# Spectrum-Adapted Tight Graph Wavelet and Vertex-Frequency Frames

arXiv: https://arxiv.org/abs/1311.0897 · HTML: https://mlsanity.com/paper/1311.0897/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Nov 2013.

## Built on by (most cited first, 15 of 15)

- [Graph Filters for Signal Processing and Machine Learning on Graphs](https://mlsanity.com/paper/2211.08854/) (2022)
- [Unlearning Graph Classifiers with Limited Data Resources](https://mlsanity.com/paper/2211.03216/) (2022)
- [Training Robust Graph Neural Networks with Topology Adaptive Edge Dropping](https://mlsanity.com/paper/2106.02892/) (2021)
- [Wide and Deep Graph Neural Network with Distributed Online Learning](https://mlsanity.com/paper/2107.09203/) (2021)
- [Handling Missing Data via Max-Entropy Regularized Graph Autoencoder](https://mlsanity.com/paper/2211.16771/) (2022)
- [Adaptive Gaussian Processes on Graphs via Spectral Graph Wavelets](https://mlsanity.com/paper/2110.12752/) (2021)
- [A General Graph Spectral Wavelet Convolution via Chebyshev Order Decomposition](https://mlsanity.com/paper/2405.13806/) (2024)
- [Analysis of the Spatio-temporal Dynamics of COVID-19 in Massachusetts via Spectral Graph Wavelet Theory](https://mlsanity.com/paper/2208.01749/) (2022)
- [Graph Scattering beyond Wavelet Shackles](https://mlsanity.com/paper/2301.11456/) (2023)
- [On the Trade-Off between Stability and Representational Capacity in Graph Neural Networks](https://mlsanity.com/paper/2312.02372/) (2023)
- [Covariance Scattering Transforms](https://mlsanity.com/paper/2511.08878/) (2025)
- [DPC: Unsupervised Deep Point Correspondence via Cross and Self Construction](https://mlsanity.com/paper/2110.08636/) (2021)
- [Frames for Graph Signals on the Symmetric Group: A Representation Theoretic Approach](https://mlsanity.com/paper/2203.03036/) (2022)
- [Windowed Fourier Analysis for Signal Processing on Graph Bundles](https://mlsanity.com/paper/2302.05592/) (2023)
- [Data-Driven Graph Filters via Adaptive Spectral Shaping](https://mlsanity.com/paper/2602.03698/) (2026)

## Abstract (arXiv)

We consider the problem of designing spectral graph filters for the construction of dictionaries of atoms that can be used to efficiently represent signals residing on weighted graphs. While the filters used in previous spectral graph wavelet constructions are only adapted to the length of the spectrum, the filters proposed in this paper are adapted to the distribution of graph Laplacian eigenvalues, and therefore lead to atoms with better discriminatory power. Our approach is to first characterize a family of systems of uniformly translated kernels in the graph spectral domain that give rise to tight frames of atoms generated via generalized translation on the graph. We then warp the uniform translates with a function that approximates the cumulative spectral density function of the graph Laplacian eigenvalues. We use this approach to construct computationally efficient, spectrum-adapted, tight vertex-frequency and graph wavelet frames. We give numerous examples of the resulting spectrum-adapted graph filters, and also present an illustrative example of vertex-frequency analysis using the proposed construction.