Deep Subspace Clustering (DSC)

• Deep Subspace Clustering (DSC) is an unsupervised method that represents each sample as a linear combination of others in a learned latent space, typically via an autoencoder.
• It integrates a self-expressive layer within deep autoencoders to jointly optimize feature representation and the construction of an affinity matrix for spectral clustering.
• Recent advances in DSC address scalability, multi-layer fusion, and optimization challenges to improve accuracy and robustness in clustering high-dimensional data.

Deep Subspace Clustering (DSC) is a family of unsupervised clustering methods built on the union-of-subspaces model and the self-expressiveness principle: each sample is represented as a linear combination of other samples from the same subspace. In its deep form, DSC replaces direct self-expression in the input space by self-expression in a learned feature space, typically produced by an autoencoder or related deep backbone, and then converts the learned coefficient matrix into an affinity for spectral clustering (Ji et al., 2017). This formulation was introduced in several early deep variants, notably Deep Sparse Subspace Clustering (DSSC), which learned nonlinear mappings and sparse self-expression with a unit-sphere regularizer, and Deep Subspace Clustering Networks (DSC-Nets), which embedded a self-expressive layer directly inside a deep autoencoder (Peng et al., 2017, Ji et al., 2017).

1. Mathematical formulation and self-expressiveness

The standard DSC problem begins with data drawn from a union of low-dimensional subspaces. In the classical formulation, with samples stacked in a matrix $X \in \mathbb{R}^{D \times N}$, self-expressiveness is written as $X \approx XC$, where $C \in \mathbb{R}^{N \times N}$ is a coefficient matrix and $\operatorname{diag}(C)=0$ is used to avoid trivial self-representation (Ji et al., 2017). A common regularized objective is

$$\min_C \|C\|_p + \frac{\lambda}{2}\|X-XC\|_F^2,\quad \text{subject to } \operatorname{diag}(C)=0,$$

with $\|\cdot\|_p$ instantiated by the $\ell_1$ norm in SSC, the nuclear norm in LRR/LRSC, or the Frobenius norm in LSR/EDSC (Ji et al., 2017).

Deep formulations replace the raw data by learned latent features. If $Z=f_{\mathrm{enc}}(X;\theta_e)$ denotes the encoder output, DSC enforces self-expression in latent space through $Z \approx ZC$ and uses the learned $C$ to construct an affinity matrix for spectral clustering (Ji et al., 2017). A common choice is $X \approx XC$0 or $X \approx XC$1, followed by normalized spectral clustering such as Ng–Jordan–Weiss (Kheirandishfard et al., 2020). Some papers use a row-wise sample convention rather than the column-wise one above, but the central object remains the same: a sample-to-sample self-representation matrix whose block-diagonal structure is intended to reveal cluster membership (Seo et al., 2019).

This formulation is attractive because it couples feature learning and graph learning. The latent space is expected to make the union-of-subspaces prior more accurate than in the input space, while the self-expressive coefficients provide a graph on which spectral clustering can operate (Ji et al., 2017). A plausible implication is that the quality of DSC depends simultaneously on representation geometry, coefficient regularization, and the post-processing used to transform $X \approx XC$2 into an affinity.

2. Canonical deep self-expressive networks

The canonical autoencoder-based instantiation is DSC-Net. Its defining architectural move is the insertion of a self-expressive layer between encoder and decoder: a linear, fully-connected layer with weights $X \approx XC$3, no bias, and no nonlinearity, mapping $X \approx XC$4 to $X \approx XC$5 (Ji et al., 2017). The complete objective is

$X \approx XC$6

with optional $X \approx XC$7 (Ji et al., 2017). Two reported variants are DSC-Net-L1, using $X \approx XC$8, and DSC-Net-L2, using $X \approx XC$9; the paper reports that DSC-Net-L2 was typically easier to optimize and slightly more robust (Ji et al., 2017).

Training in DSC-Net is explicitly two-stage. First, the autoencoder is pretrained using only reconstruction loss. Second, the self-expressive layer is inserted and the full model is jointly fine-tuned on the entire dataset as a single batch with Adam at learning rate $C \in \mathbb{R}^{N \times N}$0, while zeroing the diagonal of $C \in \mathbb{R}^{N \times N}$1 after updates if the diagonal constraint is enforced (Ji et al., 2017). This full-batch requirement follows directly from the fact that $C \in \mathbb{R}^{N \times N}$2 couples all samples to all other samples. It is also the main source of the method’s memory and scalability limitations.

