Spectral Lifting (Graph to Hypergraph)

[AS-L-C]

This PR implements a new Graph ➡️ Hypergraph lifting method inspired by the spectral clustering algorithm proposed by Ng, Jordan, and Weiss (2002) [1], which leverages the graph's spectral properties.

We provide a formal definition of the spectral lifting method below, following the notation introduced by von Luxburg in [2].

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Algorithm: Spectral Lifting (Graph ➡️ Hypergraph)

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Input: Graph (weighted) adjacency matrix $W \in \mathbb{R}^{n \times n}$, where $n$ is the number of nodes

• Compute the Laplacian matrix $L = D - W$, where D is the degree matrix.
• Compute the normalized Laplacian $L_{\text{sym}} = D^{-1/2}LD^{-1/2}$.
• Compute the first $k$ eigenvectors $u_1, \ldots, u_k$ of $L_{\text{sym}}$.
• Let $U \in \mathbb{R}^{n \times k}$ be the matrix containing the vectors $u_1, \ldots, u_k$ as columns.
• Form the matrix $T \in \mathbb{R}^{n \times k}$ from $U$ by normalizing the rows to norm 1, that is set $t_{ij} = \frac{u_{ij}}{(\sum_{c} u_{ic}^2)^{1/2}}$.
• For $i = 1, \ldots, n$, let $y_i \in \mathbb{R}^k$ be the vector corresponding to the $i$-th row of $T$.
• Cluster the points $(y_i)_{i=1, \ldots, n}$ into clusters $C_1, \ldots, C_k$ with a soft or hard clustering algorithm (e.g., k-means).
• For $i=1,\ldots,k$ define $H_i$ as the hyperedge connecting all the points belonging to cluster $C_i$
• Build the hypergraph incidence matrix $M$ of the hypergraph ${\it H}:=\{H_i\}_{{i=1, \ldots, k}}$.

Output: Hypergraph incidence matrix $M \in \mathbb{R}^{n \times k}$, where $k$ is the number of hyperedges

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Notes

Note₁: the number of hyperedges $k$ can be automatically determined by using a heuristic based on the eigengaps (i.e., spectral gaps), which estimates the number of connected components in the original graph, or provided by the user.

Note₂: hyperedges with overlapping hypernodes can be obtained by using soft clustering methods, while some degree of randomness can be introduced by using non-deterministc clutering methods.

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References

[1] Ng, Andrew, Michael Jordan, and Yair Weiss. "On spectral clustering: Analysis and an algorithm." Advances in neural information processing systems 14 (2001).
[2] Von Luxburg, Ulrike. "A tutorial on spectral clustering." Statistics and computing 17 (2007): 395-416.

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