Computing Partial Quaternion Eigenpairs with Quaternion Shift

Abstract

The existing shift technique for quaternion matrix computation is to use real shifts rather than quaternion shifts, because quaternions are multiplicatively non-commutative in general. This becomes the obstacle to the development of fast and stable algorithms for computing partial quaternion eigenpairs. To overcome this difficulty, a new inverse quaternion iteration with quaternion shift is derived for solving general right quaternion eigenvalue problem. Fixed and updated quaternion shift techniques are developed to speed up the convergence. The advantage of quaternion shifts is explained geometrically, as well as the fact that real shifts may lead to wrong results. The convergence theory is built in a different way to that in real and complex cases. Numerical experiments on simulated data and color face database demonstrate the efficiency and superiority of the newly proposed method over the state-of-the-art algorithms.

Appendices

Quaternion Power Method and Inverse Quaternion Iteration with L$_2$-norm

New versions of quaternion power method and inverse quaternion iteration are proposed by substituting L$_\infty$-norm with L$_2$-norm. Their pseudo codes are given in Algorithms 5 and 6, respectively.

Algorithm 5

(QPM-L$_2$) For a given general quaternion matrix $\textbf{A}\in \mathbb {Q}^{n\times n}$, this algorithm computes the largest eigenpair $(\lambda , \textbf{u})$ with an initial L$_{2}$-norm unit vector $\textbf{u}_0\in \mathbb {Q}^{n}$.

Algorithm 6

(IQI-L$_2$) For a given general quaternion matrix $\textbf{A}\in \mathbb {Q}^{n\times n}$, this algorithm computes the smallest eigenpair $(\lambda , \textbf{u})$ with an initial L$_{2}$-norm unit vector $\textbf{u}_0\in \mathbb {Q}^{n}$.

Shifted Inverse Quaternion Iteration with L$_2$-norm

We substitute L$_{\infty }$-norm with L$_2$-norm in the shifted inverse quaternion iterations (Algorithms 3 and 4). Their pseudo codes are shown in Algorithms 7 and 8. Notice that the implement of steps 5–7 in Algorithms 7 and 8 is not equal to line 2 of Algorithm 6. The reason is that we need not get the exact solution of quaternion linear system in the inner loop.

Algorithm 7

(IQIFs-L$_2$) IQI-L$_2$ with a fixed shift.

Algorithm 8

(IQIAs-L$_2$) IQI-L$_2$ with an alterable shift.

Applications

Now we apply the proposed shifted inverse quaternion iteration algorithms to color image reconstruction.

1.1 Subspace Computation with Deflation

Assume that we have achieved the first projection vector ${w}_1$ that is calculated by the inverse quaternion iteration with a quaternion shift. The second projection vector ${w}_2$ is computed in a similar way

$$\begin{aligned} \textbf{F}^{deflated}=\textbf{F}(\textbf{I}-\textbf{W}\textbf{W}^{*}). \end{aligned}$$

(3.1)

It is worth to mention that the projection vector ${w}_2$ at each iteration have to be orthogonalized against the previous ${w}_1$ with a Gram-Schmidt procedure. We calculate r projection vectors by weights, and write $\textbf{W}=[{w}_1, {w}_2, \cdots , {w}_r]$, $1\le r <n$. After deflating the sample by r directions, it is natural to reach that

$$\begin{aligned} \textbf{F}^{deflated}{w}_j=\textbf{F}(\textbf{I}-\textbf{W}\textbf{W}^{*}){w}_j, \end{aligned}$$

(3.2)

where $j=1,2,\cdots ,r+1$, and $\textbf{F}^{deflated}{w}_j$ are zero, $j=1,2,\cdots ,r$. If not, the feature information may be lost due to the interference from other directions. Set $j=r+1$, (3.2) suggests that the corresponding vector is not linearly represented, i.e., the freshly computed projector is not in the subspace generated by the projected vectors. Therefore, we must orthogonalize the computed projector against the known ones.

1.2 Color Image Reconstruction with Projection

Let $\textbf{F}_1, \textbf{F}_2, \cdots , \textbf{F}_l\in \mathbb {Q}^{m\times n}$ be l training samples that are represented by quaternion matrices, whose mean is noted as

$$\begin{aligned} \mathbf{\Psi }=\frac{1}{l}\sum _{s=1}^l \textbf{F}_s\in \mathbb {Q}^{m\times n}. \end{aligned}$$

The eigenface subspace is defined by $\textbf{W}=[{w}_1, {w}_2, \cdots ,$ ${w}_r]$. With computed the projections of l training images in the subspace $\textbf{W}$, all we need to do is to make the projection to the subjection to the subspace, i.e.,

$$\begin{aligned} \textbf{F}_s^{rec}=(\textbf{F}_s-\mathbf{\Psi })\textbf{W}\textbf{W}^{*}+\mathbf{\Psi }. \end{aligned}$$

In other words, we reconstruct the image only by the first r vectors. It should be noted here that there is no need to amplify the effect of features in the process of reconstruction, so the projection matrix $\textbf{W}$ is unweighted. The process mentioned above is simplified as Algorithm 9.

Algorithm 9

Inverse quaternion iteration with a shift for color image reconstruction