通过均匀旋转输入空间提升蒙德里安核的旋转不变性
The Uniformly Rotated Mondrian Kernel
- 对输入空间施加均匀随机旋转后生成新随机特征映射
- 理论证明其逼近一个各向同性核，且收敛速度可量化
- 在去坐标轴偏差数据集上性能优于原蒙德里安核
随机特征映射被用于降低大规模问题中核机器的计算成本。蒙德里安核是拉普拉斯核的一种快速随机特征近似，由一种称为蒙德里安过程的高效层次化随机划分生成。本文研究该随机特征映射的变体：在运行蒙德里安过程前对输入空间施加均匀随机旋转，以逼近一个在旋转下不变的核。我们推导出所逼近各向同性核的闭式表达式，并获得均匀旋转蒙德里安核到该极限的收敛速率。为此，我们采用随机几何中平稳随机剖分理论的技术，证明了均匀旋转蒙德里安剖分叠加后典型胞腔几何的新结果。最后，我们在合成与真实数据集上测试该随机特征映射的性能，发现在去除标准坐标轴偏差的数据集上表现优于原蒙德里安核。
原文摘要 · Abstract (English)
Random feature maps are used to decrease the computational cost of kernel machines in large-scale problems. The Mondrian kernel is one such example of a fast random feature approximation of the Laplace kernel, generated by a computationally efficient hierarchical random partition of the input space known as the Mondrian process. In this work, we study a variation of this random feature map by applying a uniform random rotation to the input space before running the Mondrian process to approximate a kernel that is invariant under rotations. We obtain a closed-form expression for the isotropic kernel that is approximated, as well as a uniform convergence rate of the uniformly rotated Mondrian kernel to this limit. To this end, we utilize techniques from the theory of stationary random tessellations in stochastic geometry and prove a new result on the geometry of the typical cell of the superposition of uniformly rotated Mondrian tessellations. Finally, we test the empirical performance of this random feature map on both synthetic and real-world datasets, demonstrating its improved performance over the Mondrian kernel on a dataset that is debiased from the standard coordinate axes.
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