive” way have occupied researchers for quite a long time.

Efficient computation of decoherent quantum walks through eigenvalue perturbation

A number of recent studies have investigated the introduction of decoherence in quantum walks and the resulting transition to classical random walks. Interestingly,it has been shown that algorithmic properties of quantum walks with decoherence such as the spreading rate are sometimes better than their purely quantum counterparts. Not only quantum walks with decoherence provide a generalization of quantum walks that naturally encompasses both the quantum and classical case, but they also give rise to new and different probability distribution. The application of quantum walks with decoherence to large graphs is limited by the necessity of evolving state vector whose sizes quadratic in the number of nodes of the graph, as opposed to the linear state vector of the purely quantum (or classical) case. In this technical report,we show how to use perturbation theory to reduce the computational complexity of evolving a continuous-time quantum walk subject to decoherence. More specifically, given a graph over n nodes, we show how to approximate the eigendecomposition of the n2×n2 Lindblad super-operator from the eigendecomposition of the n×n graph Hamiltonian.

Capacity of nonlinear fibre channels and eigenvalue division multiplexing for optical transmissions

The nonlinear Fourier transform, also known as eigenvalue communications, is a coding, transmission and signal processing technique that makes positive use of the nonlinear Kerr effect in fibre channels. I will discuss recent progress in this field. © 2015 OSA.

Bounds on end-to-end performance of networks employing erasure control coding

Erasure control coding has been exploited in communication networks with an aim to improve the end-to-end performance of data delivery across the network. To address the concerns over the strengths and constraints of erasure coding schemes in this application, we examine the performance limits of two erasure control coding strategies, forward erasure recovery and adaptive erasure recovery. Our investigation shows that the throughput of a network using an (n, k) forward erasure control code is capped by r =k/n when the packet loss rate p ≤ (te/n) and by k(l-p)/(n-te) when p > (t e/n), where te is the erasure control capability of the code. It also shows that the lower bound of the residual loss rate of such a network is (np-te)/(n-te) for (te/n) < p ≤ 1. Especially, if the code used is maximum distance separable, the Shannon capacity of the erasure channel, i.e. 1-p, can be achieved and the residual loss rate is lower bounded by (p+r-1)/r, for (1-r) < p ≤ 1. To address the requirements in real-time applications, we also investigate the service completion time of different schemes. It is revealed that the latency of the forward erasure recovery scheme is fractionally higher than that of the scheme without erasure control coding or retransmission mechanisms (using UDP), but much lower than that of the adaptive erasure scheme when the packet loss rate is high. Results on comparisons between the two erasure control schemes exhibit their advantages as well as disadvantages in the role of delivering end-to-end services. To show the impact of the bounds derived on the end-to-end performance of a TCP/IP network, a case study is provided to demonstrate how erasure control coding could be used to maximize the performance of practical systems. © 2010 IEEE.

Periodic nonlinear Fourier transform for fiber-optic communications, Part II:eigenvalue communication

In this paper we propose the design of communication systems based on using periodic nonlinear Fourier transform (PNFT), following the introduction of the method in the Part I. We show that the famous "eigenvalue communication" idea [A. Hasegawa and T. Nyu, J. Lightwave Technol. 11, 395 (1993)] can also be generalized for the PNFT application: In this case, the main spectrum attributed to the PNFT signal decomposition remains constant with the propagation down the optical fiber link. Therefore, the main PNFT spectrum can be encoded with data in the same way as soliton eigenvalues in the original proposal. The results are presented in terms of the bit-error rate (BER) values for different modulation techniques and different constellation sizes vs. the propagation distance, showing a good potential of the technique.