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The Total Variation on Hypergraphs - Learning on Hypergraphs Revisited
Matthias Hein, Simon Setzer, Leonardo Jost, Syama Sundar Rangapuram
TL;DR
Existing hypergraph learning methods rely on graph approximations or restricted tensor methods, despite higher-order relationships helping learning. This paper directly uses hypergraph cuts through total-variation-based regularizers, and its SSL methods consistently outperform clique expansion on all but one dataset.
Problem
Existing methods either approximate hypergraphs with graphs or use tensor methods applicable only under special conditions, although higher-order relations can improve learning performance.
Method
The paper defines hypergraph total variation as the Lovasz extension of the hypergraph cut and builds regularization functionals interpolating between total variation and smoother Laplacian-type regularization.
Results
SSL methods based on ΩH,p, p = 1, 2, consistently outperform clique expansion on all datasets except 20newsgroups.
Takeaways & Limitations
The framework incorporates hypergraph structure directly into clustering and semi-supervised learning through cut-based regularization.
Takeaways & Limitations
Clique expansion can perform better on 20newsgroups because large hyperedges accumulate label noise, while the balancing-function analysis is restricted to submodular functions.
Abstract
from arXiv · showhide
Hypergraphs allow one to encode higher-order relationships in data and are thus a very flexible modeling tool. Current learning methods are either based on approximations of the hypergraphs via graphs or on tensor methods which are only applicable under special conditions. In this paper, we present a new learning framework on hypergraphs which fully uses the hypergraph structure. The key element is a family of regularization functionals based on the total variation on hypergraphs.
1 Introduction
Hypergraphs extend graph-based learning by representing higher-order relationships, but existing methods either restrict hyperedge uniformity or approximate hypergraphs with graphs. The paper proposes directly optimizing hypergraph cuts through total-variation-based regularization and scalable convex methods.
- Hypergraphs model higher-order relationships beyond pairwise graph structure, supporting applications including computer vision, bioinformatics, and information retrieval.
- Tensor-based methods are limited to k-uniform hypergraphs, preventing them from modeling mixed higher-order relationships.
- Graph-expansion methods handle arbitrary hypergraphs by approximating them with standard graphs.
- The paper directly uses hypergraph cuts and introduces total variation as the Lovasz extension of the hypergraph cut.
- A family of regularization functionals connects hypergraph total variation with smoother Laplacian-type regularization for semi-supervised learning.
2 The Total Variation on Hypergraphs
The paper defines hypergraph cuts and their total variation without reducing hypergraphs to graphs. Its Lovasz-extension formulation yields convex, edge-structured regularizers that recover graph constructions in the 2-uniform case while preserving higher-order structure.
- Hypergraphs, Graphs and Cuts: A weighted undirected hypergraph consists of vertices, subset-valued hyperedges, and non-negative hyperedge weights, without requiring uniform hyperedge cardinality.
- Hypergraphs, Graphs and Cuts: The hypergraph cut sums weights of hyperedges intersecting both sides of a partition, assigning the same penalty regardless of how many vertices are separated.
- Hypergraphs, Graphs and Cuts: Clique expansion makes cut weights depend on the partition and can bias solutions toward splitting off single vertices; large hyperedges also create nearly fully connected graphs.
- Hypergraphs, Graphs and Cuts: The figure’s weights produce a balanced minimum hypergraph cut, whereas clique expansion selects an unbalanced cut that cuts one additional hyperedge.
- Hypergraphs, Graphs and Cuts: For weighted 3-uniform hypergraphs, a graph can represent every corresponding hypergraph cut exactly.
- The Total Variation on Hypergraphs: The Lovasz extension maps the hypergraph cut set function to a convex total variation functional on real-valued vertex functions.
- The Total Variation on Hypergraphs: Hypergraph total variation reduces to graph total variation for 2-uniform hypergraphs and can be written using overlapping hyperedge-based group differences.
- The Total Variation on Hypergraphs: The family Ω_H,p includes total variation at p = 1, graph-Laplacian regularization for graphs, and the hypergraph cut on characteristic functions.
3 Semi-supervised Learning
The paper formulates two-class semi-supervised learning on hypergraphs using labeled data and a hypergraph regularizer. It recommends p = 2 generally, while noting that p = 1 can also yield reasonable solutions in some hypergraph settings.
- The label vector assigns 0 to unlabeled vertices and labels in {−1, 1} to labeled vertices.
