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Scalable Deep Subspace Clustering Network

Nairouz Mrabah, Mohamed Bouguessa, Sihem Sami

TL;DR

This work tackles the cubic time bottleneck of deep subspace clustering by introducing SDSNet, a scalable framework that uses landmark-based low-rank factorization $\mathbf{C} \approx \mathbf{P}\mathbf{P}^\top$ and a convolutional auto-encoder to learn robust latent representations. Clustering is performed via a reduced eigenproblem derived from the anchor-based affinity, yielding overall complexity $O(n)$ with a fixed anchor count $m$. The method jointly optimizes reconstruction and subspace constraints through a block-coordinate scheme, leveraging Procrustes updates and closed-form anchor computations. Experiments on five real-world datasets show SDSNet achieves competitive accuracy while significantly reducing computation, demonstrating practical scalability for large-scale subspace clustering.

Abstract

Subspace clustering methods face inherent scalability limits due to the $O(n^3)$ cost (with $n$ denoting the number of data samples) of constructing full $n\times n$ affinities and performing spectral decomposition. While deep learning-based approaches improve feature extraction, they maintain this computational bottleneck through exhaustive pairwise similarity computations. We propose SDSNet (Scalable Deep Subspace Network), a deep subspace clustering framework that achieves $\mathcal{O}(n)$ complexity through (1) landmark-based approximation, avoiding full affinity matrices, (2) joint optimization of auto-encoder reconstruction with self-expression objectives, and (3) direct spectral clustering on factorized representations. The framework combines convolutional auto-encoders with subspace-preserving constraints. Experimental results demonstrate that SDSNet achieves comparable clustering quality to state-of-the-art methods with significantly improved computational efficiency.

Scalable Deep Subspace Clustering Network

TL;DR

This work tackles the cubic time bottleneck of deep subspace clustering by introducing SDSNet, a scalable framework that uses landmark-based low-rank factorization and a convolutional auto-encoder to learn robust latent representations. Clustering is performed via a reduced eigenproblem derived from the anchor-based affinity, yielding overall complexity with a fixed anchor count . The method jointly optimizes reconstruction and subspace constraints through a block-coordinate scheme, leveraging Procrustes updates and closed-form anchor computations. Experiments on five real-world datasets show SDSNet achieves competitive accuracy while significantly reducing computation, demonstrating practical scalability for large-scale subspace clustering.

Abstract

Subspace clustering methods face inherent scalability limits due to the cost (with denoting the number of data samples) of constructing full affinities and performing spectral decomposition. While deep learning-based approaches improve feature extraction, they maintain this computational bottleneck through exhaustive pairwise similarity computations. We propose SDSNet (Scalable Deep Subspace Network), a deep subspace clustering framework that achieves complexity through (1) landmark-based approximation, avoiding full affinity matrices, (2) joint optimization of auto-encoder reconstruction with self-expression objectives, and (3) direct spectral clustering on factorized representations. The framework combines convolutional auto-encoders with subspace-preserving constraints. Experimental results demonstrate that SDSNet achieves comparable clustering quality to state-of-the-art methods with significantly improved computational efficiency.
Paper Structure (14 sections, 3 theorems, 17 equations, 8 figures, 3 tables, 1 algorithm)

This paper contains 14 sections, 3 theorems, 17 equations, 8 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Proposed Approach
  4. Feature Extraction
  5. Self-Expressiveness Property
  6. Scalable Reformulation
  7. Optimization
  8. Clustering Step
  9. Experiments
  10. Experimental Setup
  11. Clustering Quality
  12. Scalability Analysis
  13. Convergence of the Affinity Matrix
  14. Conclusion

Key Result

Proposition 1

The matrix $\mathbf{C}^{\star}$ is symmetric, positive semi-definite, and its rank is equal to the rank of $\mathbf{X}$.

Figures (8)

  • Figure 1: The evolution of different loss functions outer iterations on Yaleb data base.
  • Figure 2: The evolution of ACC, NMI, and SPE outer iterations on Yaleb database.
  • Figure 3: Performance comparison of EnSC, SSC-OMP, LMVSC, SGL, S5C, and SDSNet on Synthetic data.
  • Figure 4: The relative changes of S in successive outer iterations. Ni: #data points in each subspace, # of subspace = 10.
  • Figure 5: Visualizing the graph formed by the affinity matrix on Yaleb with 10 classes.
  • ...and 3 more figures

Theorems & Definitions (4)

  • Proposition 1
  • Definition 1
  • Proposition 2
  • Proposition 3