Algorithm 2: SSE subspace matching.

Require: q: window size, z: window-sliding size, $N_a$: the number of amino acids.

1: Create empty data matrices ${\mathbf{W}}_H$, ${\mathbf{W}}_B$, and ${\mathbf{W}}_L$.

2: Form first possible window.

3: for all $i\le N_a$ do

4:      Form $\left(qd\right)\times 1$ column vector $w_i={\left[C\alpha (i-q/2+1)\cdots C\alpha \left(i\right)\cdots C\alpha (i+q/2)\right]}^T$.

5:      Expand the corresponding data matrix by adding $w_i$ as a new column vector.

6:      Slide the window by z.

7: end for

8: for all data matrices do

9:      Compute SVD. For example, ${\mathbf{W}}_H=U_H{\Sigma }_H{V}_H^T$.

10:      Estimate the rank (using rank estimation technique in [40]). For example, $rank\left({\mathbf{W}}_H\right)=r_H$.

11:      Compute a subspace. For example, ${\mathcal{S}}_H=Span(u_{H_1},\cdots ,u_{H_{r_H}})$, i.e., the span                   of the first $r_H$ columns of $U_H$.

12: end for