135 citations · 282 across the 11 of their papers we have counts for
12 papers
Optimal Recovery Meets Minimax Estimation
Ronald DeVore, Robert D. Nowak, Rahul Parhi +2
A fundamental problem in statistics and machine learning is to estimate a function from possibly noisy observations of its point samples. The goal is to design a numerical algo…
Random ReLU Neural Networks as Non-Gaussian Processes
Rahul Parhi, Pakshal Bohra, Ayoub El Biari +2
We consider a large class of shallow neural networks with randomly initialized parameters and rectified linear unit activation functions. We prove that these random neural networks…
Function-Space Optimality of Neural Architectures with Multivariate Nonlinearities
Rahul Parhi, Michael Unser
We investigate the function-space optimality (specifically, the Banach-space optimality) of a large class of shallow neural architectures with multivariate nonlinearities/activatio…
Distributional Extension and Invertibility of the -Plane Transform and Its Dual
Rahul Parhi, Michael Unser
We investigate the distributional extension of the -plane transform in and of related operators. We parameterize the -plane domain as the Cartesian product of…
Weighted variation spaces and approximation by shallow ReLU networks
Ronald DeVore, Robert D. Nowak, Rahul Parhi +1
We investigate the approximation of functions on a bounded domain by the outputs of single-hidden-layer ReLU neural networks of width . This form of…
Variation Spaces for Multi-Output Neural Networks: Insights on Multi-Task Learning and Network Compression
Joseph Shenouda, Rahul Parhi, Kangwook Lee +1
This paper introduces a novel theoretical framework for the analysis of vector-valued neural networks through the development of vector-valued variation spaces, a new class of repr…
Deep Learning Meets Sparse Regularization: A Signal Processing Perspective
Rahul Parhi, Robert D. Nowak
Deep learning has been wildly successful in practice and most state-of-the-art machine learning methods are based on neural networks. Lacking, however, is a rigorous mathematical t…
Near-Minimax Optimal Estimation With Shallow ReLU Neural Networks
Rahul Parhi, Robert D. Nowak
We study the problem of estimating an unknown function from noisy data using shallow ReLU neural networks. The estimators we study minimize the sum of squared data-fitting errors p…
What Kinds of Functions do Deep Neural Networks Learn? Insights from Variational Spline Theory
Rahul Parhi, Robert D. Nowak
We develop a variational framework to understand the properties of functions learned by fitting deep neural networks with rectified linear unit activations to data. We propose a ne…
Banach Space Representer Theorems for Neural Networks and Ridge Splines
Rahul Parhi, Robert D. Nowak
We develop a variational framework to understand the properties of the functions learned by neural networks fit to data. We propose and study a family of continuous-domain linear i…
The Role of Neural Network Activation Functions
Rahul Parhi, Robert D. Nowak
A wide variety of activation functions have been proposed for neural networks. The Rectified Linear Unit (ReLU) is especially popular today. There are many practical reasons that m…
MP3: A More Efficient Private Presence Protocol
Rahul Parhi, Michael Schliep, Nicholas Hopper
This paper proposes MP3, the second privacy-preserving presence protocol that leaks no information about the graph structure of the social network. Several cryptographic techniques…