“…, m} for a positive integer m. We shall assume the cell-type proportions ⇡ ik 's in (2) are given in the ensuing development, and later demonstrate in experiments that our method is not sensitive to uncertainties in ⇡ ik 's; see discussions in Section 6. To infer ⇡ ik 's from bulk samples, many methods have been developed that utilize cell type marker genes (i.e., genes that are only highly expressed in one cell type of interest) with expressions profiles gathered from pure cell types (Newman et al, 2015;Li et al, 2016) or single cell RNA-seq data (Wang et al, 2019;Newman et al, 2019;Jew et al, 2020;Dong et al, 2021;Yang et al, 2021). In these methods, the proportions ⇡ ik 's are estimated by, for example, nonnegative least squares (Wang et al, 2019) or support vector regression (Newman et al, 2019).…”