Subspace Clustering of Subspaces (SCoS)

• SCoS is a framework that clusters matrix samples by analyzing their column spaces, preserving crucial matrix structure and geometric relationships.
• It leverages Block Term Decomposition on a third-order tensor of basis matrices to jointly estimate cluster memberships and partially shared subspaces.
• Its robust performance in noisy, high-dimensional settings is validated on synthetic and hyperspectral imaging datasets with superior accuracy compared to vectorized methods.

Subspace Clustering of Subspaces (SCoS) is a framework for clustering a collection of tall matrices based on their column spaces. In this formulation, each sample is itself a matrix rather than a vector, and the objects to be clustered are the subspaces $\mathcal{S}_k=\operatorname{col}(\mathbf{X}_k)$ induced by those matrices. The framework was introduced to preserve matrix structure and geometric relationships between subspaces that are typically lost under vectorization, while establishing conceptual links to both traditional subspace clustering and Generalized Canonical Correlation Analysis (GCCA). Its core machinery is a Block Term Decomposition (BTD) of a third-order tensor formed from basis matrices of the input subspaces, enabling joint estimation of cluster memberships and partially shared subspaces, together with identifiability results and scalable optimization algorithms for large datasets (Karakasis et al., 23 Sep 2025).

1. Problem setting and motivation

Traditional subspace clustering methods treat samples as vectors and seek to partition high-dimensional data vectors into clusters so that each cluster lies, approximately, in a low-dimensional linear subspace. That setting is well matched to single-view data or to data in which each entity is described by one feature vector. SCoS is motivated by settings in which each object is itself a matrix, such as multi-view data, multi-subject neuroimaging, and hyperspectral imaging, so that vectorization discards structure that is intrinsic to the data-generating process (Karakasis et al., 23 Sep 2025).

In SCoS, subspaces themselves become the primitive objects of interest. The motivation is threefold. First, directly clustering subspaces preserves matrix structure and the geometric relationships between subspaces. Second, it is more natural in scenarios in which each sample corresponds to a subspace rather than to a single point. Third, vectorization can reduce performance, especially in noisy or high-dimensional settings, a point that is supported experimentally in the reported results (Karakasis et al., 23 Sep 2025).

A recurrent misconception is to equate SCoS with ordinary subspace clustering applied after flattening each matrix into a vector. The formulation explicitly differs from that reduction: the clustering target is not a cloud of vectors but a collection of column spaces, and the relevant shared structure is a partially common component across matrices within the same cluster.

2. Generative model and clustering objective

The basic input to SCoS is a collection of data matrices

$$\{\mathbf{X}_k \in \mathbb{R}^{N \times M_k}\}_{k=1}^K,$$

where each $\mathbf{X}_k$ is a set of $M_k$ samples or measurements in $\mathbb{R}^N$. The model assumes that each matrix is drawn according to

$$\mathbf{X}_k = \mathbf{G}_{r_k}\mathbf{Q}_k^{(1)} + \mathbf{H}_k\mathbf{Q}_k^{(2)} + \mathbf{E}_k,$$

with the following components (Karakasis et al., 23 Sep 2025):

• $\mathbf{G}_{r_k} \in \mathbb{R}^{N \times L}$: cluster-specific subspace basis for cluster $r_k$.
• $\mathbf{H}_k \in \mathbb{R}^{N \times (M_k - L)}$: individual-specific orthogonal basis.
• $\mathbf{Q}_k^{(1)}$ and $$\{\mathbf{X}_k \in \mathbb{R}^{N \times M_k}\}_{k=1}^K,$$0: random mixing matrices.
• $$\{\mathbf{X}_k \in \mathbb{R}^{N \times M_k}\}_{k=1}^K,$$1: noise.

Under this model, the column space of $$\{\mathbf{X}_k \in \mathbb{R}^{N \times M_k}\}_{k=1}^K,$$2 contains a partially common component inherited from the cluster subspace and an individual component specific to the sample. The clustering objective is therefore to partition the subspaces

$$\{\mathbf{X}_k \in \mathbb{R}^{N \times M_k}\}_{k=1}^K,$$3

into $$\{\mathbf{X}_k \in \mathbb{R}^{N \times M_k}\}_{k=1}^K,$$4 groups directly on the manifold of subspaces, without vectorizing the matrices (Karakasis et al., 23 Sep 2025).

