Concept

Legendre polynomials

Related publications (26)

Nonlinear simulation of plasma turbulence using a gyrokinetic moment-based approach

Plasma turbulence plays a fundamental role in determining the performances of magnetic confinement fusion devices, such as tokamaks. Advances in computer science, combined with the development of efficient physical models, have significantly improved our u ...

EPFL2024

Unlabeled Principal Component Analysis and Matrix Completion

Yunzhen Yao

We introduce robust principal component analysis from a data matrix in which the entries of its columns have been corrupted by permutations, termed Unlabeled Principal Component Analysis (UPCA). Using algebraic geometry, we establish that UPCA is a well-de ...

Microtome Publishing2024

Gyrokinetic moment-based simulations of the Dimits shift

Paolo Ricci, Baptiste Jimmy Frei, Antoine Cyril David Hoffmann

We present a convergence study of the gyromoment (GM) approach, which is based on projecting the gyrokinetic distribution function onto a Hermite–Laguerre polynomial basis, focused on the cyclone base case (CBC) (Lin et al., Phys. Rev. Lett., vol. 83, no. ...

Cambridge University Press2023

The multivariate Serre conjecture ring

Luc Guyot

It is well-known that for any integral domain R, the Serre conjecture ring R(X), i.e., the localization of the univariate polynomial ring R[X] at monic polynomials, is a Bezout domain of Krull dimension

Academic Press Inc Elsevier Science2023

Polynomial-time universality and limitations of deep learning

Emmanuel Abbé

The goal of this paper is to characterize function distributions that general neural networks trained by descent algorithms (GD/SGD), can or cannot learn in polytime. The results are: (1) The paradigm of general neural networks trained by SGD is poly-time ...

WILEY2023

Regularization of polynomial networks for image recognition

Volkan Cevher, Grigorios Chrysos, Bohan Wang

Deep Neural Networks (DNNs) have obtained impressive performance across tasks, however they still remain as black boxes, e.g., hard to theoretically analyze. At the same time, Polynomial Networks (PNs) have emerged as an alternative method with a promising ...

Ieee Computer Soc2023

Regularization of polynomial networks for image recognition

Volkan Cevher, Grigorios Chrysos, Bohan Wang

2023

The spectral bias of polynomial neural networks

Volkan Cevher, Grigorios Chrysos, Leello Tadesse Dadi

Polynomial neural networks (PNNs) have been recently shown to be particularly effective at image generation and face recognition, where high-frequency information is critical. Previous studies have revealed that neural networks demonstrate a spectral bias ...

2022

Finite free convolutions of polynomials

Adam Wade Marcus

We study three convolutions of polynomials in the context of free probability theory. We prove that these convolutions can be written as the expected characteristic polynomials of sums and products of unitarily invariant random matrices. The symmetric addi ...

SPRINGER HEIDELBERG2022

Sound and Complete Verification of Polynomial Networks

Volkan Cevher, Grigorios Chrysos, Fanghui Liu, Elias Abad Rocamora, Mehmet Fatih Sahin

Polynomial Networks (PNs) have demonstrated promising performance on face and image recognition recently. However, robustness of PNs is unclear and thus obtaining certificates becomes imperative for enabling their adoption in real-world applications. Exist ...

2022

Hidden-nucleons neural-network quantum states for the nuclear many-body problem

Giuseppe Carleo

We generalize the hidden-fermion family of neural network quantum states to encompass both continuous and discrete degrees of freedom and solve the nuclear many-body Schrodinger equation in a systematically improvable fashion. We demonstrate that adding hi ...

AMER PHYSICAL SOC2022

A PCP Theorem for Interactive Proofs and Applications

Alessandro Chiesa

The celebrated PCP Theorem states that any language in NP can be decided via a verifier that reads O(1) bits from a polynomially long proof. Interactive oracle proofs (IOP), a generalization of PCPs, allow the verifier to interact with the prover for multi ...

SPRINGER INTERNATIONAL PUBLISHING AG2022

Functional estimation of anisotropic covariance and autocovariance operators on the sphere

Victor Panaretos, Julien René Fageot, Matthieu Martin Jean-André Simeoni, Alessia Caponera

We propose nonparametric estimators for the second-order central moments of possibly anisotropic spherical random fields, within a functional data analysis context. We consider a measurement framework where each random field among an identically distribute ...

2022

The completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: BAO and RSD measurements from anisotropic clustering analysis of the quasar sample in configuration space between redshift 0.8 and 2.2

Cheng Zhao

We measure the anisotropic clustering of the quasar sample from Data Release 16 (DR16) of the Sloan Digital Sky Survey IV extended Baryon Oscillation Spectroscopic Survey (eBOSS). A sample of 343 708 spectroscopically confirmed quasars between redshift 0.8 ...

OXFORD UNIV PRESS2021

Computation of Al-Salam Carlitz and Askey-Wilson moments using Motzkin paths

Gaspard Ohlmann

In this paper we study the moments of polynomials from the Askey scheme, and we focus on Askey-Wilson polynomials. More precisely, we give a combinatorial proof for the case where d = 0. Their values have already been computed by Kim and Stanton in 2015, h ...

ELECTRONIC JOURNAL OF COMBINATORICS2021

On The Singular Set In The Thin Obstacle Problem: Higher-Order Blow-Ups And The Very Thin Obstacle Problem

Xavier Fernandez-Real Girona

We consider the singular set in the thin obstacle problem with weight vertical bar x(n +1)vertical bar(a) for a epsilon (-1, 1), which arises as the local extension of the obstacle problem for the fractional Laplacian (a nonlocal problem). We develop a ref ...

MATHEMATICAL SCIENCE PUBL2021

Strictly Real Fundamental Theorem Of Algebra Using Polynomial Interlacing

Soham Basu

Without resorting to complex numbers or any advanced topological arguments, we show that any real polynomial of degree greater than two always has a real quadratic polynomial factor, which is equivalent to the fundamental theorem of algebra. The proof uses ...

CAMBRIDGE UNIV PRESS2021

ON WAVEFRONT ABERRATIONS IN ASYMMETRIC AND MULTIPLE APERTURE OPTICAL SYSTEMS

Alessandro Grosso

The present work deals with monochromatic wavefront aberrations in optical systems without symmetries. The treatment begins with a class of systems characterized by misaligned spherical surfaces whose behavior is analyzed using the wavefront aberration exp ...

EPFL2021

Frequency dependence in GW made simple using a multipole approximation

Tommaso Chiarotti

In the GW approximation, the screened interaction W is a nonlocal and dynamical potential that usually has a complex frequency dependence. A full description of such a dependence is possible but often computationally demanding. For this reason, it is still ...

AMER PHYSICAL SOC2021

Markov cubature rules for polynomial processes

Damir Filipovic, Martin Larsson, Sergio Andres Pulido Nino

We study discretizations of polynomial processes using finite state Markov processes satisfying suitable moment matching conditions. The states of these Markov processes together with their transition probabilities can be interpreted as Markov cubature rul ...

ELSEVIER2020