# What is Smoothness?

arXiv: https://arxiv.org/abs/2609.03246 · HTML: https://mlsanity.com/paper/2609.03246/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 46 references that are in the map.
- First posted on arXiv 3 Sep 2026.

## Builds on (most cited first, 16 of 16)

- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Explaining Explanations: An Overview of Interpretability of Machine Learning](https://mlsanity.com/paper/1806.00069/) (2018)
- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks](https://mlsanity.com/paper/1901.06523/) (2019)
- [Interpretable machine learning: definitions, methods, and applications](https://mlsanity.com/paper/1901.04592/) (2019)
- [Graph signal processing for machine learning: A review and new perspectives](https://mlsanity.com/paper/2007.16061/) (2020)
- [Exploiting symmetry in variational quantum machine learning](https://mlsanity.com/paper/2205.06217/) (2022)
- [Geometric Deep Learning and Equivariant Neural Networks](https://mlsanity.com/paper/2105.13926/) (2021)
- [Group-Invariant Quantum Machine Learning](https://mlsanity.com/paper/2205.02261/) (2022)
- [Graph Signal Processing: History, Development, Impact, and Outlook](https://mlsanity.com/paper/2303.12211/) (2023)
- [Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces I: the compact case](https://mlsanity.com/paper/2208.14960/) (2022)
- [Spectral methods: crucial for machine learning, natural for quantum computers?](https://mlsanity.com/paper/2603.24654/) (2026)
- [Signal Processing on the Permutahedron: Tight Spectral Frames for Ranked Data Analysis](https://mlsanity.com/paper/2103.04150/) (2021)
- [Frames for signal processing on Cayley graphs](https://mlsanity.com/paper/2303.02812/) (2023)

## Abstract (arXiv)

Smoothness of a function on the real line is reflected in the decay of its Fourier transform, which suggests that smoothness of a function in L2(G) for a group G should mean concentration of the Fourier coefficients at low frequency. Such a reading presupposes an ordering of the irreducible representations of G, but for non-abelian G, no ordering is canonical. Given a symmetric generating set S, the Laplacian of the associated Cayley graph is block diagonal over the dual, and we order the irreps by the mean of the eigenvalues in each block. This produces an ordering function ω:widehatG→ℝ that depends only on the pair (G,S). This function is bounded between zero and two, vanishing only at the trivial representation and achieving the upper bound exactly when the Cayley graph is bipartite. We then ask how much freedom the construction has. Within the class of operators satisfying natural axioms, the induced orderings are exactly the real functions on the dual vanishing at the trivial representation and agreeing on conjugate pairs, and the orderings coming from inversion orbits of conjugacy classes form a basis for them. We cut the freedom down further by requiring two additional inputs: nonnegativity of the class weights and a declaration of which group elements count as uniform incremental changes, which pins the operator to the Cayley-Laplacian up to positive scale. We observe that the construction persists for compact groups even though the Cayley graph does not, and we extend the theory to finite sets carrying a transitive group action, where the acting group selects which frequencies exist and the generating set orders them. The answer to the title question is therefore that smoothness is a property of a function together with a choice of group and generating set, not of the function alone.