# Subspace Clustering by Block Diagonal Representation

arXiv: https://arxiv.org/abs/1805.09243 · HTML: https://mlsanity.com/paper/1805.09243/

## Key facts

- Cited by 38 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 May 2018.

## Built on by (most cited first, 30 of 38)

- [Effective and Efficient Graph Learning for Multi-view Clustering](https://mlsanity.com/paper/2108.06734/) (2021)
- [Learning a Self-Expressive Network for Subspace Clustering](https://mlsanity.com/paper/2110.04318/) (2021)
- [Towards Task Sampler Learning for Meta-Learning](https://mlsanity.com/paper/2307.08924/) (2023)
- [A Simple Approach to Automated Spectral Clustering](https://mlsanity.com/paper/2107.12183/) (2021)
- [Consistency and Diversity induced Human Motion Segmentation](https://mlsanity.com/paper/2202.04861/) (2022)
- [Semi-Supervised Subspace Clustering via Tensor Low-Rank Representation](https://mlsanity.com/paper/2205.10481/) (2022)
- [Exploring a Principled Framework for Deep Subspace Clustering](https://mlsanity.com/paper/2503.17288/) (2025)
- [Enhanced Latent Multi-view Subspace Clustering](https://mlsanity.com/paper/2312.14763/) (2023)
- [Nonlinear subspace clustering by functional link neural networks](https://mlsanity.com/paper/2402.02051/) (2024)
- [Graph Constrained Data Representation Learning for Human Motion Segmentation](https://mlsanity.com/paper/2107.13362/) (2021)
- [Adaptive Graph Convolutional Subspace Clustering](https://mlsanity.com/paper/2305.03414/) (2023)
- [Smoothed Multi-View Subspace Clustering](https://mlsanity.com/paper/2106.09875/) (2021)
- [Multi-Tensor Network Representation for High-Order Tensor Completion](https://mlsanity.com/paper/2109.04022/) (2021)
- [Learning idempotent representation for subspace clustering](https://mlsanity.com/paper/2207.14431/) (2022)
- [A Restarted Large-Scale Spectral Clustering with Self-Guiding and Block Diagonal Representation](https://mlsanity.com/paper/2306.15138/) (2023)
- [Block-Diagonal Guided DBSCAN Clustering](https://mlsanity.com/paper/2404.01341/) (2024)
- [Scale-Free Graph-Language Models](https://mlsanity.com/paper/2502.15189/) (2025)
- [Auto-weighted low-rank representation for clustering](https://mlsanity.com/paper/2104.12308/) (2021)
- [PMSSC: Parallelizable multi-subset based self-expressive model for subspace clustering](https://mlsanity.com/paper/2111.12232/) (2021)
- [Attentive Multi-View Deep Subspace Clustering Net](https://mlsanity.com/paper/2112.12506/) (2021)
- [Greedier is Better: Selecting Multiple Neighbors per Iteration for Sparse Subspace Clustering](https://mlsanity.com/paper/2204.02572/) (2022)
- [Semi-Supervised Clustering via Dynamic Graph Structure Learning](https://mlsanity.com/paper/2209.02513/) (2022)
- [Fast and Robust Sparsity-Aware Block Diagonal Representation](https://mlsanity.com/paper/2312.01137/) (2023)
- [Multilayer Graph Approach to Deep Subspace Clustering](https://mlsanity.com/paper/2401.17033/) (2024)
- [Subspace Clustering of Subspaces: Unifying Canonical Correlation Analysis and Subspace Clustering](https://mlsanity.com/paper/2509.18653/) (2025)
- [Graph-based Clustering Revisited: A Relaxation of Kernel k-Means Perspective](https://mlsanity.com/paper/2509.18826/) (2025)
- [Beyond Linear Subspace Clustering: A Comparative Study of Nonlinear Manifold Clustering Algorithms](https://mlsanity.com/paper/2103.10656/) (2021)
- [Learning Deep Representation with Energy-Based Self-Expressiveness for Subspace Clustering](https://mlsanity.com/paper/2110.15037/) (2021)
- [Fusion Subspace Clustering for Incomplete Data](https://mlsanity.com/paper/2205.10872/) (2022)
- [Revisiting data augmentation for subspace clustering](https://mlsanity.com/paper/2207.09728/) (2022)

## Abstract (arXiv)

This paper studies the subspace clustering problem. Given some data points approximately drawn from a union of subspaces, the goal is to group these data points into their underlying subspaces. Many subspace clustering methods have been proposed and among which sparse subspace clustering and low-rank representation are two representative ones. Despite the different motivations, we observe that many existing methods own the common block diagonal property, which possibly leads to correct clustering, yet with their proofs given case by case. In this work, we consider a general formulation and provide a unified theoretical guarantee of the block diagonal property. The block diagonal property of many existing methods falls into our special case. Second, we observe that many existing methods approximate the block diagonal representation matrix by using different structure priors, e.g., sparsity and low-rankness, which are indirect. We propose the first block diagonal matrix induced regularizer for directly pursuing the block diagonal matrix. With this regularizer, we solve the subspace clustering problem by Block Diagonal Representation (BDR), which uses the block diagonal structure prior. The BDR model is nonconvex and we propose an alternating minimization solver and prove its convergence. Experiments on real datasets demonstrate the effectiveness of BDR.