Meta Clustering (metasnf) Framework

• Meta clustering (metasnf) is a framework that clusters multiple SNF solutions generated under varied hyperparameters to robustly explore data subtypes.
• It employs the Adjusted Rand Index to measure similarity between solutions, facilitating the selection of representative, stable clustering outcomes.
• The method integrates batch SNF execution with visual analytics, such as ARI heatmaps and alluvial plots, to validate and interpret multi-modal biomedical clusters.

Meta clustering, as operationalized in the metasnf R package, is a methodological framework for searching the space of clustering solutions by clustering the solutions themselves. It is specifically designed to address challenges inherent in multi-modal biomedical data integration, where conventional approaches relying on a single run of similarity network fusion (SNF) with fixed hyperparameters may not adequately capture the diversity and context-specific relevance of possible clusterings. By systematically sampling a large number of SNF configurations and organizing these solutions with respect to their mutual similarity, meta clustering facilitates rigorous exploration, validation, and selection of data-driven subtyping solutions (Velayudhan et al., 2024).

1. Mathematical Foundation of Similarity Network Fusion

Let $X^{(v)} \in \mathbb{R}^{N \times p_v}$, $v = 1, \ldots, V$ denote $V$ data-type matrices, each representing a different “view” (e.g., gene expression, methylation, imaging). SNF proceeds by computing a view-specific pairwise distance $D^{(v)}_{ij} = d(x^{(v)}_i, x^{(v)}_j)$ (with common metrics such as Euclidean or Gower), from which an affinity matrix $W^{(v)}$ is constructed:

$W^{(v)}_{ij} = \begin{cases} \exp(-D^{(v)}_{ij}/\alpha) & \text{if } i \text{ in %%%%5%%%%-NN of } j \text{ or vice versa} \ 0 & \text{otherwise} \end{cases}$

Each affinity matrix is symmetrized and normalized into a stochastic matrix $$P^{(v)} = D^{-\tfrac{1}{2}} W^{(v)} D^{-\tfrac{1}{2}}$$, where $D = \operatorname{diag}(W^{(v)}\mathbf{1})$. SNF then performs $T$ iterations of multi-view fusion, where at each step $t$:

$v = 1, \ldots, V$0

$v = 1, \ldots, V$1

with $v = 1, \ldots, V$2 ensuring row stochasticity. The final fused similarity network is $v = 1, \ldots, V$3. Clustering (e.g., spectral, hierarchical) is then applied to $v = 1, \ldots, V$4, with the number of clusters $v = 1, \ldots, V$5 typically determined by eigengap or rotation-cost heuristics.

2. Meta Clustering of SNF Solutions

Meta clustering, following Caruana et al. (2006), involves pooling $v = 1, \ldots, V$6 clustering solutions, each generated under a different randomization of SNF hyperparameters or data preprocessing regimes. For each $v = 1, \ldots, V$7:

• Hyperparameters $v = 1, \ldots, V$8 (e.g., $v = 1, \ldots, V$9, $V$0, SNF scheme choices, data-type dropout, clustering algorithm) are randomly sampled.
• SNF is applied, producing cluster assignments $V$1.

Pairwise solution similarity is measured by the Adjusted Rand Index (ARI):

$V$2

where $V$3 is the number of samples co-assigned to cluster $V$4 in $V$5 and $V$6 in $V$7.

This leads to an $V$8 ARI similarity matrix, which is subjected to a second-level clustering (e.g., hierarchical, using $V$9) to recover $D^{(v)}_{ij} = d(x^{(v)}_i, x^{(v)}_j)$0 "meta-clusters" of solutions. For each meta-cluster $D^{(v)}_{ij} = d(x^{(v)}_i, x^{(v)}_j)$1, the representative solution $D^{(v)}_{ij} = d(x^{(v)}_i, x^{(v)}_j)$2 maximizing within-cluster average ARI is selected:

$D^{(v)}_{ij} = d(x^{(v)}_i, x^{(v)}_j)$3

These representatives can be further analyzed with respect to domain-specific feature separation or stability.

