Latent-Space K-Means Clustering

• Latent-space K-means is a clustering method that analyzes data in transformed latent spaces, capturing both point centers and affine subspace structures.
• It leverages dimensionality reduction techniques such as autoencoders and PCA to reveal nonlinear or manifold patterns for more robust clustering.
• The approach unifies principles from k-means, PCA, and manifold learning by using a tunable weight vector to balance point-based and subspace modeling while emphasizing geometric structure.

Latent-space K-means is a collection of methodologies that generalize and extend the classical k-means clustering framework to operate effectively in latent, feature, or nonlinear spaces derived from data via embedding, dimensionality reduction, or deep learning techniques. These approaches leverage the properties of latent spaces to overcome limitations of k-means in the original input space, modeling not only cluster locations but often their geometric structures, and enhancing the interpretability and robustness of clustering results.

1. Mathematical Framework and Generalization

Latent-space K-means formalizes the idea of performing cluster analysis not directly in the original ambient data space $\mathbb{R}^N$, but in a lower-dimensional or transformed “latent” space produced by some feature extractor or mapping—such as an autoencoder, kernel function, or manifold embedding.

A central contribution is the unification of k-means clustering and principal component analysis (PCA) as special cases of a more general approximation scheme (1109.3994). Specifically, the $(\omega,k)$-means algorithm approximates the data with $k$ affine subspaces of dimension $n$ in the latent space:

• For $n=0$, this reduces to classical k-means (point centers).
• For $k=1$, it recovers PCA (single best affine subspace).

The clustering objective becomes minimizing the sum of squared “reconstruction errors”:

$$E_{\omega}\left(S_1,\ldots,S_k\right) = \sum_{i=1}^k E_{\omega}(S_i,M_n(S_i))$$

where $M_n(S_i)$ is the optimal $n$-dimensional subspace for cluster $S_i$, computed via the Karhunen–Loève Transform (PCA).

Distances within the latent space are often measured by a weighted composite:

$$\mathrm{DIST}_{(\omega)}(x;v) = \left( \sum_{j=0}^n \omega_j \cdot dist\left(x;\mathrm{aff}(v_0,\ldots,v_j)\right)^2 \right)^{1/2}$$

with $\omega$ a tunable weight vector, providing a continuous path between classical clustering and pure subspace modeling.

2. Application of Latent-Space K-means

The practical application typically involves:

• Mapping data (e.g., via an autoencoder or nonlinear manifold method) into a latent space.
• Choosing the number of clusters $k$ and subspace dimension $n$.
• Initializing cluster prototypes, possibly with k-means++ or other advanced schemes.
• Assigning each point in the latent space to the nearest subspace/cluster.
• Iteratively recomputing subspaces and reassigning points until convergence.

Such a procedure allows clustering not only by proximity (as in k-means), but also by capturing linear or affine structures inside clusters. For instance, elongated or manifold-shaped clusters are often better approximated by subspaces than by point centers; the affine structure is explicitly modeled within each cluster (1109.3994).

When the latent space is constructed via nonlinear generative models, the geometry may be Riemannian rather than Euclidean. In these cases, clustering requires adapting both the assignment step (using geodesic—not Euclidean—distances) and the centroid update (using Fréchet means on the manifold) (1805.07632).

3. Theoretical Properties and Advantages

Latent-space K-means combines characteristics of clustering and dimensionality reduction:

• Richer cluster modeling: Each cluster can capture intrinsic geometry, not just location.
• Flexibility: The weight vector $\omega$ allows interpolation between point-based and subspace-based clustering.
• Unified perspective: Bridges techniques from manifold learning, PCA, kernel methods, and clustering.
• Adaptability: Particularly effective when the latent space is designed to reveal or linearize structure (as in manifold methods or carefully learned neural embeddings).

However, these advantages are accompanied by several practical limitations:

• Increased computational cost: Estimating a subspace (e.g., via PCA) for each cluster per iteration is more expensive than computing mean centers (1109.3994).
• Initialization sensitivity: As with classical k-means, but sometimes more severe due to local minima in the increased parameter space.
• Parameter selection: The choice of $n$ and $\omega$ can dramatically impact outcomes, with no universally optimal setting.

When the true underlying latent structure is highly nonlinear, affine subspace approximations may remain inadequate, even if the latent space attempts to linearize the data.

4. Implementation Strategies and Computational Considerations

Implementing latent-space K-means requires efficient estimation of both cluster assignments and subspaces for potentially high-dimensional latent spaces. Key steps include:

• Latent space extraction (e.g., via deep learning, PCA, or kernel trick).
• Cluster/affine subspace initialization, with k-means++ or stochastic seeding to improve convergence.
• Looping over cluster assignments: For each point, compute the squared error to each cluster’s subspace using $\mathrm{DIST}_{(\omega)}(\cdot)$, and assign to the closest.
• Subspace update: For each cluster, perform PCA on assigned points to estimate the optimal $n$-subspace or affine component.
• Stopping criterion: Iteration terminates when decreases in total energy $E_{\omega}$ fall below a threshold.

This process can be computationally intensive, especially as both the number of clusters and the subspace dimension increase. Practical acceleration techniques include:

• Approximate nearest subspace methods for assignment.
• Efficient implementations of PCA updates, possibly leveraging incremental or batch processing.
• Parallelization across clusters or data partitions.

5. Comparison to Related Clustering Paradigms

Latent-space K-means extends and generalizes several traditional approaches:

• Compared to classical k-means, it can capture not just tight spherical clusters but also distributed, elongated, or manifold-like groups.
• Unlike global PCA, which provides only a single subspace for the entire dataset, latent-space K-means affords multiple local subspace models.
• With appropriate choices of $\omega$ and $n$, the method interpolates between these paradigms—offering a spectrum of models to accommodate various data characteristics (1109.3994).

When compared to kernelized or manifold-centric clustering, latent-space K-means can either employ the kernel trick to operate in high-dimensional feature spaces, or be adapted for Riemannian manifolds, using geodesic distances and Fréchet means to update centroids (1805.07632).

6. Practical Implications and Use Cases

Latent-space K-means finds broad application in domains where latent representations are meaningful—such as unsupervised learning with autoencoders, manifold learning, scientific data analysis, and exploratory data mining. The approach is especially well-suited when:

• Clusters have significant internal structure.
• The data is embedded via representation learning into latent spaces specifically designed to contain more separable, structured, or semantically meaningful groupings.

Notable impacts include:

• Enhanced clustering quality for data with affine or manifold cluster organization.
• Improved data exploration and visualization due to richer cluster modeling.
• More effective compression and summarization, capturing both mean and direction of variation per cluster.

Limitations remain concerning scalability and parameter dependence. Nevertheless, the approach serves as a bridge between modern representation learning and classical clustering, enabling richer, more nuanced grouping in latent spaces.