Power Spectral Clustering

Abstract

Spectral clustering is one of the most important image processing tools, especially for image segmentation. This specializes at taking local information such as edge weights and globalizing them. Due to its unsupervised nature, it is widely applicable. However, traditional spectral clustering is O(n 3/2). This poses a challenge, especially given the recent trend of large datasets. In this article, we propose an algorithm by using ideas from Γ −convergence, which is an amalgamation of Maximum Spanning Tree (MST) clustering and spectral clustering. This algorithm scales as O(nlog(n)) under certain conditions, while producing solutions which are similar to that of spectral clustering. Several toy examples are used to illustrate the similarities and differences. To validate the proposed algorithm, a recent state-of-the-art technique for segmentation-multiscale combinatorial grouping is used, where the normalized cut is replaced with the proposed algorithm and results are analyzed.

Enhanced Affinity for Spectral Clustering using Topological Node Features (TNFS)

“…The complexity is arrived at by calculating the number of unit calculations done in the algorithm. The complexity of SC-TNF2 is similar to the computational complexities of the Powered Gaussian proposed by Nataliani et al, 2017 [3] and LPCA proposed by Castro et al, 2017 [4]. Besides, the proposed method uses sparse matrices for Laplacian matrix calculation, making it time and space efficient.…”

“…The proposed techniques are compared with the following methods: spectral clustering algorithm (NJW) by Ng et al [19] (2002), Neighbour Propagation (NP) (2012) proposed by Li and Guo [16], Shared Nearest Neighbours (SNN) (2016) proposed by Ye and Sakurai. [31], Powered Gaussian (PG) (2017) by Nataliani et al [18], Spectral clustering using Local PCA (LPCA) [1] (2017), Powered Ratio Cut (PRCUT) [4] (2018). In the experiments conducted, three metrics were used for comparison: Adjusted Rand Index (ARI) [20], Normalized Mutual Information (NMI) [24] and Clustering Error (CE) [11].…”

“…Challa et al [4] proposed spectral clustering using Power Ratio cut or PRcut. They aimed to come up with an efficient implementable algorithm which is faster than traditional spectral clustering and also preserves the accuracy.…”

Revisiting the Isoperimetric Graph Partitioning Problem

“…The above theorem provides an initial step to explain the MST approximation in [31]. It has been shown earlier [12], [13], [19], [23], [24] that the limit of minimizers preserves the essential properties of solutions, thus giving useful results. Theorem 2 states that computing the limit of the minimizers of the relaxed seeded isoperimetric partitioning problem on UMST is same as on the original graph.…”

“…The maximum spanning tree is instrumental in doing so. Recall that an MST of a graph G = (V , E, w) is a connected subgraph of G spanning V , with no cycles such that weight of the MST = e ij ∈MST w ij (12) is maximized. The UMST is the weighted graph induced by the union of all the maximum spanning trees.…”