SCIENCE CHINA Information Sciences, Volume 60, Issue 11: 112101(2017) https://doi.org/10.1007/s11432-016-9021-9

## A generalized power iteration method for solving quadratic problem on the Stiefel manifold

• AcceptedJan 10, 2017
• PublishedMay 5, 2017
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### Abstract

In this paper, we first propose a novel generalized power iteration (GPI) method to solve the quadratic problem on the Stiefel manifold (QPSM) as $\min_{W^{\rm~T}W=I}$ ${\rm~Tr}(W^{\rm~T}AW-2W^{\rm~T}B)$ along with the theoretical analysis. Accordingly, its special case known as the orthogonal least square regression (OLSR) is under further investigation. Based on the aforementioned studies, we then majorly focus on solving the unbalanced orthogonal procrustes problem (UOPP). As a result, not only a general convergent algorithm is derived theoretically but the efficiency of the proposed approach is verified empirically as well.

### References

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• Figure 1

(Color online) Comparisons of 6 different values of $\delta$ are performed under the GPI method with 3 different data matrices. (a) (50, 100, 30); (b) (80, 170, 80); (c) (40, 120, 60).

• Figure 2

(Color online) Comparisons of the convergence rate are performed for 6 approaches including EB [7], RSR [8], LSR [9], SP [10]LR [12]and our GPI method under 3 different data matrices. (a) (100, 10, 100); (b) (100, 15, 100); (c) (200, 15, 200).

• Figure 3

(Color online) Comparisons of PMCT [11]and GPI are performed over 2 different data matrices. (a) (900, 1000); (b) (2000, 1700).

• Figure 4

(Color online) CPU time comparison under case 3, dimension($n$, $m$).

• Table 1   Orders of complexity for 6 algorithms$^{\rm~a)}$
 RSR [8] LSR [9] SP [10] Order of the complexity $O(mnk+m^3kt)$ $O(mnk+m^3kt)$ $O(mnk+(m^2n+m^3)t)$ LR [12] EB [7] GPI (ours) Order of the complexity $O(m^2n+nk^2+m^2kt)$ $O(m^3+(m^2n+m^3)t)$ $O(m^2n+m^2kt)$ a) $t$ stands for the iteration number and $(n,m,k)$ stands for the dimension.
•

Algorithm 1 Generalized power iteration method (GPI)

Input: The symmetric matrix $A\in\mathbb{R}^{m\times~m}$ and the matrix $B\in\mathbb{R}^{m\times~k}$.

Initialize a random $W\in\mathbb{R}^{m\times~k}$ satisfying $W^{\rm~T}W=I_k$ and $\alpha$ via power method such that ${A}=\alpha~I_m-A\in\mathbb{R}^{m\times~m}$ is a positive definite matrix.

(

1) Update $M\leftarrow~2{A}W+2B$.

(

2) STATE Calculate $USV^{\rm~T}=M$ via the compact SVD method of $M$ where $U\in\mathbb{R}^{m\times~k}$, $S\in\mathbb{R}^{k\times~k}$ and $V\in\mathbb{R}^{k\times~k}$.

(

3) STATE Update $W\leftarrow~UV^{\rm~T}$.

Iteratively perform the steps (1)–(3) until the algorithm converges.

• Table 2   Comparisons of CPU time (s) under thesquare matrix $E$ for case 2$^{\rm~b)}$
 $(n=m=200)$ RSR[8] LSR[9] SP[10] LR[12] EB[7] GPI (ours) $k=10$ 64.940 23.426 2.386 0.541 0.337 0.228 $k=15$ 136.020 21.635 3.221 1.134 0.347 0.226 $k=20$ 229.851 20.560 5.054 1.806 0.445s 0.273 $(n=m=1000)$ RSR[8] LSR[9] SP[10] LR[12] EB[7] GPI (ours) $k=10$ – 842.849 132.232 3.869 11.440 1.290 $k=15$ – 851.231 196.761 5.180 12.534 1.434 $k=20$ – 860.746 260.132 7.700 12.625 1.575 b) Iteration stops when $\Vert~EQ_{i-1}-G\Vert_F^2-\Vert~EQ_i-G\Vert_F^2\leq\tau$ where $~\tau=10^{-3}$.
•

Algorithm 2 GPI for solving UOPP in (sect. 4.2)

Input: The matrix $E\in\mathbb{R}^{n\times~m}$ and the matrix $G\in\mathbb{R}^{n\times~k}$ where $m>k$.

Initialize $Q\in\mathbb{R}^{m\times~k}$ and $\gamma$ such that $Q^{\rm~T}Q=I_k$ and the matrix $\gamma~I_m-E^{\rm~T}E$ is positive definite, respectively.

While not converge do

(

1) Update matrix $M\leftarrow~2(\gamma~I_m-E^{\rm~T}E)Q+2E^{\rm~T}G$.

(

2) Calculate $U\in\mathbb{R}^{m\times~k}$ and $V\in\mathbb{R}^{~k\times~k}$ via the compact SVD of $M$ as $M=USV^{\rm~T}$.

(

3) Update $Q\leftarrow~UV^{\rm~T}$.

End while

Return $Q$.

• Table 3   Comparisons of CPU time (s) under the general dimension for case 2$^{\rm~b)}$
 Dimension $(n,m,k)$ RSR[8] LSR[9] SP[10] LR[12] EB[7] GPI (ours) $(5000,500,15)$ 713.156 528.034 450.028 20.709 16.554 3.581 $(10000,1000,30)$ – – – 56.772 191.970 9.384 $(3000,3000,90)$ – – – 186.125 395.401 17.320 $(30000,1500,30)$ – – – 306.132 1056.311 19.440 $(5000,4000,100)$ – – – 405.937 1187.512 30.128 $(100000,3000,50)$ – – – – – 215.173
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