Hypergraph $p$-Laplacian equations for data interpolation and semi-supervised learning
This work addresses computational bottlenecks in hypergraph learning for data interpolation and semi-supervised learning, offering an incremental improvement in efficiency and performance.
The paper tackled the challenge of fast numerical implementation for hypergraph p-Laplacian regularization by deriving a hypergraph p-Laplacian equation and proposing a simplified, well-posed alternative, which suppressed spiky solutions in data interpolation and improved classification accuracy in semi-supervised learning with remarkably low computational cost.
Hypergraph learning with $p$-Laplacian regularization has attracted a lot of attention due to its flexibility in modeling higher-order relationships in data. This paper focuses on its fast numerical implementation, which is challenging due to the non-differentiability of the objective function and the non-uniqueness of the minimizer. We derive a hypergraph $p$-Laplacian equation from the subdifferential of the $p$-Laplacian regularization. A simplified equation that is mathematically well-posed and computationally efficient is proposed as an alternative. Numerical experiments verify that the simplified $p$-Laplacian equation suppresses spiky solutions in data interpolation and improves classification accuracy in semi-supervised learning. The remarkably low computational cost enables further applications.