\Ell_p$ Norm

The Lp norm, a generalization of the familiar Euclidean distance, measures the magnitude of a vector in a way that adjusts the emphasis placed on individual components depending on the value of p. Current research focuses on its applications in diverse fields, including deep learning for approximating solutions to partial differential equations (using architectures like ReLU networks), adversarial attacks and defenses in machine learning (exploring optimization schemes and robustness against multiple Lp norms), and efficient algorithms for problems like metric nearness and fair clustering. These investigations are significant because they advance both theoretical understanding of approximation capabilities and the development of robust and efficient algorithms for various machine learning and scientific computing tasks.

Papers

September 30, 2024

Multilevel Picard approximations and deep neural networks with ReLU, leaky ReLU, and softplus activation overcome the curse of dimensionality when approximating semilinear parabolic partial differential equations in $L^p$-sense
Ariel Neufeld, Tuan Anh Nguyen
Neural Network Deep Neural Network \Ell_p$ Norm Parabolic Partial Differential Equation Leaky ReLU Kolmogorov Flow

August 20, 2024

Approximation Rates for Shallow ReLU$^k$ Neural Networks on Sobolev Spaces via the Radon Transform
Tong Mao, Jonathan W. Siegel, Jinchao Xu
Shallow Neural Network ReLU Activation Shallow ReLU \Ell_p$ Norm Approximation Rate Radon Transform Sobolev Norm

July 3, 2024

$L_p$-norm Distortion-Efficient Adversarial Attack
Chao Zhou, Yuan-Gen Wang, Zi-jia Wang, Xiangui Kang
Adversarial Perturbation Distortion Metric \Ell_p$ Norm Threshold Network Distortion Efficient Adversarial Attack

February 9, 2024

RAMP: Boosting Adversarial Robustness Against Multiple $l_p$ Perturbations for Universal Robustness
Enyi Jiang, Gagandeep Singh
Adversarial Attack Adversarial Training Adversarial Robustness \Ell_p$ Norm Norm Attack Robustness Trade

September 24, 2023

Deep neural networks with ReLU, leaky ReLU, and softplus activation provably overcome the curse of dimensionality for Kolmogorov partial differential equations with Lipschitz nonlinearities in the $L^p$-sense
Julia Ackermann, Arnulf Jentzen, Thomas Kruse, Benno Kuckuck, Joshua Lee Padgett
Deep Neural Network High Dimensional Partial Differential Equation \Ell_p$ Norm Leaky ReLU Kolmogorov Flow Canonical Lipschitz Mechanism

May 15, 2023

Uniform-PAC Guarantees for Model-Based RL with Bounded Eluder Dimension
Yue Wu, Jiafan He, Quanquan Gu
Sample Complexity Sub Linear Regret \Ell_p$ Norm

March 31, 2023

Differentially Private Stochastic Convex Optimization in (Non)-Euclidean Space Revisited
Jinyan Su, Changhong Zhao, Di Wang
Convex Function Non Euclidean Private Gradient Convex Loss Function \Ell_p$ Norm

February 20, 2023

Seasoning Model Soups for Robustness to Adversarial and Natural Distribution Shifts
Francesco Croce, Sylvestre-Alvise Rebuffi, Evan Shelhamer, Sven Gowal
Native Robustness Adversarial Training \Ell_p$ Norm Model Soup Norm Attack Natural Distribution Shift

December 5, 2022

Multiple Perturbation Attack: Attack Pixelwise Under Different $\ell_p$-norms For Better Adversarial Performance
Ngoc N. Tran, Anh Tuan Bui, Dinh Phung, Trung Le
Adversarial Attack Adversarial Example \Ell_p$ Norm Perturbation Attack Better Adversarial Robustness ImageNet Attack

November 25, 2022

Optimal Approximation Rates for Deep ReLU Neural Networks on Sobolev and Besov Spaces
Jonathan W. Siegel
ReLU Network Function Space Deep ReLU Network \Ell_p$ Norm Approximation Rate Sharp Bound Sobolev Space

November 2, 2022

An efficient algorithm for the $\ell_{p}$ norm based metric nearness problem
Peipei Tang, Bo Jiang, Chengjing Wang
Metric Learning Primal Dual Efficient Algorithm Geometric Constraint \Ell_p$ Norm Nonsmooth Objective

