tensor of $\mathbf{WCM}(\mathbf{0})$ from the tensor of $\mathbf{WCM}(\Delta \mathbf{p})$, the resultant tensor, $\mathbf{Y}$ consists of coefficient values for pixel intensities. Most importantly, $\mathbf{Y}$ is now independent of pixel intensities.

Symbolic Terms {#subsec:symb_terms}
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Referring to (\[eq:jd\_solve\]), two terms have to be computed to learn the LP, which are (1) $\mathbf{P} \mathbf{E}^T$, which is a linear combination of pixel intensities, and (2) $\mathbf{E} \mathbf{E}^T$, which is a quadratic combination of pixel intensities. In order to create a symbolic linear predictor, we replace the error matrix $\mathbf{E}$ with the tensor $\mathbf{Y}$ from Sec. \[subsec:symb\_err\]. We first define the following index variables:

-   $a \in {1, \cdots, 6} \rightarrow$ index for the parameter update $\Delta \mathbf{p}$,

-   $b \in 1, \cdots, n \rightarrow$ index for the pixel location in the n-vector $\mathbf{t}$ (the rasterized template),

-   $c \in 1, \cdots, m \rightarrow$ index for the sample warps, and

-   $d \in 1, \cdots, l \rightarrow$ index for the l-vector $\mathbf{u}$ (the rasterized bounding box).

With these indices, we can then compute a symbolic linear tensor $\mathbf{L} \in \mathbb{R}^{6 \times n \times l}$ as $$\mathbf{L}_{a,b,d} = \sum_{c=1}^m \mathbf{P}_{a,c} \mathbf{Y}_{b,c,d}.$$

For the quadratic term, $\mathbf{E} \mathbf{E}^T$, we can compute a symbolic quadratic tensor, $\mathbf{Q} \in 