Linear-Algebraic Audit of Biomedical Matrices: Five Empirical Case Studies in Regularized Inference

High-throughput biomedical assays—ranging from gene expression microarrays to mass spectrometry and clinical registries—routinely exhibit severe rank deficiency (N ≪ P), zero-inflation, and extreme sample outliers. Standard chemometric and bioinformatics pipelines apply variance auto-scaling, simplex normalization, and bilinear projections (PCA/PLS-DA) without verifying operator conditioning or modal localization. Grounded in inverse problem theory, this technical report subjects five public biomedical datasets to a sequential four-step linear-algebraic audit evaluating residual Gram conditioning (κ), Effective Degrees of Freedom (DoF), singular spectra (Σ), and Participation Ratios (PR). Across five empirical regimes—Golub leukemia microarrays (72 × 7129), a large-scale oncology cohort (1545 × 10935), zero-inflated NCI Arcene mass spectra (200 × 10000), Pomeroy central nervous system tumor profiles (60 × 7129), and the Cleveland cardiology registry (270 × 13)—we demonstrate how uncalibrated preprocessing inflates background noise threefold, collapses leading components onto fewer than 7 outlier samples, and bends multi-batch artifacts into parabolic horseshoes. Removing just 0.6% of leverage outliers identified by PR restores sample coherence nearly ninety-fold (PR: 6.4 → 548.1), providing a rigorous diagnostic standard for biomedical data hygiene.