Deep Closed-Form Subspace Clustering

Jamyoung Koo; Junghoon Seo; Taegyun Jeon

REVIEW 4 major objections 5 minor 41 references

Deep Closed-Form Subspace Clustering

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A closed-form matrix can replace the trainable self-expressive layer in deep subspace clustering.

desk verdict A simple idea with a serious citation gap and an uncontrolled large-N experiment; the closed-form layer is worth knowing about, but the paper overclaims. read the letter →

arxiv 1908.09419 v1 pith:2SMX5ZTV submitted 2019-08-26 cs.CV cs.LGstat.ML

Junghoon Seo , Jamyoung Koo , Taegyun Jeon This is my paper · ORCID

classification cs.CVcs.LGstat.ML

keywords subspaceclusteringself-expressivenessclosed-formsolutiondeepautoencodermemoryefficiencyspectralunsupervisedlearninglarge-scale

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes a deep subspace clustering model, DCFSC, whose self-expressive layer has zero learnable parameters: each training step computes the self-representation matrix directly from the current latent features in closed form instead of optimizing an $N\times N$ parameter matrix. The authors aim to remove the memory bottleneck that keeps deep subspace clustering stuck with shallow autoencoders on large datasets. They report that on COIL-100 this lets them train a five-layer encoder-decoder from scratch and reach a 27.3% clustering error, below the earlier deep subspace clustering baseline, while using only 0.16% of its parameters. A sympathetic reader cares because the result suggests that expensive trainable self-expression may be replaceable by a data-derived closed-form layer, opening larger datasets to deep subspace clustering.

What carries the argument

The load-bearing object is the closed-form self-expression matrix $B$ derived from the solution to $\min_{\Theta}\,\|X'-\Theta X'\|_F^2 + \lambda\|\Theta\|_F^2$ subject to $\mathrm{diag}(\Theta)=0$. The formula $B_{ij}=-P_{ij}/P_{ii}$ with $P=(X'X'^T+\lambda I)^{-1}$ turns the former trainable layer into a differentiable but stop-gradient function of the latent features, so the autoencoder can be made deeper without storing or optimizing an $N\times N$ matrix. This is what carries the argument: it converts memory cost into a per-iteration matrix inversion and removes the joint optimization of the DSC objective.

What would settle it

Track the self-expression residual $\|Y - BY\|_F$ during training on a dataset with strong distribution shift between encoder updates. If the residual grows while reconstruction loss falls, the closed-form layer is being computed against stale features and the method's premise fails. More directly, compare final clustering error against a version that backpropagates through the closed-form $B$ or jointly optimizes it; if backpropagation changes results materially, the stop-gradient simplification is not neutral.

Watch

Extended reading notes

Core claim

On its own terms, the paper claims that the self-expressive layer of deep subspace clustering can be replaced by the closed-form minimizer of the self-expression objective. Given latent features $X'$, the matrix $B$ is $I - P\,\mathrm{diagMat}(\mathbf{1}\oslash \mathrm{diag}(P))$ with $P=(X'X'^T+\lambda I)^{-1}$, i.e. off-diagonal entries $B_{ij} = -P_{ij}/P_{ii}$ and zero diagonal. During training the autoencoder is updated only by the reconstruction loss $\|X - \mathrm{Dec}(BX')\|_F^2$ with gradients stopped at $B$; the affinity matrix for spectral clustering is built from the final $B$. The paper's experimental claim is that this zero-parameter self-expressive layer yields face-clustering errors on ORL close to DSC and somewhat higher errors on Extended Yale B, yet is the only deep method tested to reach 27.3% error on COIL-100 with a deeper architecture, beating all compared baselines including DSC.

Load-bearing premise

The method's load-bearing premise is that alternately recomputing $B$ as the exact closed-form solution for the current latent features and then updating the autoencoder only by reconstruction loss keeps $B$ a valid self-expression as the encoder changes; if the encoder drifts too fast, the final $B$ can lose the block structure needed for good clustering.

