New Metric NVE Validates Biclustering with Separability and Coverage

Biclustering, or co-clustering, is a technique that simultaneously groups rows and columns of a data matrix to identify coherent submatrices. Unlike traditional clustering, where validation metrics often rely on compactness and separation in a single feature space, biclustering validation is more complex due to its two-dimensional structure. Existing popular internal measures, such as Mean Squared Residue (MSR) and Virtual Error (VE), primarily focus on the coherence within a bicluster. However, these coherence-based metrics do not directly evaluate whether the discovered biclusters are distinct from each other or if they collectively explain a significant portion of the data. To address these gaps, this research proposes Normalised Virtual Error (NVE). NVE extends the Virtual Error by introducing a super-bicluster normalization strategy, which allows it to assess the relative separability and redundancy among biclusters. Furthermore, a coverage-adjusted variant, NVE_cov, is introduced to penalize solutions that achieve low error by only selecting very small, unrepresentative submatrices. Through synthetic benchmarks and yeast gene-expression datasets, the study demonstrates that NVE is sensitive to redundant and poorly separated biclusters, while NVE_cov helps re-rank solutions based on their data coverage. These NVE-based measures offer valuable complementary criteria for internal biclustering validation, especially when considering coherence, separability, and coverage together.