Open Problem: Kernel methods on manifolds and metric spaces. What is the probability of a positive definite geodesic exponential kernel?
Open Problem: Kernel methods on manifolds and metric spaces. What is the probability of a positive definite geodesic exponential kernel?
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开放问题：流形和度量空间上的核方法。
DOI：
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
Søren Hauberg
中科院分区：
文献类型：
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作者：
Aasa Feragen;Søren Hauberg
径向核非常适合一般测地度量空间上的机器学习，其中成对距离通常是唯一可用的可计算量。我们最近已经证明，测地指数核仅在输入空间具有强线性性质时对所有带宽都是正定的。这个否定结果暗示了径向核可能并不适用于测地度量空间。然而，在这里，我们提出的证据表明，存在较大的带宽间隔，其中测地线指数内核在有限数据集上具有很高的正定义概率，同时仍然具有显着的预测能力。由此，我们根据数据空间的几何形状和样本的分布，对有限个随机样本的正定核矩阵的概率进行了公式化。
Radial kernels are well-suited for machine learning over general geodesic metric spaces, where pairwise distances are often the only computable quantity available. We have recently shown that geodesic exponential kernels are only positive definite for all bandwidths when the input space has strong linear properties. This negative result hints that radial kernel are perhaps not suitable over geodesic metric spaces after all. Here, however, we present evidence that large intervals of band-widths exist where geodesic exponential kernels have high probability of being positive definite over finite datasets, while still having significant predictive power. From this we formulate conjectures on the probability of a positive definite kernel matrix for a finite random sample, depending on the geometry of the data space and the spread of the sample.