Cayley-transform-based gradient and conjugate gradient algorithms on Grassmann manifolds
Cayley-transform-based gradient and conjugate gradient algorithms on Grassmann manifolds
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DOI：
10.1007/s10444-021-09880-9
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发表时间：
2021-07
影响因子：
1.7
通讯作者：
Xiaojing Zhu;Hiroyuki Sato
中科院分区：
文献类型：
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作者：
Xiaojing Zhu;Hiroyuki Sato
本文研究了基于cayley变换的梯度算法和共轭梯度算法在Grassmann流形上的优化问题。在由李群作用构造的商流形框架中，我们重新审视了Grassmann流形上的Cayley变换作为一种缩回，并得到了这种缩回在低秩情况下的有效公式。我们还证明了这种缩回是Stiefel流形上的Cayley变换对水平空间的限制。为了发展格拉斯曼流形上的矢量传输，我们引入了商流形上的诱导矢量传输的概念。基于这一概念，得到了与Cayley变换相关的三个矢量传输。第一个矢量传输是传统的在水平空间上的正交投影，而另外两个矢量传输是本文新提出的。我们证明了其中一个新的矢量传输满足环与非展开条件，另一个是等距的。我们还简化了低秩情况下新向量输运的公式。通过Cayley变换和上述三种矢量传输建立了黎曼梯度和共轭梯度算法。对两个子空间均值问题的数值实验证明了所提算法的有效性。
In this paper, we study Cayley-transform-based gradient and conjugate gradient algorithms for optimization on Grassmann manifolds. We revisit the Cayley transform on Grassmann manifolds as a retraction in the framework of quotient manifolds constructed by Lie group actions and obtain an efficient formula for this retraction in low-rank cases. We also prove that this retraction is the restriction of the Cayley transform on Stiefel manifolds to horizontal spaces. To develop vector transports on Grassmann manifolds, we introduce a concept called induced vector transports on quotient manifolds. Based on this concept, three vector transports associated with the Cayley transform are obtained. The first vector transport is the traditional orthogonal projection onto horizontal spaces, whereas the other two vector transports are newly proposed herein. We show that one of the new vector transports satisfies the Ring–Wirth non-expansion condition and that the other is isometric. We also simplify the formulae of the new vector transports in low-rank cases. Riemannian gradient and conjugate gradient algorithms are established via the Cayley transform and the three abovementioned vector transports. Numerical experiments on two mean-of-subspaces problems demonstrate the effectiveness of the proposed algorithms.