Spectral Clustering Using Multilinear SVD: Analysis, Approximations and Applications

Ghoshdastidar, Debarghya; Dukkipati, Ambedkar

doi:10.1609/aaai.v29i1.9556

Cited by 16 publications

(14 citation statements)

References 25 publications

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“…Cutting one hyperedge will produce three singletons which we consider as three partitions. A similar definition of normalized associativity can be seen in literature [20,40].…”

Section: Partition Cost Is Given Bymentioning

confidence: 81%

Hypergraph partitioning using tensor eigenvalue decomposition

Maurya

Ravindran

2023

PLoS ONE

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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 tensor-based representation of hypergraphs, which enables us to capture actual super-dyadic interactions. We extend the notion of minimum ratio-cut and normalized-cut from graphs to hypergraphs and show that the relaxed optimization problem can be solved using eigenvalue decomposition of the Laplacian tensor. This novel formulation also enables us to remove a hyperedge completely by using the “hyperedge score” metric proposed by us, unlike the existing reduction approaches. We propose a hypergraph partitioning algorithm inspired from spectral graph theory and 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 synthetic hypergraphs generated by stochastic block model. We also show improvement for the min-cut solution on 2-uniform hypergraphs (graphs) over the standard spectral partitioning algorithm.

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“…Cutting one hyperedge will produce three singletons which we consider as three partitions. A similar definition of normalized associativity can be seen in literature [20,40].…”

Section: Partition Cost Is Given Bymentioning

confidence: 81%

Hypergraph partitioning using tensor eigenvalue decomposition

Maurya

Ravindran

2023

PLoS ONE

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show abstract

“…In this section, we compare HLloyd + HSC with two other classic tensor clustering algorithms: HOSVD + Kmeans: apply high‐order SVD (HOSVD) on

𝒴

(De Lathauwer et al, 2000a; Ghoshdastidar & Dukkipati, 2015), then perform

k

‐means on the outcome factors of HOSVD; CP + Kmeans: apply CANDECOMP/PARAFAC (CP) decomposition on

𝒴

(Carroll & Chang, 1970), then perform

k

‐means on the outcome factors of CP decomposition. …”

Section: Numerical Studiesmentioning

confidence: 99%

“…HOSVD + Kmeans: apply high‐order SVD (HOSVD) on

𝒴

(De Lathauwer et al, 2000a; Ghoshdastidar & Dukkipati, 2015), then perform

k

‐means on the outcome factors of HOSVD;…”

Section: Numerical Studiesmentioning

confidence: 99%

Exact Clustering in Tensor Block Model: Statistical Optimality and Computational Limit

Han

Luo

Wang

et al. 2022

Journal of the Royal Statistical Society Series B: Statistical Methodology

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High‐order clustering aims to identify heterogeneous substructures in multiway datasets that arise commonly in neuroimaging, genomics, social network studies, etc. The non‐convex and discontinuous nature of this problem pose significant challenges in both statistics and computation. In this paper, we propose a tensor block model and the computationally efficient methods, high‐order Lloyd algorithm (HLloyd), and high‐order spectral clustering (HSC), for high‐order clustering. The convergence guarantees and statistical optimality are established for the proposed procedure under a mild sub‐Gaussian noise assumption. Under the Gaussian tensor block model, we completely characterise the statistical‐computational trade‐off for achieving high‐order exact clustering based on three different signal‐to‐noise ratio regimes. The analysis relies on new techniques of high‐order spectral perturbation analysis and a ‘singular‐value‐gap‐free’ error bound in tensor estimation, which are substantially different from the matrix spectral analyses in the literature. Finally, we show the merits of the proposed procedures via extensive experiments on both synthetic and real datasets.

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“…SC relies on an affinity matrix to characterize pairwise similarities, which fall short in providing satisfactory clustering performance for such complex data [8], [9]. Converging evidence [10], [11] suggests that dealing with highdimensional and noisy data requires characterizing more complex similarities, and thereafter tensor spectral clustering (TSC) [12] has been developed recently as a promising solution. TSC employs high-order affinity tensors to characterize multi-wise similarities among samples instead of merely pairwise similarities as in previous methods.…”

Section: Introductionmentioning

confidence: 99%

Deep Tensor Spectral Clustering Network via Ensemble of Multiple Affinity Tensors

Cai,

Hu,

et al. 2024

IEEE Trans. Pattern Anal. Mach. Intell.

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Tensor spectral clustering (TSC) is an emerging approach that explores multi-wise similarities to boost learning. However, two key challenges have yet to be well addressed in the existing TSC methods: (1) The construction and storage of high-order affinity tensors to encode the multi-wise similarities are memory-intensive and hampers their applicability, and (2) they mostly employ a two-stage approach that integrates multiple affinity tensors of different orders to learn a consensus tensor spectral embedding, thus often leading to a suboptimal clustering result. To this end, this paper proposes a tensor spectral clustering network (TSC-Net) to achieve one-stage learning of a consensus tensor spectral embedding, while reducing the memory cost. TSC-Net employs a deep neural network that learns to map the input samples to the consensus tensor spectral embedding, guided by a TSC objective with multiple affinity tensors. It uses stochastic optimization to calculate a small part of the affinity tensors, thereby avoiding loading the whole affinity tensors for computation, thus significantly reducing the memory cost. Through using an ensemble of multiple affinity tensors, the TSC can dramatically improve clustering performance. Empirical studies on benchmark datasets demonstrate that TSC-Net outperforms the recent baseline methods.

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Spectral Clustering Using Multilinear SVD: Analysis, Approximations and Applications

Cited by 16 publications

References 25 publications

Hypergraph partitioning using tensor eigenvalue decomposition

Hypergraph partitioning using tensor eigenvalue decomposition

Exact Clustering in Tensor Block Model: Statistical Optimality and Computational Limit

Deep Tensor Spectral Clustering Network via Ensemble of Multiple Affinity Tensors

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