September 23, 2022

Achieve the Minimum Width of Neural Networks for Universal Approximation
Yongqiang Cai
Neural Network Universal Approximation Wide Neural Network Universal Approximators \Ell_p$ Norm Minimum Width Approximation Power

August 16, 2022

$L^p$ sampling numbers for the Fourier-analytic Barron space
Felix Voigtlaender
Numerical Data Optimal Reconstruction \Ell_p$ Norm Barron Space Barron Function

June 9, 2022

A general approximation lower bound in $L^p$ norm, with applications to feed-forward neural networks
El Mehdi Achour, Armand Foucault, Sébastien Gerchinovitz, François Malgouyres
Neural Network Financial Application Average Approximation Lower Bound Upper Bound Rapid Norm Growth \Ell_p$ Norm Polynomial Neural Network

May 21, 2022

Pessimism for Offline Linear Contextual Bandits using $\ell_p$ Confidence Sets
Gene Li, Cong Ma, Nathan Srebro
Linear Contextual Bandit \Ell_p$ Norm Confidence Set Offline Contextual Bandit Optimistic Learning

May 5, 2022

Optimal Algorithms for Mean Estimation under Local Differential Privacy
Hilal Asi, Vitaly Feldman, Kunal Talwar
Local Differential Privacy Private Algorithm Optimal Algorithm Mean Estimation \Ell_p$ Norm

March 25, 2022

On the performance of preconditioned methods to solve $L^p$-norm phase unwrapping
Ricardo Legarda-Saenz, Carlos Brito-Loeza, Arturo Espinosa-Romero
System Performance NCD Method Adaptive Preconditioner \Ell_p$ Norm Phase Angle

March 18, 2022

Deterministic Bridge Regression for Compressive Classification
Kar-Ann Toh, Giuseppe Molteni, Zhiping Lin
Pattern Recognition Compact Representation \Ell_p$ Norm Analytic Solution Compressive Learning

March 16, 2022

$\ell_p$ Slack Norm Support Vector Data Description
Shervin R. Arashloo
\Ell_p$ Norm Support Vector Sparsity Inducing Norm Penalty

March 9, 2022

Reverse Engineering $\ell_p$ attacks: A block-sparse optimization approach with recovery guarantees
Darshan Thaker, Paris Giampouras, René Vidal
Adversarial Attack Reverse Engineering Attack Method Recovery Guarantee \Ell_p$ Norm Imperceptible Perturbation Norm Attack

\Ell_p$ Norm

Papers

Multilevel Picard approximations and deep neural networks with ReLU, leaky ReLU, and softplus activation overcome the curse of dimensionality when approximating semilinear parabolic partial differential equations in $L^p$-sense

Approximation Rates for Shallow ReLU$^k$ Neural Networks on Sobolev Spaces via the Radon Transform

$L_p$-norm Distortion-Efficient Adversarial Attack

RAMP: Boosting Adversarial Robustness Against Multiple $l_p$ Perturbations for Universal Robustness

Deep neural networks with ReLU, leaky ReLU, and softplus activation provably overcome the curse of dimensionality for Kolmogorov partial differential equations with Lipschitz nonlinearities in the $L^p$-sense

Uniform-PAC Guarantees for Model-Based RL with Bounded Eluder Dimension

Differentially Private Stochastic Convex Optimization in (Non)-Euclidean Space Revisited

Seasoning Model Soups for Robustness to Adversarial and Natural Distribution Shifts

Multiple Perturbation Attack: Attack Pixelwise Under Different $\ell_p$-norms For Better Adversarial Performance

Optimal Approximation Rates for Deep ReLU Neural Networks on Sobolev and Besov Spaces

An efficient algorithm for the $\ell_{p}$ norm based metric nearness problem

Achieve the Minimum Width of Neural Networks for Universal Approximation

$L^p$ sampling numbers for the Fourier-analytic Barron space

A general approximation lower bound in $L^p$ norm, with applications to feed-forward neural networks

Pessimism for Offline Linear Contextual Bandits using $\ell_p$ Confidence Sets

Optimal Algorithms for Mean Estimation under Local Differential Privacy

On the performance of preconditioned methods to solve \(L^p\)-norm phase unwrapping

Deterministic Bridge Regression for Compressive Classification

$\ell_p$ Slack Norm Support Vector Data Description

Reverse Engineering $\ell_p$ attacks: A block-sparse optimization approach with recovery guarantees