Spatial Transcriptomics Tensor Decomposition
Suppose we rearrange the gene expression matrix by expanding the first (spatial) dimension into 2-dimensional grid, so that
\[ A_{ijk} = \text{probability of a read being of gene type } k \text{ at coordinate } (i,j). \]
For each coordinate \((i,j) \in \mathbb R^{m_1 \times m_2}\) , \(A_{ij:} \in \mathbb R^n\) is the corresponding gene expression vector. We expect that though each gene expression vector contains information about relatively many (\(n\)) genes, there are relatively few genes that actually characterize the cell. As such, we assume that any \(A_{ij:}\) can be expressed as a linear combination of relatively few (\(r_g << n\)) basis vectors \(u_1, \dots, u_{r_g}\). Namely, \[ A_{ij:} = \sum_{l =1}^{r_g} v_{ijl} u_l. \]
In other words, \[ A = VU^T, \] where \(U = \matrix{ [u_1 & \dots & u_{r_g} ]} \in \mathbb R^{n \times r_g}\) is the basis matrix and \(V\) is the tensor made up of \(r_g\) horizontally stacked matrices.
For every \(l\), \(v_{::l}\) is a form of image, and is therefore spatially low rank. Question, this is now an “image” in gene space but is this still going to be low rank? perhaps not?
Now suppose we want to doubly compress \(A\), i.e. use the assumption about the gene space as well as the image space being low dimensional. We can write this as full Tucker decomposition:
\[ A = G \times_1 K \times_2 J \times_3 U \] where \(G \in \mathbb R^{r_x \times r_y \times r_g}, U \in \mathbb R^{n \times n_g}_{(+)}, K \in \mathbb R^{m_1 \times r_x} , J \in \mathbb R^{m_2 \times r_y}\).
Suppose \(Y\) is the observed noisy tensor which we pre-regularize using a KDE transformation step.
Then we model the problem as follows
\[ \min_{G, K, J, U} ||Y - G \times_1 K \times_2 J \times_3 U ||_2^2. \]
and solve it using block coordinate descent. We might want to introduce some scaling constraint to make sure the factors don’t get too small/large. Additionally we may want to enforce non-negativity of \(U\), depending what the biological interpretation is of the basis vectors that comprise the columms of \(U\).