# Multiscale Laplacian Learning

arXiv: https://arxiv.org/abs/2109.03718 · HTML: https://mlsanity.com/paper/2109.03718/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 124 references that are in the map.
- First posted on arXiv 8 Sep 2021.

## Builds on (most cited first, 14 of 14)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Spectral Networks and Locally Connected Networks on Graphs](https://mlsanity.com/paper/1312.6203/) (2013)
- [Deeper Insights into Graph Convolutional Networks for Semi-Supervised Learning](https://mlsanity.com/paper/1801.07606/) (2018)
- [Revisiting Semi-Supervised Learning with Graph Embeddings](https://mlsanity.com/paper/1603.08861/) (2016)
- [Label Propagation for Deep Semi-supervised Learning](https://mlsanity.com/paper/1904.04717/) (2019)
- [On Deep Multi-View Representation Learning: Objectives and Optimization](https://mlsanity.com/paper/1602.01024/) (2016)
- [Attention-based Graph Neural Network for Semi-supervised Learning](https://mlsanity.com/paper/1803.03735/) (2018)
- [Bayesian graph convolutional neural networks for semi-supervised classification](https://mlsanity.com/paper/1811.11103/) (2018)
- [FALKON: An Optimal Large Scale Kernel Method](https://mlsanity.com/paper/1705.10958/) (2017)
- [GMNN: Graph Markov Neural Networks](https://mlsanity.com/paper/1905.06214/) (2019)
- [N-GCN: Multi-scale Graph Convolution for Semi-supervised Node Classification](https://mlsanity.com/paper/1802.08888/) (2018)
- [Graph Partition Neural Networks for Semi-Supervised Classification](https://mlsanity.com/paper/1803.06272/) (2018)
- [Active Semi-Supervised Learning Using Sampling Theory for Graph Signals](https://mlsanity.com/paper/1405.4324/) (2014)
- [Convex variational methods for multiclass data segmentation on graphs](https://mlsanity.com/paper/1605.01443/) (2016)

## Built on by (most cited first, 5 of 5)

- [Integrating Transformer and Autoencoder Techniques with Spectral Graph Algorithms for the Prediction of Scarcely Labeled Molecular Data](https://mlsanity.com/paper/2211.06759/) (2022)
- [Persistent Laplacian-enhanced Algorithm for Scarcely Labeled Data Classification](https://mlsanity.com/paper/2305.16239/) (2023)
- [Higher-Order Regularization Learning on Hypergraphs](https://mlsanity.com/paper/2510.26533/) (2025)
- [Graph-Based Bidirectional Transformer Decision Threshold Adjustment Algorithm for Class-Imbalanced Molecular Data](https://mlsanity.com/paper/2406.06479/) (2024)
- [Topology-Aware Active Learning on Graphs](https://mlsanity.com/paper/2510.25892/) (2025)

## Abstract (arXiv)

Machine learning methods have greatly changed science, engineering, finance, business, and other fields. Despite the tremendous accomplishments of machine learning and deep learning methods, many challenges still remain. In particular, the performance of machine learning methods is often severely affected in case of diverse data, usually associated with smaller data sets or data related to areas of study where the size of the data sets is constrained by the complexity and/or high cost of experiments. Moreover, data with limited labeled samples is a challenge to most learning approaches. In this paper, the aforementioned challenges are addressed by integrating graph-based frameworks, multiscale structure, modified and adapted optimization procedures and semi-supervised techniques. This results in two innovative multiscale Laplacian learning (MLL) approaches for machine learning tasks, such as data classification, and for tackling diverse data, data with limited samples and smaller data sets. The first approach, called multikernel manifold learning (MML), integrates manifold learning with multikernel information and solves a regularization problem consisting of a loss function and a warped kernel regularizer using multiscale graph Laplacians. The second approach, called the multiscale MBO (MMBO) method, introduces multiscale Laplacians to a modification of the famous classical Merriman-Bence-Osher (MBO) scheme, and makes use of fast solvers for finding the approximations to the extremal eigenvectors of the graph Laplacian. We demonstrate the performance of our methods experimentally on a variety of data sets, such as biological, text and image data, and compare them favorably to existing approaches.