The original experiments established the empirical profile that shaped later DSC work. On Extended Yale B with 38 subjects, DSC-Net-L2 achieved a mean clustering error of $C \in \mathbb{R}^{N \times N}$3, compared with approximately $C \in \mathbb{R}^{N \times N}$4 for EDSC; DSC-Net-L1 achieved $C \in \mathbb{R}^{N \times N}$5 (Ji et al., 2017). The same study reported strong gains on ORL, COIL20, and COIL100, and emphasized robustness as the number of clusters increased (Ji et al., 2017).

A closely related early line is DSSC, which did not insert a trainable self-expressive layer into an autoencoder. Instead, it learned a fully connected nonlinear mapping, optimized sparse self-expression in the learned feature space, and regularized the top-layer features by a unit-sphere distribution assumption:

$C \in \mathbb{R}^{N \times N}$6

DSSC then formed $C \in \mathbb{R}^{N \times N}$7 and applied spectral clustering (Peng et al., 2017). In historical terms, DSSC and DSC-Net mark two early deep routes into subspace clustering: alternating sparse coding on learned features, and end-to-end self-expressive autoencoders.

3. Representation enrichment, multilevel fusion, and supervision

A major line of work extends DSC beyond a single latent layer. Multi-Level Representation Learning for Deep Subspace Clustering (MLRDSC) inserts multiple fully connected linear connection layers between encoder levels and their corresponding decoder levels, so that self-expressiveness is enforced at several depths through $C \in \mathbb{R}^{N \times N}$8 (Kheirandishfard et al., 2020). Here $C \in \mathbb{R}^{N \times N}$9 is a consistency matrix shared across levels and $\operatorname{diag}(C)=0$0 is level-specific. The method also introduces a clustering-guided regularizer

$\operatorname{diag}(C)=0$1

where $\operatorname{diag}(C)=0$2 contains pseudo-labels updated by spectral clustering every $\operatorname{diag}(C)=0$3 epochs (Kheirandishfard et al., 2020). On Extended Yale B with 38 subjects, MLRDSC reduced the clustering error from DSC’s $\operatorname{diag}(C)=0$4 to $\operatorname{diag}(C)=0$5, and it also achieved the best reported results on COIL20 and COIL100 in that study (Kheirandishfard et al., 2020).

A related observation is that useful subspace information is not confined to the last encoder layer. The multilayer graph approach to DSC learns a separate representation matrix from each encoder layer, including the input, then fuses their shifted Laplacians into a modified multilayer Laplacian

$\operatorname{diag}(C)=0$6

before a single spectral clustering step (Sindičić et al., 2024). Label-independent hyperparameter-free self-supervised single-view DSC pushes the same intuition further by enforcing a joint layer-wise self-expression loss

$\operatorname{diag}(C)=0$7

with a sequential training schedule that avoids weighted multi-loss tuning and uses a label-independent relative-error stopping rule (Sindicic et al., 25 Apr 2025).

Another large class of variants enriches the representation itself. Overcomplete Deep Subspace Clustering Networks (ODSC) fuse undercomplete and overcomplete autoencoder branches before self-expression, reporting lower clustering error than DSC-L2 on MNIST, ORL, COIL20, and Extended Yale B, as well as reduced sensitivity to the pretraining stopping point and improved robustness to noise (Valanarasu et al., 2020). Deep Structure and Attention aware Subspace Clustering (DSASC) replaces convolutional encoders by a frozen DINO Vision Transformer for content features and uses KNN graphs with a GCN to produce structure features, learning separate self-representation matrices $\operatorname{diag}(C)=0$8 and $\operatorname{diag}(C)=0$9 and fusing them as $$\min_C \|C\|_p + \frac{\lambda}{2}\|X-XC\|_F^2,\quad \text{subject to } \operatorname{diag}(C)=0,$$0 (Wu et al., 2023). On STL-10, DSASC reported ACC/NMI of $$\min_C \|C\|_p + \frac{\lambda}{2}\|X-XC\|_F^2,\quad \text{subject to } \operatorname{diag}(C)=0,$$1; on CIFAR-100 it reported $$\min_C \|C\|_p + \frac{\lambda}{2}\|X-XC\|_F^2,\quad \text{subject to } \operatorname{diag}(C)=0,$$2 (Wu et al., 2023).