Editorial extensions

If this is right

On the large-scale COIL-100 benchmark, the method reaches 27.3% clustering error with a five-layer autoencoder trained from scratch, beating the compared methods including DSC, while using only 0.16% of DSC's parameters.
On the small face benchmarks under DSC's own settings, DCFSC uses 0.25% and 0.44% of DSC's parameters and ends at 6.13% versus 2.67% mean error on Extended Yale B and 14.8% versus 14.0% on ORL, still beating all non-deep baselines.
Because the self-expressive term has no parameters and no gradient, the model can train a substantially deeper autoencoder at modest memory cost: the COIL-100 model used 10.8 GB of GPU memory versus 8.6 GB for the shallow DSC model.
The absence of a trainable self-expressive layer makes DCFSC a drop-in replacement for the self-expressive component in other deep subspace clustering variants, so it can likely be combined with adversarial, self-supervised, or distribution-preserving extensions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the stop-gradient alternating update converges, it behaves like a coordinate-descent analogue: the encoder is trained only to make latent features reconstruct well, while the self-expression matrix is re-fit exactly at each step. A testable prediction is that the quality of the final clustering tracks the stability of the latent features across iterations, not just the reconstruction loss.
The closed-form $B$ requires a dense $N\times N$ matrix inversion and storage, so the model's memory savings are about optimizer state and trainable parameters, not about the affinity matrix. The next bottleneck for scaling beyond $N\approx 10^4$ is likely the cost of forming and inverting the Gram matrix and of spectral clustering, not the autoencoder.
A direct way to separate the contribution of the closed-form layer from that of the deeper architecture would be to train DSC or a parameter-reduced variant on the same five-layer architecture; if the accuracy gap persists, the closed-form layer itself, rather than depth, is what helps on COIL-100.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know about this one because it is a simple, honest attempt to remove the big N×N trainable matrix from deep subspace clustering. The idea: compute a closed-form self-expression matrix from the current latent features at each iteration, stop gradients, and train the autoencoder only on reconstruction. The closed-form solution itself is not new—it is the EDSC/EASER ridge regression solution—but putting it as a stop-gradient layer inside a deep network is a genuine variant. The paper derives it cleanly and the TensorFlow snippet is enough to reimplement it in an afternoon.

What it does well: the math is mostly right (Equation 6 is the correct form; Equation 5 has a row/column scaling typo), the memory argument is plausible in spirit, and the small-N experiments at least show the method converges and beats the non-deep baselines, just not DSC. The idea of using the memory savings to train a deeper encoder on COIL-100 is attractive.

Where it is soft. First and most importantly, the closed-form solution is already the core of EDSC [11], which the paper cites but never credits for the formula. The "very first attempt" claim is wrong and should be fixed. Second, the COIL-100 comparison is not controlled. With N=7200 and the five-layer encoder producing about 2,592-dim features, N > d, so the equation BZ=Z becomes underdetermined per row, and the closed-form B is just the minimal-norm solution among many—it does not have to carry any block-diagonal subspace structure. Comparing the deep DCFSC against the shallow DSC does not isolate whether the closed-form layer or the depth is responsible for the error drop. A same-architecture ablation would be needed. Third, no error bars are reported, even though means of ten trials are given, and the small-N numbers are consistently worse than DSC. The abstract's "new reference model" claim is not supported. Finally, inverting an N×N matrix every iteration is computationally heavy; the paper does not discuss runtime or the memory of P and B beyond the parameter count.

All that said, the paper is not a waste of time. The stop-gradient closed-form layer is an interesting baseline that people in the subfield should know about. It deserves a serious referee, but I would ask for an EDSC attribution, a controlled large-N experiment, and error bars before publication. For my own work, I would be more likely to cite EDSC or EASER than this paper, but I would read a revised version.

Referee Report

4 major / 5 minor

Summary. The paper proposes Deep Closed-Form Subspace Clustering (DCFSC), a variant of deep subspace clustering (DSC) in which the trainable N×N self-expressive layer is replaced by a matrix B computed in closed form from the current latent features by solving Eq. (4) with a Frobenius-norm regularizer and a zero-diagonal constraint. B has no trainable parameters, and training alternates between recomputing B from the current latent features (with the gradient stopped) and updating the autoencoder on the reconstruction loss only. The method is evaluated on Extended Yale B, ORL, and COIL-100; the paper reports clustering error rates and claims a state-of-the-art result on COIL-100 while using 0.16% of the parameters of DSC.

