# Non-Negative Local Sparse Coding for Subspace Clustering

## TL;DR

This paper introduces a novel non-negative local sparse coding framework for subspace clustering that enhances data separability, incorporates kernel methods for nonlinear data, and employs a link-restore post-processing step to improve clustering accuracy.

## Contribution

It proposes a new local separability term in SSC, a kernel-based nonlinear extension, and a link-restore post-processing to address redundancies and improve clustering results.

## Key findings

- NLSSC outperforms state-of-the-art methods on benchmark datasets.
- Kernel extension captures nonlinear data structures effectively.
- Link-restore improves graph connectivity and clustering accuracy.

## Abstract

Subspace sparse coding (SSC) algorithms have proven to be beneficial to clustering problems. They provide an alternative data representation in which the underlying structure of the clusters can be better captured. However, most of the research in this area is mainly focused on enhancing the sparse coding part of the problem. In contrast, we introduce a novel objective term in our proposed SSC framework which focuses on the separability of data points in the coding space. We also provide mathematical insights into how this local-separability term improves the clustering result of the SSC framework. Our proposed non-linear local SSC algorithm (NLSSC) also benefits from the efficient choice of its sparsity terms and constraints. The NLSSC algorithm is also formulated in the kernel-based framework (NLKSSC) which can represent the nonlinear structure of data. In addition, we address the possibility of having redundancies in sparse coding results and its negative effect on graph-based clustering problems. We introduce the link-restore post-processing step to improve the representation graph of non-negative SSC algorithms such as ours. Empirical evaluations on well-known clustering benchmarks show that our proposed NLSSC framework results in better clusterings compared to the state-of-the-art baselines and demonstrate the effectiveness of the link-restore post-processing in improving the clustering accuracy via correcting the broken links of the representation graph.