Cluster-Preserving Representations

• Cluster-preserving representations are data embeddings that retain intrinsic cluster structures by ensuring intra-cluster compactness and clear inter-cluster separation.
• Methods such as spectral regularization, cluster-specific neural architectures, and contrastive losses are employed to enforce these properties with theoretical guarantees.
• Empirical results demonstrate enhanced clustering accuracy, improved visualization, and superior transfer learning performance across diverse data domains.

Cluster-preserving representations refer to data embeddings or transformations that retain the intrinsic cluster structure of the original data—ensuring that points belonging to the same underlying cluster in the data space remain close (intra-cluster compactness), while points from different clusters are well separated (inter-cluster separation) in the learned space. The preservation of cluster structure is critical for subsequent clustering, visualization, and downstream inference, as it guarantees operational consistency across representation learning and cluster discovery.

1. Theoretical Foundations and Definitions

Cluster-preserving representations have been defined both constructively and theoretically. At the most basic level, a representation $g$ is cluster-preserving if, for each cluster $c$, all data points $x, x'$ in that cluster satisfy $g(x) = g(x')$ or, for continuous embeddings, if $g(x)$ and $g(x')$ are close under an appropriate metric. The theoretical analysis in "InfoNCE Loss Provably Learns Cluster-Preserving Representations" establishes that, subject to intertwined-augmentation assumptions, any global minimizer of the InfoNCE loss is necessarily cluster-preserving and uniform—meaning that data samples from the same cluster are not only identically embedded, but clusters fill the space equitably (Parulekar et al., 2023).

In "Cluster Specific Representation Learning," a practical meta-algorithm combines joint optimization over both cluster assignments and cluster-specific representation heads: each point is assigned to a cluster, and its representation is derived from a cluster-specific encoder. This alternating minimization framework directly operationalizes the notion of cluster preservation by requiring the learned representation to be "specific" within each cluster, while soft/hard assignments ensure partitioning fidelity (Sabanayagam et al., 2024).

2. Explicit Cluster-Preserving Representation Learning Methods

Multiple paradigms have been engineered to enforce cluster preservation during representation learning. The approaches can be broadly grouped as follows:

• Self-expression and spectral regularization: The Similarity Preserving Clustering (SPC) framework simultaneously learns a similarity graph $Z$ (subject to nonnegativity) and a cluster-indicator matrix $F$, imposing the constraint that the Laplacian $L = D - Z$ has exactly $c$ zero eigenvalues (where $c$0 is the number of clusters). The objective enforces $c$1's closeness to a kernel matrix $c$2 (encoding ground-truth similarities) and strictly forces connected component structure via trace regularization on the $c$3 smallest Laplacian eigenvectors. The result is that Z decomposes as a block-diagonal matrix at convergence, each block corresponding to a cluster; cluster assignments can be read off directly, and no downstream k-means or spectral rounding is needed (Kang et al., 2019).
• Cluster-specific neural architectures: The tensorized or partial-tensorized meta-algorithm introduced in "Cluster Specific Representation Learning" partitions the embedding function into a backbone shared by all data and per-cluster heads. Embedding assignment alternates with re-clustering, ensuring that each head specializes to its cluster and the optimization bias suppresses spurious inter-cluster coincidences. The approach generalizes to autoencoders, VAEs, contrastive learners, and RBMs. It empirically boosts clustering accuracy, de-noising performance, and latent separation compared to single-head baselines (Sabanayagam et al., 2024).
• Contrastive clustering-friendly methods: Methods such as RoNID incorporate explicit intra- and inter-cluster contrastive losses: intra-cluster contrast pulls each embedding toward its prototype center, while inter-cluster contrast pushes prototypes apart on the unit sphere. This joint objective enforces compactness and maximal separation, with additional EM-like re-assignment steps for robust pseudo-labeling of unknown classes (Zhang et al., 2024). Other methods (e.g., PIPCDR) integrate neighbor-based alignment with cluster-dispersion regularization within a majorize–minimize framework, directly combating class collision and clustering collapse (Kumar et al., 2023).
• Distribution- and topology-preserving autoencoders: Several works introduce explicit geometric or topological constraints. For example, RTD-AE includes a Representation Topology Divergence loss derived from persistent homology, ensuring that homological features (clusters, loops) are matched between the original and embedding space (Trofimov et al., 2023). DPNE enforces correspondence between high-dimensional and latent densities via KL-divergence, guaranteeing that high-density regions map to concentrated latent clusters (Qin et al., 2020).
• Graph filtering and spectral propagation: Graph-based methods (e.g., SCGF) apply graph low-pass filters, which systematically enhance within-cluster similarity by suppressing high-frequency (noisy or inter-cluster) components. This process increases intra-cluster homogeneity and smooths class boundaries, making standard subspace clustering methods much more effective even on non-linearly separable data (Ma et al., 2021).
• Curvature-based and alignment-preserving embedding: EmbedOR leverages discrete Ricci curvature to augment pairwise distances by penalizing "shortcut" edges that bridge clusters, ensuring shortest-path distances respect global manifold structure and strictly suppress inter-cluster connectivity. When coupling this with stochastic neighbor embedding, the approach yields provably cluster-preserving visualizations resistant to fragmentation (Saidi et al., 3 Sep 2025).