The central structural notion is the partially common subspace. For cluster $$\{\mathbf{X}_k \in \mathbb{R}^{N \times M_k}\}_{k=1}^K,$$5, this is

$$\{\mathbf{X}_k \in \mathbb{R}^{N \times M_k}\}_{k=1}^K,$$6

Subspaces assigned to the same cluster share $$\{\mathbf{X}_k \in \mathbb{R}^{N \times M_k}\}_{k=1}^K,$$7 as an intersection, possibly together with individual variation. This framing distinguishes SCoS from models in which every sample in a cluster is assumed to lie entirely in one fixed low-dimensional subspace.

3. Position relative to subspace clustering and GCCA

SCoS is defined by explicit comparison with two established paradigms: traditional subspace clustering and GCCA. Traditional subspace clustering operates on vector samples and groups points whose linear spans are consistent. By contrast, SCoS clusters subspaces themselves, that is, the spans of the matrices $$\{\mathbf{X}_k \in \mathbb{R}^{N \times M_k}\}_{k=1}^K,$$8, based on their affinity (Karakasis et al., 23 Sep 2025).

Aspect	Traditional subspace clustering	SCoS
Data mode	Vectors	Matrices or blocks
Objective	Cluster vectors or samples	Cluster subspaces or matrices
Shared structure	Points cluster by approximately shared subspace	Cluster-specific partially shared subspaces

Within SCoS, each cluster is characterized by a partially common subspace $$\{\mathbf{X}_k \in \mathbb{R}^{N \times M_k}\}_{k=1}^K,$$9, and the subspaces in the same cluster all share that common subspace as an intersection, possibly with individual variation. This is the mechanism by which the framework extends the union-of-subspaces viewpoint from points to matrix-defined subspaces.

GCCA appears as a special case. GCCA seeks a common subspace shared by all views, and in the SCoS notation it corresponds to the single-cluster case $\mathbf{X}_k$0:

$\mathbf{X}_k$1

Accordingly, SCoS generalizes GCCA from one globally shared subspace to multiple clusters, each with its own partially common subspace (Karakasis et al., 23 Sep 2025).

A second misconception is to identify SCoS with GCCA plus a post hoc clustering step. The distinction stated in the formulation is stronger: GCCA assumes one common subspace across all matrices, whereas SCoS models multiple cluster-specific partially common subspaces and treats cluster assignment as part of the estimation problem.

4. Block-term tensor formulation and scalable estimation

The computational core of SCoS is a BTD model on a third-order tensor built from orthonormal basis matrices. Let $\mathbf{X}_k$2 be an orthonormal basis for $\mathbf{X}_k$3, where $\mathbf{X}_k$4 is the estimated subspace dimension, and stack these bases into a tensor

$\mathbf{X}_k$5

whose frontal slices are the matrices $\mathbf{X}_k$6. SCoS models this tensor through the approximation

$\mathbf{X}_k$7

where $\mathbf{X}_k$8 are the cluster-$\mathbf{X}_k$9 subspace bases to be estimated, $M_k$0 are selection or assignment matrices indicating membership, and $M_k$1 denotes the tensor outer product. Cluster membership is encoded by an assignment vector $M_k$2 (Karakasis et al., 23 Sep 2025).

The scalable workflow summarized for SCoS has three main steps (Karakasis et al., 23 Sep 2025). First, estimate subspace bases $M_k$3 for each $M_k$4, for example via SVD. Second, cluster the subspaces $M_k$5 using affinity or similarity measures such as principal angles or Grassmannian distances. Third, for each cluster, estimate the dimension and basis of the common subspace $M_k$6.

Two estimation routes are specified for the third step. One forms the sum of projectors

$M_k$7

and eigendecomposes $M_k$8. The other stacks the $M_k$9 column-wise and performs SVD, which is preferable when $\mathbb{R}^N$0 is large and the total column dimension is smaller. The supplementary algorithmic guideline is explicit: use stacking and SVD when $\mathbb{R}^N$1, where $\mathbb{R}^N$2 is the sum of column dimensions, and use sum of projectors and eigendecomposition otherwise. Model order selection through AIC, MDL, or other data-driven criteria determines the dimension $\mathbb{R}^N$3 of each partially common subspace (Karakasis et al., 23 Sep 2025).