Supervision has also been weakened rather than added. Pseudo-supervised Deep Subspace Clustering (PSSC) replaces pointwise autoencoder reconstruction by a similarity-weighted reconstruction

$$\min_C \|C\|_p + \frac{\lambda}{2}\|X-XC\|_F^2,\quad \text{subject to } \operatorname{diag}(C)=0,$$3

where the graph is derived from the learned self-expression matrix, and augments training with pseudo-graph and pseudo-label losses (Lv et al., 2021). Self-Supervised deep Subspace Clustering with Entropy-norm (S$$\min_C \|C\|_p + \frac{\lambda}{2}\|X-XC\|_F^2,\quad \text{subject to } \operatorname{diag}(C)=0,$$4CE) discards decoder training during clustering, freezes a contrastively pretrained encoder, and optimizes

$$\min_C \|C\|_p + \frac{\lambda}{2}\|X-XC\|_F^2,\quad \text{subject to } \operatorname{diag}(C)=0,$$5

subject to row-stochastic constraints on $$\min_C \|C\|_p + \frac{\lambda}{2}\|X-XC\|_F^2,\quad \text{subject to } \operatorname{diag}(C)=0,$$6, in order to promote dense within-subspace connectivity (Zhao et al., 2022). Deep Double Self-Expressive Subspace Clustering adds a second self-expression stage on the coefficient matrix itself, $$\min_C \|C\|_p + \frac{\lambda}{2}\|X-XC\|_F^2,\quad \text{subject to } \operatorname{diag}(C)=0,$$7, then fuses $$\min_C \|C\|_p + \frac{\lambda}{2}\|X-XC\|_F^2,\quad \text{subject to } \operatorname{diag}(C)=0,$$8 and $$\min_C \|C\|_p + \frac{\lambda}{2}\|X-XC\|_F^2,\quad \text{subject to } \operatorname{diag}(C)=0,$$9 for spectral clustering (Zhao et al., 2023).

Multi-view settings generalize the same pattern. Multi-view Deep Subspace Clustering Networks (MvDSCN) use a diversity network that learns view-specific self-representation matrices $\|\cdot\|_p$0, a universality network that learns a common matrix $\|\cdot\|_p$1, and HSIC-based diversity regularization together with universality alignment (Zhu et al., 2019). This allows either multiple modalities or multiple pretrained backbones such as VGG16 and ResNet50 to be treated as separate views (Zhu et al., 2019).

4. Scalability, mini-batch training, and deployment variants

The dominant systems issue in DSC is the $\|\cdot\|_p$2 coefficient matrix. In the original DSC-Net, the self-expressive layer alone scales as $\|\cdot\|_p$3 in parameters and forces full-batch fine-tuning (Ji et al., 2017). Later work attacked this bottleneck from several directions.

One route removes the explicit affinity matrix. Scalable Deep $\|\cdot\|_p$4-Subspace Clustering learns an embedding and $\|\cdot\|_p$5 low-dimensional subspaces directly, minimizing reconstruction residuals to the nearest latent subspace instead of constructing a self-representation matrix (Zhang et al., 2018). It updates subspace bases by SVD or Grassmannian steps and performs nearest-subspace assignment at inference time, thereby avoiding both $\|\cdot\|_p$6 affinities and spectral clustering (Zhang et al., 2018). On full MNIST, the SVD-based variant reported ACC/NMI/ARI of $\|\cdot\|_p$7 (Zhang et al., 2018).

A second route retains self-expression but factorizes or approximates it. Deep Closed-Form Subspace Clustering (DCFSC) replaces a trainable self-expressive layer by a closed-form ridge solution computed from the current latent codes:

$\|\cdot\|_p$8

The network then reconstructs from $\|\cdot\|_p$9 rather than learning $\ell_1$0 by backpropagation (Seo et al., 2019). Scalable Deep Subspace Network (SDSNet) replaces a full $\ell_1$1 affinity by a projector $\ell_1$2 obtained from landmark-based factorization and solves a reduced eigenproblem, claiming $\ell_1$3 complexity when the number of landmarks is fixed (Mrabah et al., 24 Dec 2025). Scalable Context-Preserving Model-Aware Deep Clustering for hyperspectral images also abandons $\ell_1$4 self-representation in favor of basis representation and a one-stage clustering loss with local and non-local structural constraints, likewise reporting $\ell_1$5 time and space complexity (Li et al., 12 Jun 2025).

