Matrix Recovery Algorithm for Reconstructing Mixing Matrices From Raw Observations and Ordinary Least Squares Unmixed...

Flow cytometric analysis requires accurate spectral unmixing using a mixing matrix ( $M$ ) to deconvolve overlapping fluorescent signals into individual fluorochrome abundances. However, current Flow Cytometry Standard (FCS) data formats inadequately support the archival storage of $M$ alongside experimental data, significantly compromising analytical reproducibility and method transparency. We present a novel matrix recovery (MR) algorithm that computationally reconstructs the original $M$ from archived raw detector observations and previously calculated unmixed abundance values. For ordinary least squares (OLS) unmixing methodologies, our algorithm achieves mathematically exact recovery with numerical errors below ${10}^{-4}$ using the closed-form solution $M={\left(A\cdot O^{+}\right)}^{+}$ , where matrix $A$ contains unmixed abundances and matrix $O$ contains raw observations. Comprehensive validation across six commercial cytometric platforms, encompassing detector arrays ranging from 55 to 182 channels and endmember panels from 10 to 47 fluorochromes, confirmed algorithmic accuracy for OLS-based compensation systems. While weighted least squares (WLS) recovery remains theoretically feasible, computational complexity renders current implementations intractable for practical applications. This matrix recovery approach provides a critical tool for retrospective analysis when manufacturers fail to provide standardized $M$ storage, though we emphasize that systematic inclusion of $M$ within FCS file specifications remains the optimal long-term solution for ensuring analytical reproducibility in flow cytometry.