# Interpretable nonconvex submodule clustering algorithm using ℓr-induced tensor nuclear norm and ℓ2,p column sparse norm with global convergence guarantees

## TL;DR

This paper introduces a new clustering method for 2D images that preserves data structure and improves performance.

## Contribution

The novel 2D-NLRSC method uses tensor nuclear norm and ℓ2,p regularization for nonconvex clustering with global convergence guarantees.

## Key findings

- 2D-NLRSC outperforms existing methods on real image datasets.
- The ℓr-induced tensor nuclear norm accurately approximates tensor rank.
- The framework preserves higher-order correlations in 2D image data.

## Abstract

Tensor-based subspace clustering algorithms have garnered significant attention for their high efficiency in clustering high-dimensional data. However, when dealing with 2D image data, traditional vectorization operations in most algorithms tend to undermine the correlations of higher-order tensor terms. To tackle this limitation, this paper proposes a non-convex submodule clustering approach (2D-NLRSC) that leverages sparse and low-rank representations for 2D image data. An ℓr-induced tensor nuclear norm is introduced to approximate the tensor rank precisely. Instead of vectorizing each 2D image, the framework arranges samples as lateral slices of a third-order tensor. It employs the t-product operation to generate an optimal representation tensor with low-rank constraint. The proposed method combines ℓq-norm induced clustering awareness with laplacian regularization to obtain a representation tensor with a diagonal structure. Additionally, 2D-NLRSC incorporates the ℓ2,p-norm as a regularization term, taking advantage of its excellent invariance, continuity, and differentiability. Experimental results on real image datasets validate the superior performance of the 2D-NLRSC model.