Metric Embedding Initialization

• Metric embedding initialization is a set of strategies that construct mappings from metric spaces to Euclidean or tree-structured spaces, facilitating effective clustering.
• The SDP-based formulations preserve graph structure and enable noise addition to meet differential privacy, ensuring low clustering error.
• HST-based initialization coupled with privacy-preserving k-median clustering yields interpretable, multi-scale clusters that support reliable analysis.

Metric embedding initialization refers to the set of mathematical and algorithmic strategies used to construct an initial mapping of elements from a metric space (such as the vertex set of a graph, data points in a network, or nodes in a similarity graph) into a target metric space—most commonly a Euclidean or tree-structured metric—so as to better facilitate downstream tasks such as clustering, particularly under constraints of differential privacy and interpretability. In the context of differentially private graph clustering, metric embedding initialization is crucial for finding effective representations that allow clustering algorithms to achieve low error with provable privacy guarantees and interpretable cluster configurations (You et al., 7 Sep 2025).

1. Semidefinite Programming for Metric Embedding

The approach begins by formulating an SDP whose solution yields a vectorial embedding of the graph nodes designed to preserve similarity and structural relations. Each node $u$ is represented as an embedding vector $\bar{u} \in \mathbb{R}^d$, and the pairwise similarity is modeled by the inner product $\langle \bar{u}, \bar{v} \rangle$, with pairwise Euclidean distances $\|\bar{u} - \bar{v}\|_2$ capturing dissimilarity.

The SDP optimization is given by:

$\begin{aligned} \min_{\{\bar{u}\}} \quad &\sum_{(u,v) \in E} \|\bar{u}-\bar{v}\|_2^2 +\frac{2\sum_{u,v \in V}\langle \bar{u},\bar{v}\rangle^2\,d_G(u)\,d_G(v)}{\lambda m}\[1mm] \text{s.t.} \quad &\sum_{u,v\in V} \|\bar{u}-\bar{v}\|_2^2\,d_G(u)\,d_G(v) \ge 2bm^2,\[1mm] &\langle\bar{u},\bar{v}\rangle \ge 0,\quad \forall\,u,v\in V,\[1mm] &\|\bar{u}\|_2^2 = 1,\quad \forall\,u\in V, \end{aligned}$

where $d_G(u)$ is node degree, $m$ the number of edges, and hyperparameters $\lambda, b$ control regularization and volume balance.

This can be equivalently formulated in terms of the Gram matrix $\mathbf{X}$, yielding a matrix SDP:

$$\min_{\mathbf{X} \in \mathcal{D}} \;\langle L_G, \mathbf{X} \rangle + \frac{n}{\lambda m}\|D_G^{1/2}\mathbf{X}D_G^{1/2}\|_F^2,$$

where $\mathcal{D}$ requires $\mathbf{X} \succeq 0$, $\mathbf{X} \geq 0$ entrywise, and normalized diagonals.

This SDP embeds the graph structure into a low-dimensional metric space in a way that preserves relevant connectivity while enforcing geometric and nonnegativity constraints.

2. Differential Privacy in the Embedding Stage

In order to provide strong privacy guarantees (specifically, $(\varepsilon, \delta)$-differential privacy), noise is injected into the embedding process. This is achieved by adding Gaussian noise to the computed Gram matrix or the resulting spectral embeddings, leveraging the bounded sensitivity of these objects in the Frobenius norm. Calibration of the scale of the added noise is determined by privacy parameters $\varepsilon, \delta$, the node set size, and the embedding dimension.

By introducing controlled Gaussian perturbation, the structure of the graph is preserved as much as possible while obfuscating any single individual’s presence or connectivity, meeting differential privacy requirements.

3. HST-Based Initialization for Clustering

Building upon the metric embedding, the next stage employs a Hierarchically Well-Separated Tree (HST) to initialize cluster centers. An HST recursively partitions the embedded metric space into clusters at multiple scales, producing a tree whose nodes correspond to nested subsets of the data. For each tree node $v$, a score is computed:

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

where $N_v$ is the (noisy) number of points in the subtree rooted at $v$ (Laplace noise is added here for increased privacy), and $h_v$ is the depth. The $k$ cluster centers correspond to the highest scoring nodes, provided no two are in an ancestor-descendant relation.

This HST-based initialization achieves two objectives: (1) clustering seeds are well separated and capture multi-scale density structure, and (2) the initialization step, being tree-based, is naturally interpretable—tree partitions can be visualized and traced hierarchically.

4. Privacy-Preserving k-Median Clustering Integration

The metric embedding and HST initialization produce candidate centers for k-median clustering, which aims to minimize the sum of distances from data points to their assigned centers. Privacy is maintained in the assignment phase by means of the exponential mechanism, which ranks possible cluster centers for each data point on a privatized utility score. The entire process allows clustering to operate in a lower-dimensional, well-behaved metric space, with both initialization and iterative assignment steps protected by formal privacy mechanisms.

This configuration significantly reduces errors due to privacy-induced noise, since initialization already finds "informative" clusters and subsequent private assignment is made easier by the geometry established in the initial embedding.

5. Interpretability and Comparative Explanations

The tree-based initialization and clustering result in an inherently interpretable model. Interpretability arises from the explicit hierarchical decomposition: clusters defined in the HST correspond to nested, inspectable regions. For any query or analysis of cluster membership, comparative explanations are generated by evaluating the difference between the cluster assignment cost (using the clustering centers) and a counterfactual cost (using fixed centroids), thus providing transparent reasoning about why specific points are grouped as they are.

Such contrastive explanations leverage the underlying metric structure and the explicit, modular form of the clusters, making this approach particularly suitable for applications in sensitive domains demanding explainability.

6. Experimental Outcomes and Performance

On several benchmark datasets (USPS, Reuters, DBLP, ACM, CiteSeer, HHAR), this integrated pipeline consistently achieves improvements in both clustering metrics (NMI, Purity, Accuracy, ARI, F1) and initial/final clustering cost compared to alternative differentially private algorithms such as DPFN and BR-DP. The embedded, HST-initialized configuration ensures that even under high privacy constraints, the clustering remains both efficient and accurate, while providing clear interpretability and robust privacy (You et al., 7 Sep 2025).

7. Implications and Future Directions

This methodology demonstrates that the careful orchestration of SDP-based metric embedding, HST initialization, and private k-median clustering forms an effective foundation for privacy-preserving, explainable graph clustering. The explicit, interpretable structure and robust privacy guarantees suggest extensions to more complex data domains, and further research may focus on scaling these methods, tightening privacy bounds, or enriching the interpretability mechanisms for even richer explanations.