A third route keeps the original self-expressive objective but changes the training protocol. A mini-batch training strategy for DSC introduces a memory bank $\ell_1$6 that stores global latent features, so that for a mini-batch $\ell_1$7 the self-expressiveness loss becomes

$\ell_1$8

This preserves global relations while allowing encoder and decoder activations to be computed only for the current mini-batch (Jiang et al., 26 Jul 2025). The same paper also proposes a decoder-free variant, CLBDSC, that combines mini-batch self-expression with contrastive learning for fine-tuning pretrained encoders (Jiang et al., 26 Jul 2025). Empirically, BDSC matched or exceeded the authors’ full-batch DSC reproduction on ORL, COIL20, and COIL100, and deeper variants improved further on high-resolution inputs (Jiang et al., 26 Jul 2025).

Deployment-oriented variants extend DSC beyond a single machine or modality. Federated Deep Subspace Clustering (FDSC) keeps the encoder shared across clients and the self-expressive layer plus decoder private, aggregates only encoder parameters by a FedAvg-style rule, and adds a locality-preserving alignment term between the local self-expression matrix and a client-side $\ell_1$9-NN adjacency (Zhang et al., 2024). On MNIST, ORL, COIL20, and COIL100, the locality-enhanced variant FDSC2 outperformed both its federated ablation and centralized DSCN baselines (Zhang et al., 2024). In application-specific form, SCDSC shows how basis-representation ideas can be specialized to hyperspectral images through jointly optimized local spatial smoothing and non-local mini-cluster consistency (Li et al., 12 Jun 2025).

5. Critique, collapse, and principled reformulations

A central controversy in DSC concerns whether jointly learning the embedding and self-expressive coefficients is well-posed. “A Critique of Self-Expressive Deep Subspace Clustering” argues that many deep self-expressive formulations can be ill-posed because, with positively homogeneous encoder/decoder architectures, the self-expressive term can be driven toward zero by shrinking the embedding magnitude while leaving reconstruction unchanged (Haeffele et al., 2020). Under several normalization schemes, the paper further proves that globally optimal embeddings can degenerate into rank-1 structures, duplicate points, or configurations in which only two nonzero columns remain, none of which correspond to a meaningful union of subspaces (Haeffele et al., 2020). It also reports that a substantial portion of earlier performance gains can be attributed to post-processing of $Z=f_{\mathrm{enc}}(X;\theta_e)$0 rather than the deep self-expressive model itself (Haeffele et al., 2020).

Subsequent work targeted these objections directly. PRO-DSC augments the standard self-expressive objective with a log-determinant representation regularizer,

$Z=f_{\mathrm{enc}}(X;\theta_e)$1

and proves that, under an explicit condition on $Z=f_{\mathrm{enc}}(X;\theta_e)$2 and $Z=f_{\mathrm{enc}}(X;\theta_e)$3, the learned representation avoids rank collapse and can lie on a union of orthogonal subspaces (Meng et al., 21 Mar 2025). The paper also proves an eigenspace-alignment result between $Z=f_{\mathrm{enc}}(X;\theta_e)$4 and $Z=f_{\mathrm{enc}}(X;\theta_e)$5, and introduces a scalable reparameterization in which the coefficient matrix is produced by Sinkhorn projection of pairwise similarities between normalized cluster embeddings (Meng et al., 21 Mar 2025). Empirically, PRO-DSC reported ACC/NMI of $Z=f_{\mathrm{enc}}(X;\theta_e)$6 on CIFAR-10 and $Z=f_{\mathrm{enc}}(X;\theta_e)$7 on ImageNet-1k using pretrained CLIP features (Meng et al., 21 Mar 2025).