3. Cluster-Preservation in Contrastive and Self-Supervised Learning

Provable cluster preservation in contrastive learning has been addressed in recent theoretical and empirical literature. Under finite-batch InfoNCE loss, with intertwined augmentations and a class with bounded expressivity, minimizers produce embeddings that are exactly constant on clusters and uniform across the space of possible embeddings; thus, intrinsic cluster geometry is guaranteed to be retained (Parulekar et al., 2023). Clustering-friendly extensions of standard contrastive frameworks actively exclude features explained away by backgrounds or nuisance variables, as in cIDFD, which uses background-aware weighting in contrastive instance discrimination loss to ensure clusters reflect semantic rather than background structure. Empirical benchmarks reveal dramatic accuracy and ARI/NMI gains, demonstrating that these mechanisms do, in effect, yield cluster-preserving representations (Oshima et al., 2024).

ClusterFit presents a simple but powerful two-stage procedure: given a pre-trained network, first cluster its features via k-means, then re-train a new network to predict cluster pseudo-labels. This enforces the removal of task-specific artifacts and results in features with superior transfer and intra-cluster compactness for a variety of downstream tasks (Yan et al., 2019).

4. Applications and Empirical Evidence

Cluster-preserving representations enable robust and accurate unsupervised clustering (particularly with kernel or deep representations), visualization, transfer learning, and robust generalization:

Method	Domain	Mechanism	Reported Impact
SPC (Kang et al., 2019)	General data	Kernel+graph learning	>90% clustering ACC (two-moons), superior to k-means
cIDFD (Oshima et al., 2024)	Vision	Reference-based contrastive	+0.55 ACC and +0.65 ARI over baseline on MNIST
PIPCDR (Kumar et al., 2023)	Deep clusters	Pos. proxy + disp. regularizer	NMI 0.897/ACC 0.948 on CIFAR-10
ClusterFit (Yan et al., 2019)	Vision	k-means+retraining	+7–8% ACC over self-supervised baselines
RTD-AE (Trofimov et al., 2023)	Geometry	Persistent homology loss	Outperforms t-SNE/UMAP on all topological metrics
RoNID (Zhang et al., 2024)	NLP/intent	Intra/inter-cluster contrastive	+1–4 ACC/ARI over SOTA on three intent datasets

These approaches consistently outperform counterparts lacking explicit cluster-preservation, especially in challenging situations (fine-grained classes, heavy background clutter, label noise, or manifold fragmentation). The inclusion of cluster-preservation objectives also boosts the interpretability and transferability of features in both supervised and unsupervised regimes.