Significance. The memory-efficiency motivation is significant and timely: eliminating the N×N trainable self-expressive matrix removes the dominant parameter count in deep subspace clustering and permits deeper autoencoders. The closed-form derivation and the TensorFlow implementation (Listing 1) are clear and reproducible. However, the experimental evidence is confounded by architecture and training differences, and the paper provides no convergence analysis of the alternating scheme. If the mechanism were properly isolated, the method could be a useful addition, but in its current form the performance claims are not supported by the evidence presented.

major comments (4)

[Section 3.3, Algorithm 2] The alternating scheme lacks convergence guarantees. At each iteration, B is the exact minimizer of Eq. (4) for the current latent Z, but the gradient through B is stopped and the autoencoder is updated only on the reconstruction loss (Eq. 1). This is neither coordinate descent on a single objective nor a stationary-point method for any joint loss involving self-expressiveness; a fixed point of the alternation does not necessarily satisfy any optimality condition related to subspace clustering. The statement in Section 4.1 that 'convergence is experimentally guaranteed' is not a substitute for such an analysis and does not characterize what the method converges to.
[Section 4.2, Table 4] The COIL-100 state-of-the-art claim is not supported by the experimental design. DCFSC uses a five-layer encoder/decoder trained from scratch for 175 epochs with λ=10, while DSC uses a one-layer architecture with pre-trained weights; the comparison changes architecture, pre-training, and hyperparameters simultaneously. The 27.3% error rate therefore cannot be attributed to the closed-form self-expressive layer. This is especially concerning because the small-N experiments in Section 4.1, which hold the architecture fixed, show that DCFSC is consistently worse than DSC (mean 6.13% vs 2.67% on Extended Yale B).
[Section 3.2, Eqs. (5)-(6)] The self-expressiveness of B on the training set is tautological: B is constructed so that Z≈BZ for the final latent features, regardless of whether those features lie on a union of subspaces. In the N≫d regime of COIL-100, the linear system Z=BZ with diag(B)=0 is highly underdetermined, and the particular Frobenius-regularized solution is not necessarily block-diagonal. Since the autoencoder update never explicitly rewards union-of-subspaces structure, the paper does not establish that the closed-form layer is the operative mechanism behind the reported clustering accuracy; the result could be driven by the deep features alone.
[Section 4.2] The paper lacks a control experiment that isolates the effect of the closed-form self-expressive layer. For example, replacing B in Algorithm 2 with the identity matrix, or with a fixed random zero-diagonal matrix, while keeping the same deep architecture, would show whether the self-expression computation contributes to the clustering performance. Without such an ablation, it is unclear whether DCFSC is a subspace clustering method or simply an autoencoder followed by spectral clustering on a data-dependent kernel.

minor comments (5)

[Figures 1 and 2] The captions say 'mean of ten trails' (typo for 'trials') and do not report any variance; please provide standard deviations or confidence intervals, since the differences discussed (e.g., ORL 14.8% vs 14.0%, COIL-100 27.3% vs 26.6%) may be within trial-to-trial noise.
[Section 4.1] The statement that DCFSC's mean error 'was significantly worse' than DSC's is not supported by any statistical test; please either provide such tests or soften the wording.
[Abstract and Section 4.2] The claim of a 'state-of-the-art result' is inconsistent with the body's acknowledgment that the S3C method in [37] reports 26.6% on COIL-100, which is better than DCFSC's 27.3%; the wording should be calibrated accordingly.
[Section 3.2] Equation (5) is presented without derivation; the text refers to Section 3.1 of [30] but does not reproduce the key steps. Reproducing the Lagrange-multiplier derivation would make the paper more self-contained and help readers understand the role of the diag(P) normalization.
[Section 3.3] It should be clarified that B is still an N×N matrix that must be materialized to compute the forward pass and the final affinity matrix; the memory benefit is in trainable parameters and optimizer state, not in removing the O(N^2) object entirely.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: B is by construction the closed-form minimizer of Eq. 4, but the reported clustering errors are external evaluations and neither B nor lambda is fitted to cluster labels.

