Approximate spectral clustering using both reference vectors and topology of the network generated by growing neural gas

Abstract

Spectral clustering (SC) is one of the most popular clustering methods and often outperforms traditional clustering methods. SC uses the eigenvectors of a Laplacian matrix calculated from a similarity matrix of a dataset. SC has serious drawbacks: the significant increases in the time complexity derived from the computation of eigenvectors and the memory space complexity to store the similarity matrix. To address the issues, I develop a new approximate spectral clustering using the network generated by growing neural gas (GNG), called ASC with GNG in this study. ASC with GNG uses not only reference vectors for vector quantization but also the topology of the network for extraction of the topological relationship between data points in a dataset. ASC with GNG calculates the similarity matrix from both the reference vectors and the topology of the network generated by GNG. Using the network generated from a dataset by GNG, ASC with GNG achieves to reduce the computational and space complexities and improve clustering quality. In this study, I demonstrate that ASC with GNG effectively reduces the computational time. Moreover, this study shows that ASC with GNG provides equal to or better clustering performance than SC.

An efficient and straightforward online vector quantization method for a data stream through remove-birth updating

“…Our future work will focus on investigating the effectiveness of the proposed method in a two-step approach that includes an approximate clustering method ( Vesanto & Alhoniemi, 2000 ; Fujita, 2021 ). In the first step of this approach, a dataset is transformed into sub-clusters.…”

“…In the second step, these sub-clusters are treated as individual objects and combined into larger clusters. This approach is known to be effective in handling a large dataset ( Vesanto & Alhoniemi, 2000 ; Fujita, 2021 ) and a data stream Mousavi et al (2020) . In the first step of this approach, the reduction of dead units is critical because dead units potentially become outliers.…”

“…The algorithm of online k-means is referred to in the Appendix section labeled ‘Online k-means’. The detailed descriptions of SOM, NG, and GNG are covered in Fujita (2021) . Notably, online k-means, SOM, and NG have parameters that decay over iterations, but in this study they were kept static in order to process data streams.…”

“…This makes the network remarkably flexible and reflective of the data structure. GNG has a wide range of applications, including topology learning, such as landmark extraction ( Fatemizadeh, Lucas & Soltanian-Zadeh, 2003 ), cluster analysis ( Canales & Chacón, 2007 ; Costa & Oliveira, 2007 ; Fujita, 2021 ), reconstruction of 3D models ( Holdstein & Fischer, 2008 ), object tracking ( Frezza-Buet, 2008 ), extraction of the two-dimensional outline of an image ( Angelopoulou, Psarrou & García-Rodríguez, 2011 ; Angelopoulou et al, 2018 ), and anomaly detection ( Sun, Liu & Harada, 2017 ).…”

Characteristics of Networks Generated by Kernel Growing Neural Gas

“…The network of the GNG is flexible, and its structure represents the data structure. GNG has been widely applied to topology learning, such as the extraction of the twodimensional outline of an image [15,16,17], the reconstruction of 3D models [18], landmark extraction [19], object tracking [20], anomaly detection [12], and cluster analysis [21,22,23].…”