Searcharxiv⌕ Search

arXiv subjects

Manuela Girotti

Publications and source records attributed to Manuela Girotti.

13 recordsLinked to original sources
arXiv2025

Soliton Synchronization with Randomness: Rogue Waves and Universality

We consider an $N$-soliton solution of the focusing nonlinear Schr\"{o}dinger equations. We give conditions for the synchronous collision of these $N$ solitons. When the solitons velocities are well separated and the solitons have equal amplitude, we show that the local wave profile at the collision point scales as the $\operatorname{sinc}(x)$ function. We show that this behaviour persists when the amplitudes of the solitons are i.i.d. sub-exponential random variables. Namely the central collision peak exhibits universality: its spatial profile converges to the $\operatorname{sinc}(x)$ function, independently of the distribution. We derive Central Limit Theorems for the fluctuations of the profile in the near-field regime (near the collision point) and in the far-regime.

nlin.PS↗
arXiv2024

Law of Large Numbers and Central Limit Theorem for random sets of solitons of the focusing nonlinear Schr\"odinger equation

We study a random configuration of $N$ soliton solutions $\psi_N(x,t;\boldsymbol{\lambda})$ of the cubic focusing Nonlinear Schr\"odinger (fNLS) equation in one space dimension. The $N$ soliton solutions are parametrized by $2N$ complex numbers $(\boldsymbol{\lambda}, \boldsymbol{c})$ where $\boldsymbol{\lambda}\in\mathbb{C}_+^N$ are the eigenvalues of the Zakharov-Shabat linear operator, and $ \boldsymbol{c}\in\mathbb{C}^N\backslash \{0\}$ are the norming constants of the corresponding eigenfunctions. The randomness is obtained by choosing the complex eigenvalues to be i.i.d. random variables sampled from a probability distribution with compact support in the complex plane. The corresponding norming constants are interpolated by a smooth function of the eigenvalues. Then we consider the expectation of the random measure associated to this random spectral data. Such expectation uniquely identifies, via the Zakharov-Shabat inverse spectral problem, a solution $\psi_\infty(x,t)$ of the fNLS equation. This solution can be interpreted as a soliton gas solution. We prove a Law of Large Numbers and a Central Limit Theorem for the differences $\psi_N(x,t;\boldsymbol{\lambda})-\psi_\infty(x,t)$ and $|\psi_N(x,t;\boldsymbol{\lambda})|^2-|\psi_\infty(x,t)|^2$ when $(x,t)$ are in a compact set of $\mathbb R\times\mathbb R^+$; we additionally compute the correlation functions.

math-ph↗
arXiv2022

Neural Networks Efficiently Learn Low-Dimensional Representations with SGD

We study the problem of training a two-layer neural network (NN) of arbitrary width using stochastic gradient descent (SGD) where the input $\boldsymbol{x}\in \mathbb{R}^d$ is Gaussian and the target $y \in \mathbb{R}$ follows a multiple-index model, i.e., $y=g(\langle\boldsymbol{u_1},\boldsymbol{x}\rangle,...,\langle\boldsymbol{u_k},\boldsymbol{x}\rangle)$ with a noisy link function $g$. We prove that the first-layer weights of the NN converge to the $k$-dimensional principal subspace spanned by the vectors $\boldsymbol{u_1},...,\boldsymbol{u_k}$ of the true model, when online SGD with weight decay is used for training. This phenomenon has several important consequences when $k \ll d$. First, by employing uniform convergence on this smaller subspace, we establish a generalization error bound of $O(\sqrt{{kd}/{T}})$ after $T$ iterations of SGD, which is independent of the width of the NN. We further demonstrate that, SGD-trained ReLU NNs can learn a single-index target of the form $y=f(\langle\boldsymbol{u},\boldsymbol{x}\rangle) + \epsilon$ by recovering the principal direction, with a sample complexity linear in $d$ (up to log factors), where $f$ is a monotonic function with at most polynomial growth, and $\epsilon$ is the noise. This is in contrast to the known $d^{\Omega(p)}$ sample requirement to learn any degree $p$ polynomial in the kernel regime, and it shows that NNs trained with SGD can outperform the neural tangent kernel at initialization. Finally, we also provide compressibility guarantees for NNs using the approximate low-rank structure produced by SGD.

stat.ML↗
arXiv2022

Soliton versus the gas: Fredholm determinants, analysis, and the rapid oscillations behind the kinetic equation

We analyze the case of a dense mKdV soliton gas and its large time behaviour in the presence of a single trial soliton. We show that the solution can be expressed in terms of Fredholm determinants as well as in terms of a Riemann-Hilbert problem. We then show that the solution can be decomposed as the sum of the background gas solution (a modulated elliptic wave), plus a soliton solution: the individual expressions are however quite convoluted due to the interaction dynamics. Additionally, we are able to derive the local phase shift of the gas after the passage of the soliton, and we can trace the location of the soliton peak as the dynamics evolves. Finally we show that the soliton peak, while interacting with the soliton gas, has an oscillatory velocity whose leading order average value satisfies the kinetic velocity equation analogous to the one posited by V. Zakharov and G. El.

math-ph↗
arXiv2021

Convergence Analysis and Implicit Regularization of Feedback Alignment for Deep Linear Networks

