# Implicit Neural Representations and the Algebra of Complex Wavelets

arXiv: https://arxiv.org/abs/2310.00545 · HTML: https://mlsanity.com/paper/2310.00545/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 26 references that are in the map.
- First posted on arXiv 1 Oct 2023.

## Builds on (most cited first, 9 of 9)

- [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)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [WIRE: Wavelet Implicit Neural Representations](https://mlsanity.com/paper/2301.05187/) (2023)
- [Beyond Periodicity: Towards a Unifying Framework for Activations in Coordinate-MLPs](https://mlsanity.com/paper/2111.15135/) (2021)
- [A Structured Dictionary Perspective on Implicit Neural Representations](https://mlsanity.com/paper/2112.01917/) (2021)
- [Provable approximation properties for deep neural networks](https://mlsanity.com/paper/1509.07385/) (2015)
- [Signal Processing for Implicit Neural Representations](https://mlsanity.com/paper/2210.08772/) (2022)
- [Generalised Implicit Neural Representations](https://mlsanity.com/paper/2205.15674/) (2022)

## Built on by (most cited first, 8 of 8)

- [Continuous Representation Methods, Theories, and Applications: An Overview and Perspectives](https://mlsanity.com/paper/2505.15222/) (2025)
- [Bias for Action: Video Implicit Neural Representations with Bias Modulation](https://mlsanity.com/paper/2501.09277/) (2025)
- [Physics-Guided Dual Implicit Neural Representations for Source Separation](https://mlsanity.com/paper/2507.05249/) (2025)
- [Rates and architectures for learning geometrically non-trivial operators](https://mlsanity.com/paper/2512.09376/) (2025)
- [Hypernetwork-based approach for grid-independent functional data clustering](https://mlsanity.com/paper/2602.22823/) (2026)
- [Multi-resolution Enhancement for Full Spectrum Neural Representations](https://mlsanity.com/paper/2509.15494/) (2025)
- [In Situ Training of Implicit Neural Compressors for Scientific Simulations via Sketch-Based Regularization](https://mlsanity.com/paper/2511.02659/) (2025)
- [Recurrent Sinusoidal INRs for Efficient High-Fidelity Representation](https://mlsanity.com/paper/2607.21485/) (2026)

## Abstract (arXiv)

Implicit neural representations (INRs) have arisen as useful methods for representing signals on Euclidean domains. By parameterizing an image as a multilayer perceptron (MLP) on Euclidean space, INRs effectively represent signals in a way that couples spatial and spectral features of the signal that is not obvious in the usual discrete representation, paving the way for continuous signal processing and machine learning approaches that were not previously possible. Although INRs using sinusoidal activation functions have been studied in terms of Fourier theory, recent works have shown the advantage of using wavelets instead of sinusoids as activation functions, due to their ability to simultaneously localize in both frequency and space. In this work, we approach such INRs and demonstrate how they resolve high-frequency features of signals from coarse approximations done in the first layer of the MLP. This leads to multiple prescriptions for the design of INR architectures, including the use of complex wavelets, decoupling of low and band-pass approximations, and initialization schemes based on the singularities of the desired signal.