Papers

Manuscripts Under Review

1. arXiv

   Optimal Recovery Meets Minimax Estimation

   Ronald DeVore, Robert D. Nowak , Rahul Parhi, Guergana Petrova, and Jonathan W. Siegel

   2025

   arXiv Bib

   @unpublished{devore2025optimal,
     title = {Optimal Recovery Meets Minimax Estimation},
     author = {DeVore, Ronald and Nowak, Robert D. and Parhi, Rahul and Petrova, Guergana and Siegel, Jonathan W.},
     year = {2025},
     eprinttype = {arxiv},
     eprintclass = {math.ST},
     eprint = {2502.17671},
   }

2025

1. JMLR

   Random ReLU Neural Networks as Non-Gaussian Processes

   Rahul Parhi, Pakshal Bohra, Ayoub El Biari, Mehrsa Pourya, and Michael Unser

   Journal of Machine Learning Research, 2025

   Bib HTML PDF

   @article{parhi2025random,
     title = {Random ReLU Neural Networks as Non-{G}aussian Processes},
     author = {Parhi, Rahul and Bohra, Pakshal and {El Biari}, Ayoub and Pourya, Mehrsa and Unser, Michael},
     year = {2025},
     journal = {Journal of Machine Learning Research},
     volume = {26},
     number = {19},
     pages = {1--31},
     url = {http://jmlr.org/papers/v26/24-0737.html},
   }

2. SIMODS

   Function-Space Optimality of Neural Architectures with Multivariate Nonlinearities

   Rahul Parhi, and Michael Unser

   SIAM Journal on Mathematics of Data Science, 2025

   Bib DOI PDF

   @article{parhi2025function,
     title = {Function-Space Optimality of Neural Architectures with Multivariate Nonlinearities},
     author = {Parhi, Rahul and Unser, Michael},
     year = {2025},
     journal = {SIAM Journal on Mathematics of Data Science},
     volume = {7},
     number = {1},
     pages = {110--135},
     doi = {10.1137/23M1620971},
   }

3. ACHA

   Weighted Variation Spaces and Approximation by Shallow ReLU Networks

   Ronald DeVore, Robert D. Nowak , Rahul Parhi, and Jonathan W. Siegel

   Applied and Computational Harmonic Analysis, 2025

   Bib DOI PDF

   @article{devore2025weighted,
     title = {Weighted Variation Spaces and Approximation by Shallow {ReLU} Networks},
     author = {DeVore, Ronald and Nowak, Robert D. and Parhi, Rahul and Siegel, Jonathan W.},
     year = {2025},
     journal = {Applied and Computational Harmonic Analysis},
     volume = {74},
     number = {101713},
     doi = {10.1016/j.acha.2024.101713},
   }

2024

1. JMLR

   Variation Spaces for Multi-Output Neural Networks: Insights on Multi-Task Learning and Network Compression

   Joseph Shenouda , Rahul Parhi, Kangwook Lee, and Robert D. Nowak

   Journal of Machine Learning Research, 2024

   Bib HTML PDF

   @article{shenouda2024variation,
     title = {Variation Spaces for Multi-Output Neural Networks: Insights on Multi-Task Learning and Network Compression},
     author = {Shenouda, Joseph and Parhi, Rahul and Lee, Kangwook and Nowak, Robert D.},
     year = {2024},
     journal = {Journal of Machine Learning Research},
     volume = {25},
     number = {231},
     pages = {1--40},
     url = {https://www.jmlr.org/papers/v25/23-0677.html},
   }

2. SIMA

   Distributional Extension and Invertibility of the k-Plane Transform and Its Dual

   Rahul Parhi, and Michael Unser

   SIAM Journal on Mathematical Analysis, 2024

   Bib DOI PDF

   @article{parhi2024distributional,
     title = {Distributional Extension and Invertibility of the {$k$}-Plane Transform and Its Dual},
     author = {Parhi, Rahul and Unser, Michael},
     year = {2024},
     journal = {SIAM Journal on Mathematical Analysis},
     volume = {56},
     number = {4},
     pages = {4662--4686},
     doi = {10.1137/23M1556721},
   }

