Table 2.

Symphony minimal reference elements vs. additional components of Harmony reference.

Symphony minimal reference elements
$\mathbf{\mu }\in {\mathbb{R}}^{g\times 1}$	Reference gene means used to center each gene for PCA.
$\mathbf{\sigma }\in {\mathbb{R}}^{g\times 1}$	Reference gene standard deviations used to scale each gene for PCA.
$\mathbf{U}\in {\mathbb{R}}^{g\times d}$	Gene loadings to project from expression to PCA (or CCA) space.
${\mathbf{Y}}_{\cos }\in {\mathbb{R}}^{d\times k}$	Cluster centroid locations in harmonized PC space, L2-normalized.
${\mathbf{N}}_{\mathbf{r}}\in {\mathbb{R}}^{k\times 1}$	First reference compression term. Vector containing the size of each of the $k$ clusters, effectively the number of reference cells contained within them.
$\mathbf{C}\in {\mathbb{R}}^{k\times d}$	Second reference compression term.
Additional components of a full Harmony reference
${\mathbf{G}}_{\mathbf{r}}\in {\mathbb{R}}^{g\times n}$	Input reference gene expression matrix, prior to scaling.
${\mathbf{X}}_{\mathbf{r}}\in {\left\{0,1\right\}}^{b\times n}$	Design matrix assigning reference cells (cols) to reference batches (rows).
${\mathbf{B}}_{\mathbf{r}}\in {\mathbb{R}}^{k\times \left(1+b\right)\times d}$	3D tensor of the estimated parameters (betas and intercepts) of the linear mixture model for each of $k$ clusters for the reference cells.
${\widehat{\mathbf{Z}}}_{\mathbf{r}}\in {\mathbb{R}}^{d\times n}$	Integrated embedding for reference cells in harmonized PC (hPC) space, as output by Harmony.
${\mathbf{R}}_{\mathbf{r}}\in {\left[0,1\right]}^{k\times n}$	Soft cluster assignment of reference cells (cols) to clusters (rows), as output by Harmony. Each column is a probability distribution that sums to 1.