# LogDet Rank Minimization with Application to Subspace Clustering

arXiv: https://arxiv.org/abs/1507.00908 · HTML: https://mlsanity.com/paper/1507.00908/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Jul 2015.

## Built on by (most cited first, 6 of 6)

- [ReduNet: A White-box Deep Network from the Principle of Maximizing Rate Reduction](https://mlsanity.com/paper/2105.10446/) (2021)
- [Neural Manifold Clustering and Embedding](https://mlsanity.com/paper/2201.10000/) (2022)
- [Eliminating Gradient Conflict in Reference-based Line-Art Colorization](https://mlsanity.com/paper/2207.06095/) (2022)
- [On Interpretable Approaches to Cluster, Classify and Represent Multi-Subspace Data via Minimum Lossy Coding Length based on Rate-Distortion Theory](https://mlsanity.com/paper/2302.10383/) (2023)
- [L2,1-Norm Regularized Quaternion Matrix Completion Using Sparse Representation and Quaternion QR Decomposition](https://mlsanity.com/paper/2309.03764/) (2023)
- [Depth-Aided Color Image Inpainting in Quaternion Domain](https://mlsanity.com/paper/2503.16818/) (2025)

## Abstract (arXiv)

Low-rank matrix is desired in many machine learning and computer vision problems. Most of the recent studies use the nuclear norm as a convex surrogate of the rank operator. However, all singular values are simply added together by the nuclear norm, and thus the rank may not be well approximated in practical problems. In this paper, we propose to use a log-determinant (LogDet) function as a smooth and closer, though non-convex, approximation to rank for obtaining a low-rank representation in subspace clustering. Augmented Lagrange multipliers strategy is applied to iteratively optimize the LogDet-based non-convex objective function on potentially large-scale data. By making use of the angular information of principal directions of the resultant low-rank representation, an affinity graph matrix is constructed for spectral clustering. Experimental results on motion segmentation and face clustering data demonstrate that the proposed method often outperforms state-of-the-art subspace clustering algorithms.