full rationale

The derivation chain is self-contained. Equation 5-6 define B as the exact minimizer of Eq. 4 for the current latent features, and Algorithm 2 computes B in a forward pass, stops gradients, and updates the autoencoder only on the reconstruction loss of Eq. 1. The paper never presents the self-expressiveness of B as an empirical prediction; it is imposed and solved in closed form. The final clustering accuracy is measured against ground-truth labels after spectral clustering of B, and no label information is used to fit B or to select lambda. The only lambda exploration (Section 4.3) is an explicit sensitivity study, not a fitted parameter renamed as a prediction. The closed-form solution is imported from the external EASER work [30], with stated assumptions, and does not presuppose the COIL-100 result; there is no load-bearing self-citation chain. The lack of a convergence proof for the stop-gradient alternation is a correctness and robustness concern, not circularity. Therefore none of the seven circularity patterns applies.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The method rests on the self-expressiveness assumption, the correctness of the EASER/EDSC closed-form solution, and an unproven alternating optimization with stopped gradients. The regularization parameter lambda is a per-dataset free choice. No new entities are introduced.

free parameters (1)

lambda (self-expression regularization) = 5e5 for Extended Yale B and ORL; 10 for COIL-100
Controls the Frobenius norm penalty in Equation 4. The value is chosen per dataset, and Figure 2 shows performance is sensitive to it, with large values degrading COIL-100 accuracy.

assumptions (3)

domain assumption The latent representation satisfies the self-expressiveness property: each point is a linear combination of other points in the same subspace.
This is the standard subspace clustering assumption (Equation 2 and Section 1) and is enforced indirectly through the closed-form B.
standard math Equation 5 is the exact minimizer of Equation 4 with the diagonal constraint.
The derivation is taken from Section 3.1 of EASER [30]; we verified the Lagrange multiplier algebra and it is correct.
ad hoc to paper Updating the autoencoder with the gradient stopped at B (Algorithm 2, line 7) converges to a representation useful for clustering.
The paper provides no convergence analysis for this alternating scheme; it is an empirical assumption supported only by the reported experiments.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Deep Closed-Form Subspace Clustering." pith.science (2026). https://pith.science/paper/2SMX5ZTV

@misc{pith2026190809419,
  author       = {Pith},
  title        = {Pith review of: Deep Closed-Form Subspace Clustering},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2SMX5ZTV}},
  note         = {Machine review of arXiv:1908.09419}
}

read the original abstract

We propose Deep Closed-Form Subspace Clustering (DCFSC), a new embarrassingly simple model for subspace clustering with learning non-linear mapping. Compared with the previous deep subspace clustering (DSC) techniques, our DCFSC does not have any parameters at all for the self-expressive layer. Instead, DCFSC utilizes the implicit data-driven self-expressive layer derived from closed-form shallow auto-encoder. Moreover, DCFSC also has no complicated optimization scheme, unlike the other subspace clustering methods. With its extreme simplicity, DCFSC has significant memory-related benefits over the existing DSC method, especially on the large dataset. Several experiments showed that our DCFSC model had enough potential to be a new reference model for subspace clustering on large-scale high-dimensional dataset.

Figures

Figures reproduced from arXiv: 1908.09419 by the authors.

**Figure 2.** shows variation of subspace clustering error in COIL-100 with different selection of λ. It can be seen that choosing λ from 1 to 100 guaranteed better performance than DSC, and selection from 1 to 10 gave the best performance. Conversely, too large λ (≥ 1e 3 ) degenerated performance. On the other hand, with fine-tuning the pretrained model in small N cases (Section 4.1), a relatively large λ (5e 5 ) was the appro… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages

[11]

Efﬁcient dense subspace clustering

Pan Ji, Mathieu Salzmann, and Hongdong Li. Efﬁcient dense subspace clustering. In IEEE Winter Conference on Applica- tions of Computer Vision , pages 461–468. IEEE, 2014. 2, 4

work page 2014
[1]

Tensorﬂow: A system for large-scale machine learning

Mart ´ın Abadi, Paul Barham, Jianmin Chen, Zhifeng Chen, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghe- mawat, Geoffrey Irving, Michael Isard, et al. Tensorﬂow: A system for large-scale machine learning. In USENIX Sym- posium on Operating Systems Design and Implementation , pages 265–283, 2016. 3

work page 2016
[2]

K-subspaces with missing data

Laura Balzano, Arthur Szlam, Benjamin Recht, and Robert Nowak. K-subspaces with missing data. In IEEE Statistical Signal Processing Workshop, pages 612–615. IEEE, 2012. 2

work page 2012
[3]

