HST-Based Initialization for Clustering

• HST-based initialization method is a family of algorithms that use hierarchically well-separated trees to embed complex metric spaces for effective clustering seed selection.
• It employs a two-stage center selection process that guarantees dispersed and representative initial centers, offering improved performance over traditional methods.
• By integrating differential privacy through calibrated noise, the method ensures privacy compliance while sustaining strong theoretical performance in k-median and graph clustering tasks.

The HST-based initialization method encompasses a family of algorithms employing Hierarchically Well-Separated Tree (HST) constructions to facilitate initialization for clustering and related optimization problems, especially in complex metric spaces or under privacy constraints. These methods leverage hierarchical metric embeddings to produce high-quality initial cluster configurations or grouping seeds, often outperforming conventional random or sequential initialization approaches such as k-median++. HST-based strategies are especially prominent in k-median clustering, differentially private clustering, and explainable graph clustering, offering scalability, interpretability, and strong theoretical performance guarantees.

1. Hierarchically Well-Separated Tree Construction

The central feature of HST-based initialization methods is the transformation of the data space into a metric embedding represented by a tree structure known as a Hierarchically Well-Separated Tree (HST). The procedure decomposes an arbitrary metric space (including Euclidean and graph-induced metrics) into clusters at discrete hierarchical scales:

• The dataset $U$ of diameter $\Delta$ is recursively partitioned into balls (clusters) at each tree level, with ball radii halving at each step ($\Delta/2$, $\Delta/4$, ..., $\Delta/2^{L}$, where $L = \log_2 \Delta$).
• Each internal node of the HST represents a cluster (ball), with associated edge weights determined by the cluster's scale.
• At each level, unassigned points are grouped via padded decompositions, with representatives chosen randomly or according to predefined heuristics.
• The result is a tree in which the distance between any two data points, measured as the sum of edge weights along the tree path, approximates their original metric distance up to a known distortion factor.

This embedding is pivotal as it makes explicit both point densities and hierarchical groupings, which underpin the two-stage center selection algorithms utilized in initialization.

2. HST-based Initialization Algorithm for k-Median Clustering

Following HST construction, the initialization for the $k$-median clustering problem proceeds as follows (Fan et al., 2022):

1. Subtree (Coarse) Selection:
   • For each node $v$ in the HST, compute a score

   $\text{score}(v) = N_v \cdot 2^{h_v}$

   where $N_v$ is the (possibly perturbed for privacy) count of leaves under $v$, and $h_v$ is the level of $v$. - Select a candidate set $C_1$ of the $k$ highest-scoring nodes, excluding any ancestor-descendant relationships to ensure non-overlapping subtrees.

2. Leaf (Fine) Search:

   • For each $v \in C_1$, traverse downward (greedily following the child with the maximal score) until reaching a leaf node.
   • Collect these $k$ leaves as the initialized centers $C_0$.

This hierarchical selection ensures dispersal and density representation among seeds, with the guarantee that, under the 2-HST tree metric $\rho^T$, the clustering cost with centers $C_0$ satisfies:

$\text{cost}_k^T(U) \leq 10 \cdot \text{OPT}_k^T(U)$

When translated back to the original metric, the expected cost of using these centers is bounded by:

$$\mathbb{E}[\text{cost}_k(U)] = O(\min\{\log n, \log \Delta\}) \cdot \text{OPT}_k(U)$$

where $n$ is data size and $\text{OPT}_k(U)$ is the optimal $k$-median cost.

3. Differential Privacy in HST Initialization

Differential privacy is incorporated into the HST-based initialization in two primary steps (Fan et al., 2022, You et al., 7 Sep 2025):

• During HST construction, the count $N_v$ at each node is perturbed with Laplace noise calibrated based on the level $h_v$:

$$\hat{N}_v = N_v + \text{Lap}\left( \frac{2^{L-h_v}}{\epsilon} \right)$$

where $\epsilon$ is the privacy parameter.

• In the embedding phase for graph data, Gaussian noise is added to the representation matrices for spectral metric embedding, preserving $(\epsilon, \delta)$-differential privacy.

This two-stage privacy mechanism ensures that the selection of initial centers reveals limited information about sensitive subsets or clusters within the data. The resultant DP-HST initialization method provides provable privacy guarantees with an additive approximation error of the form:

$$\text{cost}_k(D) \leq 6 \cdot \text{OPT}_k(D) + O(\epsilon^{-1}\Delta k^2 (\log \log n)\log n)$$

with high probability, where $D$ is a differentially private demand set.

4. Integration with Metric Embedding and Explainable Clustering

For graph clustering, a metric embedding phase precedes HST construction (You et al., 7 Sep 2025):

• Spectral embedding is performed via semidefinite programming (SDP), resulting in low-dimensional representations $\overline{u}$ for each node $u$.
• Noise is injected to maintain privacy.
• The HST is built on these embeddings, and candidate centers are selected using the scoring rule as above.

Interpretability is achieved with a post-initialization explanation module: after clustering, for each point, the difference between the original and the fixed-center clustering cost is reported. This comparative analysis provides a "contrastive explanation" (Editor's term), revealing the marginal contribution of different centers and addressing explainability in high-noise, differentially private settings.

5. Computational Complexity and Performance

HST-based initialization methods are designed for practical efficiency:

• HST construction and subtree selection have complexity $O(dn\log n)$ for $n$ data points in $d$-dimensional space, competitive with or surpassing the $O(nk)$ cost of k-median++ when $k \gg \log n$.
• Experimental results across domains (Euclidean, graph, and imbalanced metric spaces) affirm that HST-based initial centers consistently exhibit lower initial and final clustering costs compared to classic methods, demanding fewer iterations for convergence and displaying robustness in both non-private and private regimes (Fan et al., 2022, You et al., 7 Sep 2025).

Multiple public benchmarks (USPS, Reuters, DBLP, CiteSeer, ACM, HHAR) show the approach yields superior metrics, including Normalized Mutual Information, Purity, Accuracy, ARI, and F1-score.

6. Applications, Limitations, and Future Developments

Applications span k-median clustering, graph clustering under privacy constraints, and explainable clustering in sensitive domains such as social network analysis and healthcare data. The hierarchical initialization supports scalability to moderate graph sizes and enhances the interpretability of clustering assignments.

Challenges remain in the sensitivity of the method to the choice of tree parameters and in the management of distortion introduced by the metric embedding, particularly in data with large intrinsic diameter or complex graph structure. Ongoing development is likely to focus on expanding scalability, further tightening privacy-utility tradeoffs, and integrating advanced embedding techniques for higher-dimensional and heterogeneous data.

7. Summary Table: HST-Based Initialization in Key Settings

Setting	Metric Embedding Type	Privacy Mechanism	Main Clustering Method
Euclidean/Generic Metric	2-HST via padded decomposition	Laplace noise on subtree counts	k-median / k-means
Graph Clustering	SDP spectral embedding + HST	Gaussian + Laplace noise	k-median
Explainable Clustering	HST on metric embeddings	Laplace noise, ranking explanation	k-median with contrastive explanation

In conclusion, HST-based initialization provides an algorithmically principled, empirically justified approach to initializing clustering solutions in challenging metric and privacy-sensitive scenarios. Its strengths in initialization quality, scalability, privacy integration, and explainability position it as a foundational method in modern unsupervised learning and private data analysis (Fan et al., 2022, You et al., 7 Sep 2025).