Publications related to Critical point (mathematics)

The effect of smooth parametrizations on nonconvex optimization landscapes

We develop new tools to study landscapes in nonconvex optimization. Given one optimization problem, we pair it with another by smoothly parametrizing the domain. This is either for practical purposes (e.g., to use smooth optimization algorithms with good g ...

Springer Heidelberg2024

Photothermal spectroscopy on-chip sensor for the measurement of a PMMA film using a silicon nitride micro-ring resonator and an external cavity quantum cascade laser

Simone Iadanza

Laser-based mid-infrared (mid-IR) photothermal spectroscopy (PTS) represents a selective, fast, and sensitive analytical technique. Recent developments in laser design permits the coverage of wider spectral regions in combination with higher power, enablin ...

Walter De Gruyter Gmbh2024

Avoidance of Concave Obstacles Through Rotation of Nonlinear Dynamics

Aude Billard, Lukas Huber

Controlling complex tasks in robotic systems, such as circular motion for cleaning or following curvy lines, can be dealt with using nonlinear vector fields. This article introduces a novel approach called the rotational obstacle avoidance method (ROAM) fo ...

Ieee-Inst Electrical Electronics Engineers Inc2024

A Theory of Finite-Width Neural Networks: Generalization, Scaling Laws, and the Loss Landscape

Berfin Simsek

Deep learning has achieved remarkable success in various challenging tasks such as generating images from natural language or engaging in lengthy conversations with humans. The success in practice stems from the ability to successfully train massive neural ...

EPFL2023

Language Model Decoding as Likelihood–Utility Alignment

Boi Faltings, Robert West, Maxime Jean Julien Peyrard, Martin Josifoski, Valentin Hartmann, Debjit Paul, Jiheng Wei, Frano Rajic

A critical component of a successful language generation pipeline is the decoding algorithm. However, the general principles that should guide the choice of a decoding algorithm re- main unclear. Previous works only compare decoding algorithms in narrow sc ...

Assoc Computational Linguistics-Acl2023

Unraveling looping efficiency of stochastic Cosserat polymers

Raushan Singh, Giulio Corazza

Understanding looping probabilities, including the particular case of ring closure or cyclization, of fluctuating polymers (e.g., DNA) is important in many applications in molecular biology and chemistry. In a continuum limit the configuration of a polymer ...

AMER PHYSICAL SOC2022

Development of a new method to determine the axial void velocity profile in BWRs from measurements of the in-core neutron noise

Mathieu Hursin

Determination of the local void fraction in BWRs from in-core neutron noise measurements requires the knowledge of the axial velocity of the void. The purpose of this paper is to revisit the problem of determining the axial void velocity profile from the t ...

PERGAMON-ELSEVIER SCIENCE LTD2021

Geometry of the Loss Landscape in Overparameterized Neural Networks: Symmetries and Invariances

Wulfram Gerstner, Clément Hongler, Johanni Michael Brea, Francesco Spadaro, Berfin Simsek, Arthur Jacot

We study how permutation symmetries in overparameterized multi-layer neural networks generate `symmetry-induced' critical points. Assuming a network with

L

layers of minimal widths

r_1^*, \ldots, r_{L-1}^*

reaches a zero-loss minimum at $ r_1^*! \c ...

2021

Wavelet analysis of the Besov regularity of Levy white noise

Julien René Pierre Fageot, Shayan Aziznejad

We characterize the local smoothness and the asymptotic growth rate of the Levy white noise. We do so by characterizing the weighted Besov spaces in which it is located. We extend known results in two ways. First, we obtain new bounds for the local smoothn ...

2020

On the relaxed maximum-likelihood blind MIMO channel estimation for orthogonal space-time block codes

Kamran Kalbasi

This paper concerns the maximum-likelihood channel estimation for MIMO systems with orthogonal space-time block codes when the finite alphabet constraint of the signal constellation is relaxed. We study the channel coefficients estimation subspace generate ...

ELSEVIER2020

Complex Dynamics in Simple Neural Networks: Understanding Gradient Flow in Phase Retrieval

Florent Gérard Krzakala, Lenka Zdeborová

Despite the widespread use of gradient-based algorithms for optimizing high-dimensional non-convex functions, understanding their ability of finding good minima instead of being trapped in spurious ones remains to a large extent an open problem. Here we fo ...

Curran Associates, Inc.2020

Energy distribution of harmonic 1-forms and Jacobians of Riemann surfaces with a short closed geodesic

Jürgpeter Buser, Robert Silhol

We study the energy distribution of harmonic 1-forms on a compact hyperbolic Riemann surface S where a short closed geodesic is pinched. If the geodesic separates the surface into two parts, then the Jacobian variety of S develops into a variety that split ...

2021

Stable Manifold for the Critical Non-Linear Wave Equation: A Fourier Theory Approach

Gérôme Graf

I will try to explain, without going into too much detail, how one can consider a non-linear wave equation as a dynamical system and what it brings to the study of its solutions. We begin by considering our model case, the non-linear Klein-Gordon equation ...

EPFL2019

The index of singular zeros of harmonic mappings of anti-analytic degree one

Robert Gerhard Jérôme Luce

We study harmonic mappings of the form , where h is an analytic function. In particular, we are interested in the index (a generalized multiplicity) of the zeros of such functions. Outside the critical set of f, where the Jacobian of f is non-vanishing, it ...

TAYLOR & FRANCIS LTD2021

Criteria for the existence of a potential well

Charles Stuart

We consider a critical point u(0) of a functional f is an element of C-1 (H, R), where H is a real Hilbert space, and formulate criteria ensuring that u(0) lies in a potential well of f without supposing that f' is Frechet differentiable at u(0). The deriv ...

Pergamon-Elsevier Science Ltd2017

An Alternating Trust Region Algorithm for Distributed Linearly Constrained Nonlinear Programs, Application to the Optimal Power Flow Problem

Colin Neil Jones, Jean-Hubert Hours

A novel trust region method for solving linearly constrained nonlinear programs is presented. The proposed technique is amenable to a distributed implementation, as its salient ingredient is an alternating projected gradient sweep in place of the Cauchy po ...

2017

A generalized mountain-pass theorem: the existence of multiple critical points

Hans-Jörg Ruppen

We analyse the existence of multiple critical points for an even functional J : H -> R in the following context: the Hilbert space H can be split into an orthogonal sum H = Y circle plus Z in such a way that inf{J(u) : u is an element of Z and parallel to ...

Springer Heidelberg2016

Fixed Points of Generalized Approximate Message Passing with Arbitrary Matrices

Volkan Cevher

The estimation of a random vector with independent components passed through a linear transform followed by a componentwise (possibly nonlinear) output map arises in a range of applications. Approximate message passing (AMP) methods, based on Gaussian appr ...

2016

Multiple Points Of The Brownian Sheet In Critical Dimensions

Robert Dalang

It is well known that an N-parameter d-dimensional Brownian sheet has no k-multiple points when (k - 1)d > 2kN, and does have such points when (k - 1)d < 2kN. We complete the study of the existence of k-multiple points by showing that in the critical cases ...

Inst Mathematical Statistics2015

A New Convergence Proof For The Higher-Order Power Method And Generalizations

André Uschmajew

A proof for the point-wise convergence of the factors in the higher-order power method for tensors towards a critical point is given. It is obtained by applying established results from the theory of Lojasiewicz inequalities to the equivalent, unconstraine ...

Yokohama Publ2015