Graph regularized nonnegative matrix factorization for data representation

Deng Cai, Xiaofei He, Jiawei Han, and Thomas S Huang. Graph regularized nonnegative matrix factorization for data representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 33(8):1548–1560, 2010. 7

work page 2010
[4]

Subspace cluster- ing using a low-rank constrained autoencoder

Yuanyuan Chen, Lei Zhang, and Zhang Yi. Subspace cluster- ing using a low-rank constrained autoencoder. Information Sciences, 424:27–38, 2018. 1

work page 2018
[5]

Lorslim: Low rank sparse linear methods for top-n recommendations

Yao Cheng, Liang Yin, and Yong Yu. Lorslim: Low rank sparse linear methods for top-n recommendations. In IEEE International Conference on Data Mining , pages 90–99. IEEE, 2014. 3

work page 2014
[6]

Sparse subspace cluster- ing

Ehsan Elhamifar and Ren ´e Vidal. Sparse subspace cluster- ing. In IEEE Conference on Computer Vision and Pattern Recognition, pages 2790–2797. IEEE, 2009. 1, 2

work page 2009
[7]

Sparse subspace cluster- ing: Algorithm, theory, and applications

Ehsan Elhamifar and Rene Vidal. Sparse subspace cluster- ing: Algorithm, theory, and applications. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35(11):2765– 2781, 2013. 1, 2, 3, 4

work page 2013

Show all 41 references

[8]

Clustering appearances of objects under varying illumination conditions

Jeffrey Ho, Ming-Hsuan Yang, Jongwoo Lim, Kuang-Chih Lee, and David Kriegman. Clustering appearances of objects under varying illumination conditions. In IEEE Conference on Computer Vision and Pattern Recognition, pages 11–18,
[9]

Batch normalization: Accelerating deep network training by reducing internal co- variate shift

Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal co- variate shift. In International Conference on Machine Learn- ing, 2015. 7

2015
[10]

Robust multi-body feature tracker: a segmentation-free ap- proach

Pan Ji, Hongdong Li, Mathieu Salzmann, and Yiran Zhong. Robust multi-body feature tracker: a segmentation-free ap- proach. In IEEE Conference on Computer Vision and Pattern Recognition, pages 3843–3851, 2016. 1

2016
[12]

Deep subspace clustering networks

Pan Ji, Tong Zhang, Hongdong Li, Mathieu Salzmann, and Ian Reid. Deep subspace clustering networks. InAdvances in Neural Information Processing Systems, pages 24–33, 2017. 1, 2, 3, 4, 7

2017
[13]

Cluster- ing high-dimensional data: A survey on subspace clustering, pattern-based clustering, and correlation clustering

Hans-Peter Kriegel, Peer Kr ¨oger, and Arthur Zimek. Cluster- ing high-dimensional data: A survey on subspace clustering, pattern-based clustering, and correlation clustering. ACM Transactions on Knowledge Discovery from Data) , 3(1):1,
[14]

Ac- quiring linear subspaces for face recognition under variable lighting

Kuang-Chih Lee, Jeffrey Ho, and David J Kriegman. Ac- quiring linear subspaces for face recognition under variable lighting. IEEE Transactions on Pattern Analysis and Ma- chine Intelligence, pages 684–698, 2005. 4

2005
[15]

Projective low-rank subspace clus- tering via learning deep encoder

Jun Li and Hongfu Liu. Projective low-rank subspace clus- tering via learning deep encoder. In International Joint Con- ference on Artiﬁcial Intelligence, 2017. 2

2017
[16]

Robust recovery of subspace structures by low-rank representation

Guangcan Liu, Zhouchen Lin, Shuicheng Yan, Ju Sun, Yong Yu, and Yi Ma. Robust recovery of subspace structures by low-rank representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35(1):171–184, 2012. 1, 2, 4

2012
[17]

Columbia object image library (coil-100), 1996

Sameer A Nene, Shree K Nayar, Hiroshi Murase, et al. Columbia object image library (coil-100), 1996. 4, 7

1996
[18]

On spectral clustering: Analysis and an algorithm

Andrew Y Ng, Michael I Jordan, and Yair Weiss. On spectral clustering: Analysis and an algorithm. InAdvances in Neural Information Processing Systems, pages 849–856, 2002. 1, 3

2002
[19]

