Deepak Maurya

Deepak Maurya, Balaraman Ravindran

Link prediction in graphs is studied by modeling the dyadic interactions among two nodes. The relationships can be more complex than simple dyadic interactions and could require the user to model super-dyadic associations among nodes. Such interactions can be modeled using a hypergraph, which is a generalization of a graph where a hyperedge can connect more than two nodes. In this work, we consider the problem of hyperedge prediction in a $k-$uniform hypergraph. We utilize the tensor-based representation of hypergraphs and propose a novel interpretation of the tensor eigenvectors. This is further used to propose a hyperedge prediction algorithm. The proposed algorithm utilizes the \textit{Fiedler} eigenvector computed using tensor eigenvalue decomposition of hypergraph Laplacian. The \textit{Fiedler} eigenvector is used to evaluate the construction cost of new hyperedges, which is further utilized to determine the most probable hyperedges to be constructed. The functioning and efficacy of the proposed method are illustrated using some example hypergraphs and a few real datasets. The code for the proposed method is available on https://github.com/d-maurya/hypred_ tensorEVD

Deepak Maurya, Balaraman Ravindran

Hypergraphs have gained increasing attention in the machine learning community lately due to their superiority over graphs in capturing super-dyadic interactions among entities. In this work, we propose a novel approach for the partitioning of k-uniform hypergraphs. Most of the existing methods work by reducing the hypergraph to a graph followed by applying standard graph partitioning algorithms. The reduction step restricts the algorithms to capturing only some weighted pairwise interactions and hence loses essential information about the original hypergraph. We overcome this issue by utilizing the tensor-based representation of hypergraphs, which enables us to capture actual super-dyadic interactions. We prove that the hypergraph to graph reduction is a special case of tensor contraction. We extend the notion of minimum ratio-cut and normalized-cut from graphs to hypergraphs and show the relaxed optimization problem is equivalent to tensor eigenvalue decomposition. This novel formulation also enables us to capture different ways of cutting a hyperedge, unlike the existing reduction approaches. We propose a hypergraph partitioning algorithm inspired from spectral graph theory that can accommodate this notion of hyperedge cuts. We also derive a tighter upper bound on the minimum positive eigenvalue of even-order hypergraph Laplacian tensor in terms of its conductance, which is utilized in the partitioning algorithm to approximate the normalized cut. The efficacy of the proposed method is demonstrated numerically on simple hypergraphs. We also show improvement for the min-cut solution on 2-uniform hypergraphs (graphs) over the standard spectral partitioning algorithm.

Deepak Maurya, Sivadurgaprasad Chinta, Abhishek Sivaram et al.

Model identification is a crucial problem in chemical industries. In recent years, there has been increasing interest in learning data-driven models utilizing partial knowledge about the system of interest. Most techniques for model identification do not provide the freedom to incorporate any partial information such as the structure of the model. In this article, we propose model identification techniques that could leverage such partial information to produce better estimates. Specifically, we propose Structural Principal Component Analysis (SPCA) which improvises over existing methods like PCA by utilizing the essential structural information about the model. Most of the existing methods or closely related methods use sparsity constraints which could be computationally expensive. Our proposed method is a wise modification of PCA to utilize structural information. The efficacy of the proposed approach is demonstrated using synthetic and industrial case-studies.