We theoretically analyze the Feedback Alignment (FA) algorithm, an efficient alternative to backpropagation for training neural networks. We provide convergence guarantees with rates for deep linear networks for both continuous and discrete dynamics. Additionally, we study incremental learning phenomena for shallow linear networks. Interestingly, certain specific initializations imply that negligible components are learned before the principal ones, thus potentially negatively affecting the effectiveness of such a learning algorithm; a phenomenon we classify as implicit anti-regularization. We also provide initialization schemes where the components of the problem are approximately learned by decreasing order of importance, thus providing a form of implicit regularization.

cs.LG↗
arXiv2021

Rigorous asymptotics of a KdV soliton gas

We analytically study the long time and large space asymptotics of a new broad class of solutions of the KdV equation introduced by Dyachenko, Zakharov, and Zakharov. These solutions are characterized by a Riemann--Hilbert problem which we show arises as the limit $N\to \infty$ of a gas of $N$-solitons. We show that this gas of solitons in the limit $N \to \infty$ is slowly approaching a cnoidal wave solution for $x \to - \infty$ (up to terms of order $\mathcal{O} (1/x)$), while approaching zero exponentially fast for $x\to+\infty$. We establish an asymptotic description of the gas of solitons for large times that is valid over the entire spatial domain, in terms of Jacobi elliptic functions.

math-ph↗
arXiv2020

A Study of Condition Numbers for First-Order Optimization

The study of first-order optimization algorithms (FOA) typically starts with assumptions on the objective functions, most commonly smoothness and strong convexity. These metrics are used to tune the hyperparameters of FOA. We introduce a class of perturbations quantified via a new norm, called *-norm. We show that adding a small perturbation to the objective function has an equivalently small impact on the behavior of any FOA, which suggests that it should have a minor impact on the tuning of the algorithm. However, we show that smoothness and strong convexity can be heavily impacted by arbitrarily small perturbations, leading to excessively conservative tunings and convergence issues. In view of these observations, we propose a notion of continuity of the metrics, which is essential for a robust tuning strategy. Since smoothness and strong convexity are not continuous, we propose a comprehensive study of existing alternative metrics which we prove to be continuous. We describe their mutual relations and provide their guaranteed convergence rates for the Gradient Descent algorithm accordingly tuned. Finally we discuss how our work impacts the theoretical understanding of FOA and their performances.

cs.LG↗
arXiv2019

Fredholm determinant solutions of the Painlevé II hierarchy and gap probabilities of determinantal point processes

We study Fredholm determinants of a class of integral operators, whose kernels can be expressed as double contour integrals of a special type. Such Fredholm determinants appear in various random matrix and statistical physics models. We show that the logarithmic derivatives of the Fredholm determinants are directly related to solutions of the Painlevé II hierarchy. This confirms and generalizes a recent conjecture by Le Doussal, Majumdar, and Schehr. In addition, we obtain asymptotics at $\pm\infty$ for the Painlevé transcendents and large gap asymptotics for the corresponding point processes.

math-ph↗
arXiv2016

Large gap asymptotics at the hard edge for product random matrices and Muttalib-Borodin ensembles

We study the distribution of the smallest eigenvalue for certain classes of positive-definite Hermitian random matrices, in the limit where the size of the matrices becomes large. Their limit distributions can be expressed as Fredholm determinants of integral operators associated to kernels built out of Meijer $G$-functions or Wright's generalized Bessel functions. They generalize in a natural way the hard edge Bessel kernel Fredholm determinant. We express the logarithmic derivatives of the Fredholm determinants identically in terms of a $2\times 2$ Riemann-Hilbert problem, and use this representation to obtain the so-called large gap asymptotics.

math-ph↗
arXiv2013

Gap probabilities for the Generalized Bessel process: a Riemann-Hilbert approach

We consider the gap probability for the Generalized Bessel process in the single-time and multi-time case. We prove that the scalar and matrix Fredholm determinants of such process can be expressed in terms of determinants of Its-Izergin-Korepin-Slavnov integrable kernels and thus related to suitable Riemann-Hilbert problems. In the single-time case, we construct a Lax pair formalism, while in the multi-time case we explicitly define a new multi-time kernel to study.

math-ph↗
arXiv2013

Riemann-Hilbert approach to gap probabilities for the Bessel process

We consider the gap probability for the Bessel process in the single-time and multi-time case. We prove that the scalar and matrix Fredholm determinants of such process can be expressed in terms of determinants of integrable kernels à la Its-Izergin-Korepin-Slavnov and thus related to suitable Riemann-Hilbert problems. In the single-time case, we construct a Lax pair formalism and we derive a Painlevé III equation related to the Fredholm determinant.

math-ph↗
arXiv2011

Time relaxation of a phase-field model with entropy balance

We deal with a system of two coupled differential equations, describing the evolution of a first order phase transition. In particular, we have two non-linear parabolic equations: the first one is deduced from a balance law for entropy and it describes the evolution of the absolute temperature; the other one is an equilibrium equation for microforces and it regulates the behaviour of a scalar phase parameter. Moreover, the second equation shows a time-relaxation coefficient related to the time-derivative of the phase parameter. We prove well-posedness of solution to the given system, using a standard method of approximating problems; afterwards, we study the behaviour of the system as the time-relaxation coefficient tends to zero: as a result, we find out that the original problem converges to a new problem, with a stationary phase equation.

math.AP↗