# Towards a Sampling Theory for Implicit Neural Representations

arXiv: https://arxiv.org/abs/2405.18410 · HTML: https://mlsanity.com/paper/2405.18410/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 33 references that are in the map.
- First posted on arXiv 28 May 2024.

## Builds on (most cited first, 15 of 15)

- [Instant Neural Graphics Primitives with a Multiresolution Hash Encoding](https://mlsanity.com/paper/2201.05989/) (2022)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Neural Fields in Visual Computing and Beyond](https://mlsanity.com/paper/2111.11426/) (2021)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [Characterizing Implicit Bias in Terms of Optimization Geometry](https://mlsanity.com/paper/1802.08246/) (2018)
- [WIRE: Wavelet Implicit Neural Representations](https://mlsanity.com/paper/2301.05187/) (2023)
- [A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Case](https://mlsanity.com/paper/1910.01635/) (2019)
- [How do infinite width bounded norm networks look in function space?](https://mlsanity.com/paper/1902.05040/) (2019)
- [Seeing Implicit Neural Representations as Fourier Series](https://mlsanity.com/paper/2109.00249/) (2021)
- [Convex Geometry and Duality of Over-parameterized Neural Networks](https://mlsanity.com/paper/2002.11219/) (2020)
- [Sparsity of solutions for variational inverse problems with finite-dimensional data](https://mlsanity.com/paper/1809.05045/) (2018)
- [Deep Learning Meets Sparse Regularization: A Signal Processing Perspective](https://mlsanity.com/paper/2301.09554/) (2023)
- [Off-the-Grid Recovery of Piecewise Constant Images from Few Fourier Samples](https://mlsanity.com/paper/1510.00384/) (2015)
- [Recovery of Piecewise Smooth Images from Few Fourier Samples](https://mlsanity.com/paper/1502.00705/) (2015)

## Built on by (most cited first, 1 of 1)

- [Content-Aware Frequency Encoding for Implicit Neural Representations with Fourier-Chebyshev Features](https://mlsanity.com/paper/2603.01028/) (2026)

## Abstract (arXiv)

Implicit neural representations (INRs) have emerged as a powerful tool for solving inverse problems in computer vision and computational imaging. INRs represent images as continuous domain functions realized by a neural network taking spatial coordinates as inputs. However, unlike traditional pixel representations, little is known about the sample complexity of estimating images using INRs in the context of linear inverse problems. Towards this end, we study the sampling requirements for recovery of a continuous domain image from its low-pass Fourier coefficients by fitting a single hidden-layer INR with ReLU activation and a Fourier features layer using a generalized form of weight decay regularization. Our key insight is to relate minimizers of this non-convex parameter space optimization problem to minimizers of a convex penalty defined over an infinite-dimensional space of measures. We identify a sufficient number of samples for which an image realized by a width-1 INR is exactly recoverable by solving the INR training problem, and give a conjecture for the general width-W case. To validate our theory, we empirically assess the probability of achieving exact recovery of images realized by low-width single hidden-layer INRs, and illustrate the performance of INR on super-resolution recovery of more realistic continuous domain phantom images.