Slim: Sparse linear meth- ods for top-n recommender systems

Xia Ning and George Karypis. Slim: Sparse linear meth- ods for top-n recommender systems. In IEEE International Conference on Data Mining, pages 497–506. IEEE, 2011. 3

2011
[20]

Sparse linear methods with side information for top-n recommendations

Xia Ning and George Karypis. Sparse linear methods with side information for top-n recommendations. In ACM Con- ference on Recommender systems , pages 155–162. ACM,
[21]

Sub- space clustering for high dimensional data: a review

Lance Parsons, Ehtesham Haque, and Huan Liu. Sub- space clustering for high dimensional data: a review. ACM SIGKDD Explorations Newsletter, 6(1):90–105, 2004. 1

2004
[22]

Latent space sparse and low-rank subspace clustering

Vishal M Patel, Hien Van Nguyen, and Ren ´e Vidal. Latent space sparse and low-rank subspace clustering. IEEE Jour- nal of Selected Topics in Signal Processing , 9(4):691–701,
[23]

Kernel sparse subspace clus- tering

Vishal M Patel and Ren ´e Vidal. Kernel sparse subspace clus- tering. In IEEE International Conference on Image Process- ing, pages 2849–2853. IEEE, 2014. 4

2014
[24]

Cascade subspace clustering

Xi Peng, Jiashi Feng, Jiwen Lu, Wei-Yun Yau, and Zhang Yi. Cascade subspace clustering. In AAAI Conference on Artiﬁcial Intelligence, 2017. 2

2017
[25]

Structured autoencoders for sub- space clustering

Xi Peng, Jiashi Feng, Shijie Xiao, Wei-Yun Yau, Joey Tianyi Zhou, and Songfan Yang. Structured autoencoders for sub- space clustering. IEEE Transactions on Image Processing , 27(10):5076–5086, 2018. 1

2018
[26]

Deep subspace clustering with sparsity prior

Xi Peng, Shijie Xiao, Jiashi Feng, Wei-Yun Yau, and Zhang Yi. Deep subspace clustering with sparsity prior. In Inter- national Joint Conference on Artiﬁcial Intelligence, 2016. 1, 4

2016
[27]

Mo- tion segmentation via robust subspace separation in the pres- ence of outlying, incomplete, or corrupted trajectories

Shankar R Rao, Roberto Tron, Ren ´e Vidal, and Yi Ma. Mo- tion segmentation via robust subspace separation in the pres- ence of outlying, incomplete, or corrupted trajectories. In IEEE Conference on Computer Vision and Pattern Recogni- tion, pages 1–8. IEEE, 2008. 1, 2

2008
[28]

Parameterisation of a stochastic model for human face identiﬁcation

Ferdinando S Samaria and Andy C Harter. Parameterisation of a stochastic model for human face identiﬁcation. In IEEE Workshop on Applications of Computer Vision , pages 138–
[29]

Normalized cuts and image segmentation

Jianbo Shi and Jitendra Malik. Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2000. 1

2000
[30]

Embarrassingly shallow autoencoders for sparse data

Harald Steck. Embarrassingly shallow autoencoders for sparse data. In ACM International World Wide Web Con- ference, pages 3251–3257. ACM, 2019. 1, 3

2019
[31]

Subspace clustering

Ren ´e Vidal. Subspace clustering. IEEE Signal Processing Magazine, 28(2):52–68, 2011. 1

2011
[32]

Low rank subspace clustering (lrsc)

Ren ´e Vidal and Paolo Favaro. Low rank subspace clustering (lrsc). Pattern Recognition Letters, 43:47–61, 2014. 4

2014
[33]

Provable sub- space clustering: When lrr meets ssc

Yu-Xiang Wang, Huan Xu, and Chenlei Leng. Provable sub- space clustering: When lrr meets ssc. In Advances in Neural Information Processing Systems, pages 64–72, 2013. 1

2013
[34]

Unsupervised segmentation of natural images via lossy data compression

Allen Y Yang, John Wright, Yi Ma, and S Shankar Sastry. Unsupervised segmentation of natural images via lossy data compression. Computer Vision and Image Understanding , 110(2):212–225, 2008. 1

2008
[35]

Scalable sparse subspace clustering by orthogonal matching pursuit

Chong You, Daniel Robinson, and Ren ´e Vidal. Scalable sparse subspace clustering by orthogonal matching pursuit. In IEEE Conference on Computer Vision and Pattern Recog- nition, pages 3918–3927, 2016. 1, 4

