Fast Subspace Clustering Based on the Kronecker Product

TL;DR

This paper introduces a novel subspace clustering method leveraging the Kronecker product to significantly reduce computational complexity, enabling efficient clustering of large-scale high-dimensional data.

Contribution

The proposed model uses the Kronecker product to efficiently learn block diagonal representation matrices, improving scalability and efficiency over traditional spectral clustering methods.

Findings

Reduces computational complexity to O(kN^{3/k})

Significantly faster than state-of-the-art methods on public datasets

Demonstrates scalability on synthetic large-scale data

Abstract

Subspace clustering is a useful technique for many computer vision applications in which the intrinsic dimension of high-dimensional data is often smaller than the ambient dimension. Spectral clustering, as one of the main approaches to subspace clustering, often takes on a sparse representation or a low-rank representation to learn a block diagonal self-representation matrix for subspace generation. However, existing methods require solving a large scale convex optimization problem with a large set of data, with computational complexity reaches O(N^3) for N data points. Therefore, the efficiency and scalability of traditional spectral clustering methods can not be guaranteed for large scale datasets. In this paper, we propose a subspace clustering model based on the Kronecker product. Due to the property that the Kronecker product of a block diagonal matrix with any other matrix is still a block diagonal matrix, we can efficiently learn the representation matrix which is formed by the Kronecker product of k smaller matrices. By doing so, our model significantly reduces the computational complexity to O(kN^{3/k}). Furthermore, our model is general in nature, and can be adapted to different regularization based subspace clustering methods. Experimental results on two public datasets show that our model significantly improves the efficiency compared with several state-of-the-art methods. Moreover, we have conducted experiments on synthetic data to verify the scalability of our model for large scale datasets.