- The formulation combines a loss on labeled data with the hypergraph regularizer ΩH,p and regularization parameter λ > 0.
- The convex optimization problem can be solved efficiently for p = 1 and p = 2.
- The authors generally recommend p = 2 because it corresponds to Laplacian-type graph regularization known to work well.
- For hypergraphs with few hyperedges, p = 1 produces reasonable solutions despite producing spiky solutions on graphs with few labeled points.
4 Balanced Hypergraph Cuts
Balanced hypergraph cuts are optimized through a total-variation ratio whose exact continuous relaxation is a nonlinear eigenproblem. The resulting RatioDCA approach handles a convex-over-difference-of-convex objective through iterative convex inner problems.
- Unlike graph normalized cuts, normalized hypergraph cuts do not admit relaxation into a linear eigenproblem.
- The paper extends exact nonlinear-eigenproblem relaxation methods to general balanced hypergraph cuts.
- Balanced cuts use the hypergraph cut divided by a symmetric, non-negative balancing function such as normalized-cut or Cheeger-cut volume terms.
- Theorem 4.1 establishes an exact continuous nonlinear relaxation for finite weighted hypergraphs with symmetric, non-negative balancing functions.
- Optimal thresholding of any continuous function into a level-set partition achieves a balanced hypergraph cut no worse than the corresponding continuous value.
- RatioDCA minimizes the non-negative ratio of convex total variation to a difference-of-convex balancing extension through convex inner problems.
- For submodular balancing functions, the balancing extension is convex, simplifying the decomposition by setting its second convex component to zero.
5 Algorithms for the Total Variation on Hypergraphs
The paper solves its hypergraph regularized optimization problems with primal-dual methods whose iterations decouple across convex terms. A novel proximal-map procedure makes the key hyperedge subproblem computationally tractable.
- Semi-supervised learning and RatioDCA inner problems share a sum-of-convex-functionals structure containing the hypergraph regularizer ΩH,p.
- PDHG iteratively solves proximal problems for the separate convex terms in the objective.
- For p = 1, the method represents each hyperedge contribution using two functionals and solves associated conjugate proximal maps through simplex projections.
- The primal-dual formulation uses operators Ke that select the vertices belonging to each hyperedge.
- The simplex projections for the p = 1 subproblem can be performed in linear time.
- For p = 2, the conjugate function is not an indicator function, so its proximal problem is solved through a corresponding proximal problem for the original functional.
- The novel proximal-map result computes the p = 2 hyperedge map with O(me log me) arithmetic operations.
- After sorting, the p = 2 algorithm takes order m steps to compute the proximal map.
6 Experiments
Experiments compare the proposed hypergraph methods with clique-expansion baselines for clustering and semi-supervised learning. The results show competitive or better accuracy, while preserving substantial memory advantages on large hypergraphs.
- Experimental setup: The experiments evaluate clustering and semi-supervised learning against Zhou et al.’s clique-expansion method on UCI datasets.The datasets include categorical-feature benchmarks and two covertype tasks formed from pairs of original classes.
- Experimental setup: The clique expansion produces nearly fully connected graphs for datasets with large hyperedges, creating substantial memory demands.For covertype (6,7), the clique-expansion weight matrix requires over 10GB, whereas the original hypergraph requires 4MB.
- Computational cost: The proposed SSL method uses 2∑e∈E|e| + |V| memory, and solves the largest covertype (6,7) example in 30–100 seconds depending on λ.The clique-expansion approach can require |V|^2 memory and O(|V|^3) computations in the worst case; experiments stop at a relative duality gap of 10^-6.
- Semi-supervised learning: The SSL methods based on ΩH,p, p = 1, 2 outperform Zhou et al.’s clique-expansion technique on every dataset except 20newsgroups3.The paper attributes the exception to label noise accumulated by large hyperedges and recommends ΩH,2 because it slightly outperforms ΩH,1.
- Clustering: Our normalized-hypergraph-cut method achieves better or similar clustering errors than clique expansion, while always attaining smaller normalized hypergraph cuts.The improvement is significant on Mushrooms, whereas Zoo has a slightly higher clustering error; covertype (6,7) could not be run with Zhou et al.’s method because the weight matrix was too dense.
- Discussion: The authors conclude that incorporating hypergraph structure improves results on most datasets, while hypergraph construction remains an area for future improvement.The comparison uses a standard graph-based approach based on Hamming distance for categorical features.