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The approximation power of general feedforward neural networks with piecewise linear activation functions is investigated. First, lower bounds on the size of a network are established in terms of the approximation error and network depth and width. These bounds improve upon state-of-the-art bounds for certain classes of functions, such as strongly convex functions. Second, an upper bound is established on the difference of two neural networks with identical weights but different activation functions.

Polarization-division multiplexed (PDM) transmission based on the nonlinear Fourier transform (NFT) is proposed for optical fiber communication. The NFT algorithms are generalized from the scalar nonlinear Schr\"odinger equation for one polarization to the Manakov system for two polarizations. The transmission performance of the PDM nonlinear frequency-division multiplexing (NFDM) and PDM orthogonal frequency-division multiplexing (OFDM) are determined. It is shown that the transmission performance in terms of Q-factor is approximately the same in PDM-NFDM and single polarization NFDM at twice the data rate and that the polarization-mode dispersion does not seriously degrade system performance. Compared with PDM-OFDM, PDM-NFDM achieves a Q-factor gain of 6.4 dB. The theory can be generalized to multi-mode fibers in the strong coupling regime, paving the way for the application of the NFT to address the nonlinear effects in space-division multiplexing., Comment: 15 pages, 9 figures

The per-sample zero-dispersion channel model of the optical fiber is considered. It is shown that capacity is uniquely achieved by an input probability distribution that has continuous uniform phase and discrete amplitude that takes on finitely many values. This result holds when the channel is subject to general input cost constraints, that include a peak amplitude constraint and a joint average and peak amplitude constraint., Comment: 18 pages; Submitted for review to the IEEE Transactions on Information Theory

Achievable rates of the nonlinear frequency-division multiplexing (NFDM) and wavelength-division multiplexing (WDM) subject to the same power and bandwidth constraints are computed as a function of transmit power in the standard single-mode fiber. NFDM achieves higher rates than WDM., Comment: Invited paper to be presented at The Optical Fiber Communications Conference and Exposition (OFC), March 2017

It is shown that signal energy is the only available degree-of-freedom (DOF) for fiber-optic transmission as the input power tends to infinity. With $n$ signal DOFs at the input, $n-1$ DOFs are asymptotically lost to signal-noise interactions. The main observation is that, nonlinearity introduces a multiplicative noise in the channel, similar to fading in wireless channels. The channel is viewed in the spherical coordinate system, where signal vector $\underline{X}\in\mathbb{C}^n$ is represented in terms of its norm $|\underline{X}|$ and direction $\underline{\hat{X}}$. The multiplicative noise causes signal direction $\underline{\hat{X}}$ to vary randomly on the surface of the unit $(2n-1)$-sphere in $\mathbb{C}^{n}$, in such a way that the effective area of the support of $\underline{\hat{X}}$ does not vanish as $|\underline{X}|\rightarrow\infty$. On the other hand, the surface area of the sphere is finite, so that $\underline{\hat{X}}$ carries finite information. This observation is used to show several results. Firstly, let $\mathcal C(\mathcal P)$ be the capacity of a discrete-time periodic model of the optical fiber with distributed noise and frequency-dependent loss, as a function of the average input power $\mathcal P$. It is shown that asymptotically as $\mathcal P\rightarrow\infty$, $\mathcal C=\frac{1}{n}\log\bigl(\log\mathcal P\bigr)+c$, where $n$ is the dimension of the input signal space and $c$ is a bounded number. In particular, $\lim_{\mathcal P\rightarrow\infty}\mathcal C(\mathcal P)=\infty$ in finite-dimensional periodic models. Secondly, it is shown that capacity saturates to a constant in infinite-dimensional models where $n=\infty$., Comment: The abstract in the PDF file is longer. Arxiv limits the abstract field to 1,920 characters

Two signal multiplexing schemes for optical fiber communication are considered: Wavelength-division multiplexing (WDM) and nonlinear frequency-division multiplexing (NFDM), based on the nonlinear Fourier transform (NFT). Achievable information rates (AIRs) of NFDM and WDM are compared in a network scenario with an ideal lossless model of the optical fiber in the defocusing regime. It is shown that the NFDM AIR is greater than the WDM AIR subject to a bandwidth and average power constraint, in a representative system with one symbol per user. The improvement results from nonlinear signal multiplexing., Comment: The presentation has been improved considerably

An upper bound on the capacity of a cascade of nonlinear and noisy channels is presented. The cascade mimics the split-step Fourier method for computing waveform propagation governed by the stochastic generalized nonlinear Schroedinger equation. It is shown that the spectral efficiency of the cascade is at most log(1+SNR), where SNR is the receiver signal-to-noise ratio. The results may be applied to optical fiber channels. However, the definition of bandwidth is subtle and leaves open interpretations of the bound. Some of these interpretations are discussed., Comment: The main change is to define the noise as bandlimited already in (8) rather than before (15). This serves to clarify subsequent steps

It is shown that the capacity of the channel modeled by (a discretized version of) the stochastic nonlinear Schr\"odinger (NLS) equation is upper-bounded by $\log(1+\text{SNR})$ with $\text{SNR}=\mathcal P_0/\sigma^2(z)$, where $\mathcal P_0$ is the average input signal power and $\sigma^2(z)$ is the total noise power up to distance $z$. The result is a consequence of the fact that the deterministic NLS equation is a Hamiltonian energy-preserving dynamical system., Comment: To be presented at the 14th Canadian Workshop on Information Theory (CWIT), St. John's, NL, Canada, July 6-9, 2015. This is the final version submitted to the CWIT 2015

