Can Metadata Fix the Gauge? Calibration Turns Sparse Multi-Source Learning from Sparse PCA into Sparse Mean Recovery

Summary

Source labels tell us where a sample came from, but not how that source oriented its coordinates. In a sparse Gaussian multi-source model with source-specific sign gauges, that distinction changes the learning problem. Without calibration, pooling across sources erases the mean signal, and the source-labelled experiment reduces to sparse PCA. With task-independent relative-gauge metadata, synchronization aligns sources and the same observations become a sparse mean problem. We formalize the value of metadata through AlignedMass, which quantifies how much first-order signal a metadata-induced alignment restores. Source labels create only limited AlignedMass, whereas valid calibration can create AlignedMass proportional to the full sample size. This yields a statistical separation in micro-source regimes and, under standard sparse-PCA hardness assumptions, a computational separation as well. Synthetic experiments check the predicted scaling; controlled real-feature and physical sensor studies examine sign cancellation, separately acquired calibration, and reuse across tasks.