正交随机特征
Orthogonal Random Features
摘要：
我们提出了一个有趣的发现与随机傅立叶特征：在高斯核近似中，用适当缩放的随机正交矩阵代替随机高斯矩阵显着降低核近似误差。我们称这种技术为正交随机特征（ORF），并为这种行为提供理论和经验的理由。基于这一发现，我们进一步提出了结构化正交随机特征（SORF），它使用一类结构化离散正交矩阵来加快计算速度。该方法减少了从$\mathcal{O}(d^2)$到$\mathcal{O}(d \log d)$的时间成本，其中$d$是数据维数，与ORF相比，在核近似质量上几乎没有妥协。在多个数据集上的实验验证了ORF和SORF方法的有效性。我们还提供了讨论使用相同类型的离散正交结构的更广泛的应用。
英文摘要：
We present an intriguing discovery related to Random Fourier Features: in Gaussian kernel approximation, replacing the random Gaussian matrix by a properly scaled random orthogonal matrix significantly decreases kernel approximation error. We call this technique Orthogonal Random Features (ORF), and provide theoretical and empirical justification for this behavior. Motivated by this discovery, we further propose Structured Orthogonal Random Features (SORF), which uses a class of structured discrete orthogonal matrices to speed up the computation. The method reduces the time cost from $\mathcal{O}(d^2)$ to $\mathcal{O}(d \log d)$, where $d$ is the data dimensionality, with almost no compromise in kernel approximation quality compared to ORF. Experiments on several datasets verify the effectiveness of ORF and SORF over the existing methods. We also provide discussions on using the same type of discrete orthogonal structure for a broader range of applications.
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