A mathematical framework is presented to study the evolution of multi-point cumulants in nonlinear dispersive partial differential equations with random input data, based on the theory of weak wave turbulence (WWT). This framework is used to explain how energy is distributed among Fourier modes in the nonlinear Schr\"odinger equation. This is achieved by considering interactions among four Fourier modes and studying the role of the resonant, non-resonant, and trivial quartets in the dynamics. As an application, a power spectral density is suggested for calculating the interference power in dense wavelength-division multiplexed optical systems, based on the kinetic equation of the WWT. This power spectrum, termed the Kolmogorov-Zakharov (KZ) model, results in a better estimate of the signal spectrum in optical fiber, compared with the so-called Gaussian noise (GN) model. The KZ model is generalized to non-stationary inputs and multi-span optical systems., Comment: The final version of the paper. It has been substantially improved and better presented, compared with the previous version. The list of the authors has changed in this version

The capacity of the channel defined by the stochastic nonlinear Schr\"odinger equation, which includes the effects of the Kerr nonlinearity and amplified spontaneous emission noise, is considered in the case of zero dispersion. In the absence of dispersion, this channel behaves as a collection of parallel per-sample channels. The conditional probability density function of the nonlinear per-sample channels is derived using both a sum-product and a Fokker-Planck differential equation approach. It is shown that, for a fixed noise power, the per-sample capacity grows unboundedly with input signal. The channel can be partitioned into amplitude and phase subchannels, and it is shown that the contribution to the total capacity of the phase channel declines for large input powers. It is found that a two-dimensional distribution with a half-Gaussian profile on the amplitude and uniform phase provides a lower bound for the zero-dispersion optical fiber channel, which is simple and asymptotically capacity-achieving at high signal-to-noise ratios (SNRs). A lower bound on the capacity is also derived in the medium-SNR region. The exact capacity subject to peak and average power constraints is numerically quantified using dense multiple ring modulation formats. The differential model underlying the zero-dispersion channel is reduced to an algebraic model, which is more tractable for digital communication studies, and in particular it provides a relation between the zero-dispersion optical channel and a $2 \times 2$ multiple-input multiple-output Rician fading channel. It appears that the structure of the capacity-achieving input distribution resembles that of the Rician fading channel, i.e., it is discrete in amplitude with a finite number of mass points, while continuous and uniform in phase., Comment: Published in 2011

Motivated by the looming "capacity crunch" in fiber-optic networks, information transmission over such systems is revisited. Among numerous distortions, inter-channel interference in multiuser wavelength-division multiplexing (WDM) is identified as the seemingly intractable factor limiting the achievable rate at high launch power. However, this distortion and similar ones arising from nonlinearity are primarily due to the use of methods suited for linear systems, namely WDM and linear pulse-train transmission, for the nonlinear optical channel. Exploiting the integrability of the nonlinear Schr\"odinger (NLS) equation, a nonlinear frequency-division multiplexing (NFDM) scheme is presented, which directly modulates non-interacting signal degrees-of-freedom under NLS propagation. The main distinction between this and previous methods is that NFDM is able to cope with the nonlinearity, and thus, as the the signal power or transmission distance is increased, the new method does not suffer from the deterministic cross-talk between signal components which has degraded the performance of previous approaches. In this paper, emphasis is placed on modulation of the discrete component of the nonlinear Fourier transform of the signal and some simple examples of achievable spectral efficiencies are provided., Comment: Updated version of IEEE Transactions on Information Theory, vol. 60, no. 7, pp. 4346--4369, July, 2014

In this paper, numerical methods are suggested to compute the discrete and the continuous spectrum of a signal with respect to the Zakharov-Shabat system, a Lax operator underlying numerous integrable communication channels including the nonlinear Schr\"odinger channel, modeling pulse propagation in optical fibers. These methods are subsequently tested and their ability to estimate the spectrum are compared against each other. These methods are used to compute the spectrum of various signals commonly used in the optical fiber communications. It is found that the layer-peeling and the spectral methods are suitable schemes to estimate the nonlinear spectra with good accuracy. To illustrate the structure of the spectrum, the locus of the eigenvalues is determined under amplitude and phase modulation in a number of examples. It is observed that in some cases, as signal parameters vary, eigenvalues collide and change their course of motion. The real axis is typically the place from which new eigenvalues originate or are absorbed into after traveling a trajectory in the complex plane., Comment: Minor updates to IEEE Transactions on Information Theory, vol. 60, no. 7, pp. 4329--4345, July 2014

The nonlinear Fourier transform (NFT), a powerful tool in soliton theory and exactly solvable models, is a method for solving integrable partial differential equations governing wave propagation in certain nonlinear media. The NFT decorrelates signal degrees-of-freedom in such models, in much the same way that the Fourier transform does for linear systems. In this three-part series of papers, this observation is exploited for data transmission over integrable channels such as optical fibers, where pulse propagation is governed by the nonlinear Schr\"odinger equation. In this transmission scheme, which can be viewed as a nonlinear analogue of orthogonal frequency-division multiplexing commonly used in linear channels, information is encoded in the nonlinear frequencies and their spectral amplitudes. Unlike most other fiber-optic transmission schemes, this technique deals with both dispersion and nonlinearity directly and unconditionally without the need for dispersion or nonlinearity compensation methods. This first paper explains the mathematical tools that underlie the method., Comment: This version contains minor updates of IEEE Transactions on Information Theory, vol. 60, no. 7, pp. 4312--4328, July 2014