# Data Imputation from the Perspective of Graph Dirichlet Energy

arXiv: https://arxiv.org/abs/2304.04474 · HTML: https://mlsanity.com/paper/2304.04474/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 62 references that are in the map.
- First posted on arXiv 10 Apr 2023.

## Builds on (most cited first, 30 of 30)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Simplifying Graph Convolutional Networks](https://mlsanity.com/paper/1902.07153/) (2019)
- [Variational Graph Auto-Encoders](https://mlsanity.com/paper/1611.07308/) (2016)
- [Graph U-Nets](https://mlsanity.com/paper/1905.05178/) (2019)
- [Deep Laplacian Pyramid Networks for Fast and Accurate Super-Resolution](https://mlsanity.com/paper/1704.03915/) (2017)
- [Self-Attention Graph Pooling](https://mlsanity.com/paper/1904.08082/) (2019)
- [GAIN: Missing Data Imputation using Generative Adversarial Nets](https://mlsanity.com/paper/1806.02920/) (2018)
- [Large Scale Learning on Non-Homophilous Graphs: New Benchmarks and Strong Simple Methods](https://mlsanity.com/paper/2110.14446/) (2021)
- [A Note on Over-Smoothing for Graph Neural Networks](https://mlsanity.com/paper/2006.13318/) (2020)
- [Graph Neural Networks for Graphs with Heterophily: A Survey](https://mlsanity.com/paper/2202.07082/) (2022)
- [Traffic Graph Convolutional Recurrent Neural Network: A Deep Learning Framework for Network-Scale Traffic Learning and Forecasting](https://mlsanity.com/paper/1802.07007/) (2018)
- [Handling Missing Data with Graph Representation Learning](https://mlsanity.com/paper/2010.16418/) (2020)
- [Matrix Completion and Low-Rank SVD via Fast Alternating Least Squares](https://mlsanity.com/paper/1410.2596/) (2014)
- [Dirichlet Energy Constrained Learning for Deep Graph Neural Networks](https://mlsanity.com/paper/2107.02392/) (2021)
- [Symmetric Graph Convolutional Autoencoder for Unsupervised Graph Representation Learning](https://mlsanity.com/paper/1908.02441/) (2019)
- [Inductive Graph Neural Networks for Spatiotemporal Kriging](https://mlsanity.com/paper/2006.07527/) (2020)
- [Tackling Over-Smoothing for General Graph Convolutional Networks](https://mlsanity.com/paper/2008.09864/) (2020)
- [What's a good imputation to predict with missing values?](https://mlsanity.com/paper/2106.00311/) (2021)
- [Deep Bidirectional and Unidirectional LSTM Recurrent Neural Network for Network-wide Traffic Speed Prediction](https://mlsanity.com/paper/1801.02143/) (2018)
- [Densely Residual Laplacian Super-Resolution](https://mlsanity.com/paper/1906.12021/) (2019)
- [Scattering GCN: Overcoming Oversmoothness in Graph Convolutional Networks](https://mlsanity.com/paper/2003.08414/) (2020)
- [Semi-Supervised Graph Classification: A Hierarchical Graph Perspective](https://mlsanity.com/paper/1904.05003/) (2019)
- [MIRACLE: Causally-Aware Imputation via Learning Missing Data Mechanisms](https://mlsanity.com/paper/2111.03187/) (2021)
- [Graph Convolutional Networks for Graphs Containing Missing Features](https://mlsanity.com/paper/2007.04583/) (2020)
- [Faster Kernels for Graphs with Continuous Attributes via Hashing](https://mlsanity.com/paper/1610.00064/) (2016)
- [Deconvolutional Networks on Graph Data](https://mlsanity.com/paper/2110.15528/) (2021)
- [Using Fuzzy Logic to Leverage HTML Markup for Web Page Representation](https://mlsanity.com/paper/1606.04429/) (2016)
- [Handling Missing Data via Max-Entropy Regularized Graph Autoencoder](https://mlsanity.com/paper/2211.16771/) (2022)
- [Multiple imputation for continuous variables using a Bayesian principal component analysis](https://mlsanity.com/paper/1401.5747/) (2014)
- [Differentiable and Scalable Generative Adversarial Models for Data Imputation](https://mlsanity.com/paper/2201.03202/) (2022)

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

- [Spectral Initialization and Scheduled Graph Smoothness for Uncertain Knowledge Graph Completion](https://mlsanity.com/paper/2609.02519/) (2026)

## Abstract (arXiv)

Data imputation is a crucial task due to the widespread occurrence of missing data. Many methods adopt a two-step approach: initially crafting a preliminary imputation (the "draft") and then refining it to produce the final missing data imputation result, commonly referred to as "draft-then-refine". In our study, we examine this prevalent strategy through the lens of graph Dirichlet energy. We observe that a basic "draft" imputation tends to decrease the Dirichlet energy. Therefore, a subsequent "refine" step is necessary to restore the overall energy balance. Existing refinement techniques, such as the Graph Convolutional Network (GCN), often result in further energy reduction. To address this, we introduce a new framework, the Graph Laplacian Pyramid Network (GLPN). GLPN incorporates a U-shaped autoencoder and residual networks to capture both global and local details effectively. Through extensive experiments on multiple real-world datasets, GLPN consistently outperforms state-of-the-art methods across three different missing data mechanisms. The code is available at https://github.com/liguanlue/GLPN.