2023

1. IEEE SPM

   Deep Learning Meets Sparse Regularization: A signal processing perspective

   Rahul Parhi, and Robert D. Nowak

   IEEE Signal Processing Magazine, 2023

   Bib DOI PDF

   @article{parhi2023deep,
     title = {Deep Learning Meets Sparse Regularization: A signal processing perspective},
     author = {Parhi, Rahul and Nowak, Robert D.},
     year = {2023},
     journal = {IEEE Signal Processing Magazine},
     volume = {40},
     number = {6},
     pages = {63--74},
     doi = {10.1109/MSP.2023.3286988},
   }

2. IEEE SPL

   The Sparsity of Cycle Spinning for Wavelet-Based Solutions of Linear Inverse Problems

   Rahul Parhi, and Michael Unser

   IEEE Signal Processing Letters, 2023

   Bib DOI PDF

   @article{parhi2023sparsity,
     title = {The Sparsity of Cycle Spinning for Wavelet-Based Solutions of Linear Inverse Problems},
     author = {Parhi, Rahul and Unser, Michael},
     year = {2023},
     journal = {IEEE Signal Processing Letters},
     volume = {30},
     pages = {568--572},
     doi = {10.1109/LSP.2023.3275916},
   }

3. IEEE TIT

   Near-Minimax Optimal Estimation With Shallow ReLU Neural Networks

   Rahul Parhi, and Robert D. Nowak

   IEEE Transactions on Information Theory, 2023

   Bib DOI PDF

   @article{parhi2023near,
     title = {Near-Minimax Optimal Estimation With Shallow {ReLU} Neural Networks},
     author = {Parhi, Rahul and Nowak, Robert D.},
     year = {2023},
     journal = {IEEE Transactions on Information Theory},
     volume = {69},
     number = {2},
     pages = {1125--1140},
     doi = {10.1109/TIT.2022.3208653},
   }

4. SampTA

   Modulation Spaces and the Curse of Dimensionality

   Rahul Parhi, and Michael Unser

   In International Conference on Sampling Theory and Applications (SampTA), 2023

   Bib DOI PDF

   @inproceedings{parhi2023modulation,
     title = {Modulation Spaces and the Curse of Dimensionality},
     author = {Parhi, Rahul and Unser, Michael},
     year = {2023},
     booktitle = {International Conference on Sampling Theory and Applications (SampTA)},
     pages = {1--5},
     doi = {10.1109/SampTA59647.2023.10301395},
   }

5. SampTA

   A Continuous Transform for Localized Ridgelets

   Joseph Shenouda , Rahul Parhi, and Robert D. Nowak

   In International Conference on Sampling Theory and Applications (SampTA), 2023

   Bib DOI PDF

   @inproceedings{shenouda2023continuous,
     title = {A Continuous Transform for Localized Ridgelets},
     author = {Shenouda, Joseph and Parhi, Rahul and Nowak, Robert D.},
     year = {2023},
     booktitle = {International Conference on Sampling Theory and Applications (SampTA)},
     pages = {1--5},
     doi = {10.1109/SampTA59647.2023.10301398},
   }

2022

1. SIMODS

   What Kinds of Functions Do Deep Neural Networks Learn? Insights from Variational Spline Theory

   Rahul Parhi, and Robert D. Nowak

   SIAM Journal on Mathematics of Data Science, 2022

   Bib DOI PDF

   @article{parhi2022kinds,
     title = {What Kinds of Functions Do Deep Neural Networks Learn? Insights from Variational Spline Theory},
     author = {Parhi, Rahul and Nowak, Robert D.},
     year = {2022},
     journal = {SIAM Journal on Mathematics of Data Science},
     volume = {4},
     number = {2},
     pages = {464--489},
     doi = {10.1137/21M1418642},
   }