Procedural responses also emerged. Label-independent hyperparameter-free single-view DSC explicitly identifies several weaknesses in prior DSC practice: reliance on only the output layer, independent treatment of representation learning and clustering, dependence on held-out data for hyperparameter tuning, label-based stopping criteria, and post-processing that relies on labeled data (Sindicic et al., 25 Apr 2025). Its response is not a new coefficient regularizer but a multi-stage schedule with layer-wise self-expression, a subspace-structured norm, and a relative-error self-stopping mechanism that does not use external labels (Sindicic et al., 25 Apr 2025). The multilayer graph approach likewise rejects label-dependent thresholding and instead uses IPD truncation and multilayer Laplacian fusion (Sindičić et al., 2024).

Taken together, these papers establish that DSC is not just an architectural template but also a question of optimization geometry. This suggests that later DSC research increasingly treats collapse prevention, label-independence, and post-processing dependence as first-class design constraints rather than secondary implementation details.

6. Empirical profile, application domains, and unresolved issues

Across the literature, DSC is most heavily benchmarked on image datasets such as Extended Yale B, ORL, COIL20, COIL100, MNIST, USPS, Fashion-MNIST, CIFAR-10/100, and STL-10, with additional use in multi-modal RGB-D Object and CUB, and in hyperspectral image clustering on Houston, Trento, and PaviaU (Ji et al., 2017, Zhu et al., 2019, Wu et al., 2023, Li et al., 12 Jun 2025). In these settings, the dominant evaluation metrics are clustering error, ACC, NMI, ARI, F-measure, PUR, and sometimes subspace-preserving error or connectivity (Kheirandishfard et al., 2020, Zhao et al., 2022, Zhao et al., 2023). The empirical pattern is consistent: deep feature learning tends to outperform shallow SSC/LRR-style baselines when the data depart from a simple linear-subspace model, but gains are strongly mediated by architecture, regularization, and the handling of the coefficient matrix.

The strongest gains tend to appear when additional structure is exploited. MLRDSC improves over single-layer DSC on Extended Yale B, COIL20, and COIL100 through multi-level self-expression and clustering-guided fusion (Kheirandishfard et al., 2020). MvDSCN improves multi-view and multi-modal clustering by combining diversity regularization and universality alignment (Zhu et al., 2019). DSASC improves image clustering by fusing transformer-derived content features with graph-derived structure features (Wu et al., 2023). S$Z=f_{\mathrm{enc}}(X;\theta_e)$8CE reports leading ACC/NMI on COIL20, COIL40, and MNIST-1000 by combining contrastive features with an entropy-norm regularizer that promotes dense within-subspace connectivity (Zhao et al., 2022). SCDSC shows that basis-representation and context-preserving losses can produce state-of-the-art results on several HSI benchmarks while remaining linear in $Z=f_{\mathrm{enc}}(X;\theta_e)$9 (Li et al., 12 Jun 2025).

At the same time, recurring limitations are explicit throughout the field. The coefficient matrix remains $Z \approx ZC$0 in most self-expressive variants, spectral clustering still scales poorly, hyperparameters and architecture depth remain dataset-sensitive, and performance can depend on known cluster counts or prior knowledge such as subspace dimension (Ji et al., 2017, Kheirandishfard et al., 2020, Sindicic et al., 25 Apr 2025). Additional limitations are method-specific: MvDSCN does not explicitly handle missing views (Zhu et al., 2019); DSASC can suffer under domain shift from color-pretrained ViTs to grayscale data (Wu et al., 2023); S$Z \approx ZC$1CE remains quadratic in the size of $Z \approx ZC$2 despite freezing the encoder (Zhao et al., 2022); and federated DSC offers privacy by keeping data local but does not provide secure aggregation or differential privacy guarantees (Zhang et al., 2024).

The field’s current trajectory is therefore bifurcated. One branch continues to refine the classical self-expressive pipeline through better representations, multi-layer or multi-view fusion, and more principled regularization. The other branch seeks to preserve the subspace-clustering inductive bias while removing its computational bottlenecks through basis representations, factorized affinities, mini-batch memory banks, or direct subspace assignment (Mrabah et al., 24 Dec 2025, Jiang et al., 26 Jul 2025, Zhang et al., 2018). A plausible implication is that future DSC systems will be judged less by raw benchmark accuracy alone than by whether they can combine subspace-preserving structure, scalable optimization, and label-independent operation in a single training framework.