5. Unified and Hierarchical Extensions

Cluster-preserving principles have also been applied to hierarchical and task-coordinated settings:

• Hierarchical cluster preservation: HCRL considers data with multilevel cluster hierarchies (e.g., hierarchical categories in text or images). Its deep generative construction enables soft (probabilistic) cluster identities at each abstraction level, allowing one to reconstruct data from any chosen hierarchy depth and to infer robust level-proportion features (Shin et al., 2019). Empirically, HCRL yields best-in-class likelihood and hierarchical F1 scores across image and text domains.
• Simultaneous class and cluster preservation: FSMLP_struct interleaves a classification loss with a Sammon-stress penalty on pairwise distances, jointly optimizing for features that support both discriminative (class) and unsupervised (cluster) tasks. The result is improved clustering performance with minimal sacrifice in supervised accuracy—particularly on modalities such as hyperspectral imaging or small structured datasets (Das et al., 2023).
• Generalization under structural shift: CIT-GNN augments graph neural networks with an explicit inter-cluster transfer mechanism, systematically reassigning nodes among clusters via mean/variance re-centering while preserving their cluster-independent components. This decorrelates class assignment from cluster idiosyncrasies and significantly improves generalization to new graphs under structural perturbations (Xia et al., 2024).

6. Visualization and Geometry-Preserving Embeddings

Cluster-preserving embedding is especially vital for interpretable visualization and structure-rich data analysis. Methods such as EmbedOR and cl-MDS are designed to preserve intra-cluster proximity and avoid the frequent fragmentation and artificial separation observed in embeddings produced by t-SNE, UMAP, or vanilla MDS. These techniques explicitly bias the learning of the embedding metric so that both local and global structure survive—yielding both robust cluster integrity and transparency into multi-scale data organization (Saidi et al., 3 Sep 2025, Hernández-León et al., 2022).

7. Practical Guidelines and Trade-offs

Cluster-preserving architectures typically require careful selection of regularization strengths, mixture weights, or degree-of-tensorization (in cluster-specific models) to balance model capacity, interpretability, and computational cost. Overestimating the number of clusters in cluster-specific formulations leads to redundant heads, while underestimation causes merging of separable clusters. Empirical observations across methods consistently show that explicit cluster-preservation not only enhances downstream clustering, but also denoising, transfer tasks, and geometry-consistent visualization (Sabanayagam et al., 2024). Model-agnostic meta-algorithms such as the (partial) tensorization scheme enable retrofitting of cluster-preserving structure into existing AE, VAE, or contrastive pipelines with minimal code changes.

• "Clustering with Similarity Preserving" (Kang et al., 2019)
• "Cluster Specific Representation Learning" (Sabanayagam et al., 2024)
• "InfoNCE Loss Provably Learns Cluster-Preserving Representations" (Parulekar et al., 2023)
• "RoNID: New Intent Discovery with Generated-Reliable Labels and Cluster-friendly Representations" (Zhang et al., 2024)
• "Enhancing Clustering Representations with Positive Proximity and Cluster Dispersion Learning" (Kumar et al., 2023)
• "RTD-AE: Learning Topology-Preserving Data Representations" (Trofimov et al., 2023)
• "ClusterFit: Improving Generalization of Visual Representations" (Yan et al., 2019)
• "Subspace Clustering via Graph Filtering" (Ma et al., 2021)
• "EmbedOR: Provable Cluster-Preserving Visualizations with Curvature-Based Stochastic Neighbor Embeddings" (Saidi et al., 3 Sep 2025)
• "Feature selection simultaneously preserving both class and cluster structures" (Das et al., 2023)
• "Learning Invariant Representations of Graph Neural Networks via Cluster Generalization" (Xia et al., 2024)
• "Consistent Representation Learning for High Dimensional Data Analysis" (Li et al., 2020)
• "Cluster-based multidimensional scaling embedding tool for data visualization" (Hernández-León et al., 2022)
• "Learning a Deep Part-based Representation by Preserving Data Distribution" (Qin et al., 2020)
• "Hierarchically Clustered Representation Learning" (Shin et al., 2019)
• "Similarity Preserving Representation Learning for Time Series Clustering" (Lei et al., 2017)
• "Clustering-friendly Representation Learning for Enhancing Salient Features" (Oshima et al., 2024)