2. ICASSP

   On Continuous-Domain Inverse Problems with Sparse Superpositions of Decaying Sinusoids as Solutions

   Rahul Parhi, and Robert D. Nowak

   In IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), 2022

   Bib DOI PDF

   @inproceedings{parhi2022continuous,
     title = {On Continuous-Domain Inverse Problems with Sparse Superpositions of Decaying Sinusoids as Solutions},
     author = {Parhi, Rahul and Nowak, Robert D.},
     year = {2022},
     booktitle = {IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP)},
     pages = {5603--5607},
     doi = {10.1109/ICASSP43922.2022.9746165},
   }

3. PhD

   On Ridge Splines, Neural Networks, and Variational Problems in Radon-Domain BV Spaces

   Rahul Parhi

   University of Wisconsin–Madison, 2022

   Awarded Bib PDF

   2023 Harold Peterson Outstanding Dissertation Award from UW–Madison

   @phdthesis{parhi2022ridge,
     title = {On Ridge Splines, Neural Networks, and Variational Problems in {R}adon-Domain {BV} Spaces},
     author = {Parhi, Rahul},
     year = {2022},
     url = {https://search.library.wisc.edu/catalog/9913848927002121},
     school = {University of Wisconsin--Madison},
   }

2021

1. JMLR

   Banach Space Representer Theorems for Neural Networks and Ridge Splines

   Rahul Parhi, and Robert D. Nowak

   Journal of Machine Learning Research, 2021

   Bib HTML PDF

   @article{parhi2021banach,
     title = {Banach Space Representer Theorems for Neural Networks and Ridge Splines},
     author = {Parhi, Rahul and Nowak, Robert D.},
     year = {2021},
     journal = {Journal of Machine Learning Research},
     volume = {22},
     number = {43},
     pages = {1--40},
     url = {https://jmlr.org/papers/v22/20-583.html},
   }

2020

1. IEEE SPL

   The Role of Neural Network Activation Functions

   Rahul Parhi, and Robert D. Nowak

   IEEE Signal Processing Letters, 2020

   Bib DOI PDF

   @article{parhi2020role,
     title = {The Role of Neural Network Activation Functions},
     author = {Parhi, Rahul and Nowak, Robert D.},
     year = {2020},
     journal = {IEEE Signal Processing Letters},
     volume = {27},
     pages = {1779--1783},
     doi = {10.1109/LSP.2020.3027517},
   }

2018

1. FC

   MP3: A More Efficient Private Presence Protocol

   Rahul Parhi, Michael Schliep, and Nicholas Hopper

   In Financial Cryptography and Data Security, 2018

   Bib DOI PDF

   @inproceedings{parhi2018mp3,
     title = {{MP3}: A More Efficient Private Presence Protocol},
     author = {Parhi, Rahul and Schliep, Michael and Hopper, Nicholas},
     year = {2018},
     booktitle = {Financial Cryptography and Data Security},
     pages = {38--57},
     doi = {10.1007/978-3-662-58387-6_3},
   }

2015

1. ISCAS

   Fault-Tolerant Ripple-Carry Binary Adder Using Partial Triple Modular Redundancy (PTMR)

   Rahul Parhi, Chris H. Kim, and Keshab K. Parhi

   In IEEE International Symposium on Circuits and Systems (ISCAS), 2015

   Bib DOI PDF

   @inproceedings{parhi2015fault,
     title = {Fault-Tolerant Ripple-Carry Binary Adder Using Partial Triple Modular Redundancy (PTMR)},
     author = {Parhi, Rahul and Kim, Chris H. and Parhi, Keshab K.},
     year = {2015},
     booktitle = {IEEE International Symposium on Circuits and Systems (ISCAS)},
     pages = {41--44},
     doi = {10.1109/ISCAS.2015.7168565},
   }