2016
[36]

Low-rank tensor constrained multiview sub- space clustering

Changqing Zhang, Huazhu Fu, Si Liu, Guangcan Liu, and Xiaochun Cao. Low-rank tensor constrained multiview sub- space clustering. In IEEE International Conference on Com- puter Vision, pages 1582–1590, 2015. 1

2015
[37]

Self- supervised convolutional subspace clustering network

Junjian Zhang, Chun-Guang Li, Chong You, Xianbiao Qi, Honggang Zhang, Jun Guo, and Zhouchen Lin. Self- supervised convolutional subspace clustering network. In IEEE Conference on Computer Vision and Pattern Recog- nition, 2019. 2, 8

2019
[38]

Scalable deep k-subspace clustering

Tong Zhang, Pan Ji, Mehrtash Harandi, Richard Hartley, and Ian Reid. Scalable deep k-subspace clustering. InAsian Con- ference on Computer Vision, pages 466–481. Springer, 2018. 2

2018
[39]

Neural collaborative subspace clustering

Tong Zhang, Pan Ji, Mehrtash Harandi, Wen-bing Huang, and Hongdong Li. Neural collaborative subspace clustering. In International Conference on Machine Learning, 2019. 2, 8

2019
[40]

Latent distribution preserving deep subspace clustering

Lei Zhou, Xiao Bai, Dong Wang, Xianglong Liu, Jun Zhou, and Edwin Hancock. Latent distribution preserving deep subspace clustering. In International Joint Conference on Artiﬁcial Intelligence, 2019. 2, 8

2019
[41]

Deep adversar- ial subspace clustering

Pan Zhou, Yunqing Hou, and Jiashi Feng. Deep adversar- ial subspace clustering. In IEEE Conference on Computer Vision and Pattern Recognition, pages 1596–1604, 2018. 2, 8

2018

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [11]

Efﬁcient dense subspace clustering

Pan Ji, Mathieu Salzmann, and Hongdong Li. Efﬁcient dense subspace clustering. In IEEE Winter Conference on Applica- tions of Computer Vision , pages 461–468. IEEE, 2014. 2, 4

work page 2014

[2] [1]

Tensorﬂow: A system for large-scale machine learning

Mart ´ın Abadi, Paul Barham, Jianmin Chen, Zhifeng Chen, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghe- mawat, Geoffrey Irving, Michael Isard, et al. Tensorﬂow: A system for large-scale machine learning. In USENIX Sym- posium on Operating Systems Design and Implementation , pages 265–283, 2016. 3

work page 2016

[3] [2]

K-subspaces with missing data

Laura Balzano, Arthur Szlam, Benjamin Recht, and Robert Nowak. K-subspaces with missing data. In IEEE Statistical Signal Processing Workshop, pages 612–615. IEEE, 2012. 2

work page 2012

[4] [3]

Graph regularized nonnegative matrix factorization for data representation

Deng Cai, Xiaofei He, Jiawei Han, and Thomas S Huang. Graph regularized nonnegative matrix factorization for data representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 33(8):1548–1560, 2010. 7

work page 2010

[5] [4]

Subspace cluster- ing using a low-rank constrained autoencoder

Yuanyuan Chen, Lei Zhang, and Zhang Yi. Subspace cluster- ing using a low-rank constrained autoencoder. Information Sciences, 424:27–38, 2018. 1

work page 2018

[6] [5]

Lorslim: Low rank sparse linear methods for top-n recommendations

Yao Cheng, Liang Yin, and Yong Yu. Lorslim: Low rank sparse linear methods for top-n recommendations. In IEEE International Conference on Data Mining , pages 90–99. IEEE, 2014. 3

work page 2014

[7] [6]

Sparse subspace cluster- ing

Ehsan Elhamifar and Ren ´e Vidal. Sparse subspace cluster- ing. In IEEE Conference on Computer Vision and Pattern Recognition, pages 2790–2797. IEEE, 2009. 1, 2

work page 2009

[8] [7]

Sparse subspace cluster- ing: Algorithm, theory, and applications

Ehsan Elhamifar and Rene Vidal. Sparse subspace cluster- ing: Algorithm, theory, and applications. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35(11):2765– 2781, 2013. 1, 2, 3, 4

work page 2013