3. Implementation: metasnf Workflow and Functionality

The typical metasnf workflow consists of:

1. Data Preparation: Input is a set of tidy data frames (one row per sample, no missing values, unique sample ID). The generate_data_list() function standardizes and packages these views.
2. Random Sampling of Hyperparameters: Using generate_settings_matrix(), users specify the number of SNF runs ($D^{(v)}_{ij} = d(x^{(v)}_i, x^{(v)}_j)$4), ranges for $D^{(v)}_{ij} = d(x^{(v)}_i, x^{(v)}_j)$5 (nearest neighbors, 10–100) and $D^{(v)}_{ij} = d(x^{(v)}_i, x^{(v)}_j)$6 (decay, 0.3–0.8), dropout schemes, and other SNF or clustering parameters.
3. Batch SNF Execution: batch_snf() executes all $D^{(v)}_{ij} = d(x^{(v)}_i, x^{(v)}_j)$7 SNF runs in parallel, outputting a solutions_matrix comprising settings, $D^{(v)}_{ij} = d(x^{(v)}_i, x^{(v)}_j)$8, and sample cluster assignments per run.
4. Meta Clustering: The pairwise ARI matrix is computed (calc_aris()), ordered for visualization (get_matrix_order()), and displayed as a heatmap (adjusted_rand_index_heatmap()). Users select meta-cluster partitions, and representative solutions are extracted (get_representative_solutions()).
5. Validation and Visualization: Functions are available for statistical and visual validation (silhouette, Dunn, Davies–Bouldin indices; separation $D^{(v)}_{ij} = d(x^{(v)}_i, x^{(v)}_j)$9-values; alluvial diagrams; co-clustering heatmaps). External matrices (clinical or molecular endpoints) can be integrated via, e.g., extend_solutions().

4. Visualization, Characterization, and Validation

metasnf provides an array of visualization and analytical endpoints critical for the interpretation of both clustering solution diversity and biological or clinical relevance:

• ARI Heatmaps for interactive meta-cluster annotation.
• Silhouette, Dunn, Davies–Bouldin indices to assess compactness and separation of clusters.
• Co-clustering Stability via resampling/subsampling protocols to quantify the consistency of cluster assignments across random data perturbations.
• Feature–Cluster Association Testing through Manhattan plots visualizing $W^{(v)}$0-values.
• Alluvial Plots facilitating understanding of cluster membership evolution across different cluster counts or parameter settings.

The underlying infrastructure leverages and extends R packages such as ComplexHeatmap, ggplot2, cluster, and clv.

5. Practical Considerations and Workflow Guidance

Key input requirements are clean, fully observed data matrices with unique sample identifiers. Hyperparameter recommendations include:

• $W^{(v)}$1 (nearest neighbors): 10–100
• $W^{(v)}$2 (affinity decay): 0.3–0.8
• $W^{(v)}$3 (fusion iterations): default 20
• Clustering algorithms: spectral (default), eigengap or rotation cost for choosing the number of clusters

Computational runtime scales with the number of runs, samples, and features for SNF ($W^{(v)}$4) and quadratically with the number of runs for ARI computation. Parallelization is supported for scalability.

The recommended pipeline is:

1. Data preparation ($W^{(v)}$5 generate_data_list())
2. Settings matrix construction ($W^{(v)}$6 generate_settings_matrix())
3. Batch SNF execution ($W^{(v)}$7 batch_snf())
4. ARI computation, solution meta-clustering, representative selection
5. Validation and visualization: cluster quality indices, feature separation, stability analysis, generalizability ($W^{(v)}$8 lp_solutions_matrix())
6. Iterative review of representative solutions in domain context

6. Significance and Use Cases

metasnf enables systematic exploration of subtyping solutions in multi-modal biomedical datasets, supporting robust optimization of clustering quality under multiple criteria. The meta clustering formalism addresses the instability and subjectivity inherent in single-run SNF and responds to the need for context-specific evaluation metrics over generic solution quality measures. It is applicable whenever: (a) the underlying data are heterogeneous or multi-view; (b) the space of parameter settings is large; and (c) high-stakes cluster interpretation (e.g., in disease stratification) requires comprehensive solution validation (Velayudhan et al., 2024).

A plausible implication is that this approach generalizes to any clustering framework where solution sampling and similarity scoring are meaningful and computationally tractable, particularly in complex biomedical and multi-modal contexts.