Non-Negative Blind Source Separation and the Identifiability of Cell-Type Deconvolution in Spatial Transcriptomics: A Mathematical Review
Year: 2026
Type: preprint
Abstract: Cell-type deconvolution in spatial transcriptomics is, in signal-processing terms, a non-negative blind source separation (BSS) problem: an observed mixture \(Y = {{AS} + E}\) must be resolved into a mixing matrix of cell-type proportions \(A\) and a set of source signatures \(S\), from counts alone. The biological literature has answered this with more than twenty competing algorithms and a series of benchmarks that agree on only one thing—no method wins consistently—without a formal account of why. This review supplies that account from the source-separation side. We show that the identifiability theory developed for BSS, independent component analysis, and non-negative matrix factorisation—and matured over two decades of hyperspectral unmixing—transfers directly to the deconvolution problem, because it is literally the same model on the same equation. Three conditions organise the entire field: full column rank of the signature matrix (reference completeness), which governs supervised methods; the condition number \(\kappa{(S)}\), which bounds noise amplification when two sources are nearly collinear; and a separability or minimum-volume condition, which governs the unsupervised case. We further show that the negative-binomial nature of count data makes the squared-error objectives most methods minimise a high-count approximation rather than the maximum-likelihood estimator. We locate the major published algorithms within this framework, demonstrate through a worked simulation that recovered-proportion error tracks the \(\kappa{(S)}\) bound, and show that each condition reproduces a documented benchmark observation— including the much-noted result that a plain regression keeps pace with half the dedicated tools. The practical payoff is that all three conditions are computable on the reference before any algorithm is run, turning method selection from a benchmark lookup into a short calculation.
Source: Qeios
DOI: 10.32388/eoizjd
Author: Amit Walinjkar
Institution: EES Research (United Kingdom)
Subfield: Molecular Biology
Domain: Life Sciences
Open Access status: diamond
OpenAlex ID: W7167798511