Cell-type deconvolution

We define spatial deconvolution as the problem of inferring cell-type abundances at each location from the observed omics data, with or without cell-type reference information from external data. The task is especially relevant for spatial techniques with limited resolution, including 10x Visium, spatial-CUT&Tag [4] and spatial-ATAC-seq [5], where each profiled location might consists of cells from multiple cell types. The deconvolution model is usually determined by the generative model of the observations YG×S. Most deconvolution methods assume a linear relationship between observation and cell-type abundance:

Here, YG×S is the observed activities of G features at S spots, RG×C is the expected reference activities of G features in C cell types, XC×S is the abundance of C cell types at S spots, σ02 is the sampling variability, and g is the data generative function that introduces additional noise such as location-specific biases. Without loss of generality, we follow the same linearity assumption and extend existing deconvolution models to leverage neighborhood information by imposing the spatial prior on the abundance of each cell type:

Here, Σc is the prior covariance structure, and 1/σ12 represents the strength of prior. When the reference RG×C is known, usually from a paired single-cell dataset, we solve XC×S by minimizing the following regularized factorization problem:

where LdeconvY,Xβ is the loss specified by the corresponding deconvolution model. We implemented four spatially aware deconvolution models including nonnegative least squares (NNLS), support vector regression (SVR), dampened least squares (DWLS), and log-normal regression (LNR). Further details can be found in the Additional file 2: Supplementary Notes. When the reference is unknown, the above deconvolution can be solved via matrix factorization, which is also a special case of the dimensionality reduction task, as will be discussed in the next section.