An Efficient Exact-PGA Algorithm for Constant Curvature Manifolds
An Efficient Exact-PGA Algorithm for Constant Curvature Manifolds
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一种高效的恒曲率流形精确PGA算法
DOI：
10.1109/cvpr.2016.431
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
B. Vemuri
中科院分区：
文献类型：
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作者：
Rudrasis Chakraborty;Dohyung Seo;B. Vemuri
流形值数据集在许多计算机视觉任务中被广泛遇到。PCA算法的非线性模拟，称为主测地线分析（PGA）算法，适用于黎曼流形上的数据在十年前的文献中报道。由于PGA算法中的目标函数是高度非线性的，通常难以有效求解，研究人员提出了一种线性近似。虽然这种线性近似很容易计算，但它缺乏准确性，特别是当数据表现出较大的方差时。最近，提出了一种称为精确PGA的替代方案，该方案试图在没有任何线性化的情况下解决优化问题。对于一般的黎曼流形，虽然它产生了比原始（线性化）PGA更好的精度，但对于表现出大方差的数据，优化在计算上并不高效。本文提出了一种求解常曲率黎曼流形的精确PGA算法（CCM-EPGA）。CCM-EPGA算法在两个方面与现有的PGA算法有很大的不同，（i）给定流形值数据点与主子流形之间的距离是解析计算的，因此不需要像现有方法那样进行优化。(ii)与现有的PGA算法不同，下降到余维-1子流形不需要任何优化，而是通过使用Rimeannian逆指数映射和并行传输操作来完成。我们目前的理论和实验结果为常曲率黎曼流形描绘良好的性能相比，现有的PGA算法的CCM-EPGA算法。我们还提出了数据重建的主成分还没有在文献中报道，在这种情况下。
Manifold-valued datasets are widely encountered in many computer vision tasks. A non-linear analog of the PCA algorithm, called the Principal Geodesic Analysis (PGA) algorithm suited for data lying on Riemannian manifolds was reported in literature a decade ago. Since the objective function in the PGA algorithm is highly non-linear and hard to solve efficiently in general, researchers have proposed a linear approximation. Though this linear approximation is easy to compute, it lacks accuracy especially when the data exhibits a large variance. Recently, an alternative called the exact PGA was proposed which tries to solve the optimization without any linearization. For general Riemannian manifolds, though it yields a better accuracy than the original (linearized) PGA, for data that exhibit large variance, the optimization is not computationally efficient. In this paper, we propose an efficient exact PGA algorithm for constant curvature Riemannian manifolds (CCM-EPGA). The CCM-EPGA algorithm differs significantly from existing PGA algorithms in two aspects, (i) the distance between a given manifold-valued data point and the principal submanifold is computed analytically and thus no optimization is required as in the existing methods. (ii) Unlike the existing PGA algorithms, the descent into codimension-1 submanifolds does not require any optimization but is accomplished through the use of the Rimeannian inverse Exponential map and the parallel transport operations. We present theoretical and experimental results for constant curvature Riemannian manifolds depicting favorable performance of the CCM-EPGA algorithm compared to existing PGA algorithms. We also present data reconstruction from the principal components which has not been reported in literature in this setting.
DOI：
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发表时间：
2013-06
期刊：
Proceedings of the ... International Conference on Machine Learning. International Conference on Machine Learning
影响因子：
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作者：
J. Ho;Yuchen Xie;B. Vemuri
通讯作者：
J. Ho;Yuchen Xie;B. Vemuri