The clustering criterion itself can be expressed through an optimization over the assignment vector $\mathbb{R}^N$4. The summary states that the objective may maximize, for example, the sum of trace overlap or a likelihood model involving the $\mathbb{R}^N$5 and the estimated $\mathbb{R}^N$6s. This suggests that the framework accommodates both geometry-driven and model-based assignment rules, provided that they operate on subspaces rather than on vectorized samples.

5. Identifiability, robustness, and empirical behavior

A notable theoretical result is that SCoS provides the first identifiability results for this formulation. Under mild conditions, the partially common subspaces $\mathbb{R}^N$7 and the clustering assignment $\mathbb{R}^N$8 are identifiable almost surely. The stated rationale is the uniqueness behavior of BTD together with the geometry of the Grassmann manifold, where the intersection structure of subspaces is robust to noise and high dimensionality if it is modeled correctly (Karakasis et al., 23 Sep 2025).

This theoretical claim is presented as a significant improvement over vectorized approaches, which are described as prone to ambiguities. A plausible implication is that the move from pointwise vector models to matrix-defined subspaces does not merely alter the input representation; it changes the identifiability landscape by making shared intersection structure directly estimable.

The empirical evidence summarized for synthetic data emphasizes robustness to noise. SCoS is reported to substantially outperform SSC-OMP, ENSC, and BDRSC in clustering accuracy (ACC), Normalized Mutual Information (NMI), and Adjusted Rand Index (ARI), especially as the signal-to-noise ratio (SINR) decreases. It remains effective at lower SINR levels, such as down to $\mathbb{R}^N$9 dB or lower, where classical vectorized methods already degrade to random performance. Computational efficiency is described as generally favorable, with SCoS competitive in runtime except in extremely noisy regimes (Karakasis et al., 23 Sep 2025).

On real hyperspectral datasets, specifically Indian Pines, University of Pavia, and Salinas-A, SCoS yields more coherent and accurate segmentation maps, closely matching ground truth and outperforming vectorized baselines and K-means or spectral clustering. The abstract further emphasizes superior clustering accuracy and robustness under high noise and interference. These findings position SCoS as a method for challenging high-dimensional applications in which relevant structure exists beyond individual data vectors (Karakasis et al., 23 Sep 2025).

6. Related formulations and broader significance

The immediate significance of SCoS lies in its treatment of matrix-structured objects without flattening. The reported implications are that it preserves structure, avoids the information loss and increased sensitivity to noise caused by vectorization, scales better in the ambient dimension $$\mathbf{X}_k = \mathbf{G}_{r_k}\mathbf{Q}_k^{(1)} + \mathbf{H}_k\mathbf{Q}_k^{(2)} + \mathbf{E}_k,$$0 through projector-sum or stacked-SVD strategies, and is naturally applicable to multi-view or multi-block data such as multi-subject fMRI, multi-channel communication, and multi-hyperspectral image analyses (Karakasis et al., 23 Sep 2025).

Related work shows that the phrase and intuition behind “clustering at the level of subspaces or subspace representatives” also appear in adjacent literatures. “Subspace Clustering through Sub-Clusters” replaces direct clustering of all points with the more efficient and robust problem of “clustering sub-clusters,” where a small sample is spectrally clustered and the remaining points are labeled out of sample; the paper presents this as a concrete instantiation of the SCoS paradigm because the affinity matrix is built among sub-clusters rather than individual points (Li et al., 2018). “Filtrated Algebraic Subspace Clustering” states that its filtration construction is conceptually a process of recursively clustering subspaces within a variety—i.e., subspace clustering of subspaces—even though its primary formal setting remains a union of linear subspaces populated by data points (Tsakiris et al., 2015). “Neural Collaborative Subspace Clustering” remains a point-level union-of-subspaces method, but its summary presents a relation to SCoS in which the framework could be extended to cases where input features describe subspaces or where hierarchy or structure among subspaces is important (Zhang et al., 2019).

These neighboring usages should not obscure the more specific meaning of SCoS in the 2025 formulation. In its precise sense, SCoS is not merely any higher-level or scalable subspace clustering heuristic. It is the problem of clustering a collection of matrices according to their column spaces, with partially common subspaces as cluster-defining objects, a BTD-based tensor model for joint estimation, and almost-sure identifiability under mild conditions (Karakasis et al., 23 Sep 2025). This suggests a unifying viewpoint in which classical subspace clustering, GCCA, and matrix-native clustering can be situated within a single hierarchy, while still preserving the distinct assumptions and estimation targets of each method.