\mathbb{R}^{n\times n \times q}$, where $q \approx \frac{1}{2} l^2$ as the result of $\mathbf{E} \mathbf{E}^T$ is symmetric. The tensor $\mathbf{Q}$ is computed as $$\mathbf{Q}_{b_1, b_2, e(d_1,d_2)} = \sum_{c=1}^m \mathbf{Y}_{b_1, c, d_1} \mathbf{Y}_{b_2, c, d_2},$$ where $e = 1,2,\cdots q$, noting that every index in $e$ corresponds to a unique combination of the indices $d_1$ and $d_2$. Fig. \[fig:symb\_terms\] provides an illustration of the symbolic terms.

![An illustration of how (a) symbolic linear and (b) symbolic quadratic terms are obtained from the symbolic tensor $\mathbf{Y}$.[]{data-label="fig:symb_terms"}](SymbolicTerms.pdf){width="\columnwidth"}

Linear Predictor from Symbolic Terms
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Once the tensors $\mathbf{L}$ and $\mathbf{Q}$ are learned, they can be used to compute the Linear Predictor for different templates. The linear term, $\mathbf{P} \mathbf{E}^T$, can be computed by left-multiplying the l-vector $\mathbf{t}$ (see Sec. \[subsec:symb\_err\]) with the tensor $\mathbf{L}$: $$(\mathbf{P} \mathbf{E}^T)_{a,b} = \sum_{d=1}^l \mathbf{L}_{a,b,d} \mathbf{t}_d.$$ On the other hand, the quadratic term, $\mathbf{E} \mathbf{E}^T$, can be computed as $$(\mathbf{E} \mathbf{E}^T)_{b_1, b_2} = \sum_{d_1=1}^l \sum_{d_2=1}^l \mathbf{Q}_{b_1, b_2, e} \mathbf{t}_{d_1} \mathbf{t}_{d_2}.$$ We note that our approach is complementary with the DCT method described in Sec. \[subsec:efficient\_lin\_predictors\]. For the linear term, this is done by right multiplying $\mathbf{P} \mathbf{E}^T$ with the mapping $\mathbf{W}_r$, resulting in $\mathbf{P} \mathbf{E}^T \mathbf{W}_r^T$. For the quadratic term, 