Adaptive Localized Cayley Parametrization Technique for Smooth optimization over the Stiefel Manifold
Adaptive Localized Cayley Parametrization Technique for Smooth optimization over the Stiefel Manifold
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DOI：
10.23919/eusipco.2019.8903097
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发表时间：
2019-09
期刊：
影响因子：
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通讯作者：
Keita Kume;I. Yamada
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文献类型：
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作者：
Keita Kume;I. Yamada
我们提出了一种新颖的计算策略，称为自适应局部Cayley参数技术，用于加速在Stiefel歧管上优化。在所有偏斜材料的矢量空间上定义的转换。您对自适应的局部Cayley变换，这是正交组和偏压材料的矢量空间之间的可计算差异。奇异性问题可能导致性能退化，在双凯利参数技术[yamada- ezaki'03]以及（ii）中观察到，可以在矢量空间上享受有力的艺术，而不会遭受符号的非线性性质我们还提出了采用Armijo规则的原型算法的收敛分析，该算法显示了在局部Cayley变换范围内的复合函数的梯度，保证了数值的表现出色。主要的优化算法基本上是在Stiefel歧管的切线[Absil-Mahony-Sepulcher'08，Wen-Yin'13]上设计的。
We propose a novel computational strategy, named the adaptive localized Cayley parametrization technique for acceleration of optimization over the Stiefel manifold. The proposed optimization algorithm is designed as a gradient descent type scheme for the composite of the original cost function and the inverse of the localized Cayley transform defined on the vector space of all skew-symmetric matrices. Thanks to the adaptive localized Cayley transform which is a computable diffeomorphism between the orthogonal group and the vector space of the skew-symmetric matrices, the proposed algorithm (i) is free from the singularity issue, which can cause performance degradation, observed in the dual Cayley parametrization technique [Yamada- Ezaki’03] as well as (ii) can enjoy powerful arts for acceleration on the vector space without suffering from the nonlinear nature of the Stiefel manifold. We also present a convergence analysis, for the prototype algorithm employing the Armijo’s rule, that shows the gradient of the composite function at zero in the range space of the localized Cayley transform is guaranteed to converge to zero. Numerical experiments show excellent performance compared with major optimization algorithms designed essentially with retractions on the tangent space of the Stiefel manifold [Absil-Mahony-Sepulcher’08, Wen-Yin’13].