核低秩近似的输入稀疏时间可能吗？
Is Input Sparsity Time Possible for Kernel Low-Rank Approximation?
摘要：
低秩近似是用于加速核方法的常用工具：$n \times n$核矩阵$K$通过秩- $k$矩阵$\tilde K$来近似，该秩-$k$矩阵可以存储在更少的空间中并且更快地处理。在这项工作中，我们研究了计算效率低秩核近似的限制。我们表明，对于广泛的一类内核，包括流行的高斯和多项式内核，计算相对误差$k$秩近似$K$是至少一样困难的输入数据矩阵$A \in \mathbb{R}^{n \times d}$乘以任意矩阵$C \in \mathbb{R}^{d \times k}$。除非在快速矩阵乘法方面有所突破，否则当$k$不是太大时，这需要$Ω(nnz(A)k)$时间，其中$nnz(A)$是$A$中非零的数目。这个下限匹配，在许多参数制度，最近的工作次二次时间算法的低秩近似的一般内核[MM 16，MW 17]，表明这些算法不太可能得到显着改善，特别是$O(nnz(A))$输入稀疏运行时。同时也有希望：我们首次证明了$O(nnz(A))$时间近似对于一般径向基函数核是可能的（例如，高斯核）的核化数据集的低秩近似的密切相关的问题。
英文摘要：
Low-rank approximation is a common tool used to accelerate kernel methods: the $n \times n$ kernel matrix $K$ is approximated via a rank-$k$ matrix $\tilde K$ which can be stored in much less space and processed more quickly. In this work we study the limits of computationally efficient low-rank kernel approximation. We show that for a broad class of kernels, including the popular Gaussian and polynomial kernels, computing a relative error $k$-rank approximation to $K$ is at least as difficult as multiplying the input data matrix $A \in \mathbb{R}^{n \times d}$ by an arbitrary matrix $C \in \mathbb{R}^{d \times k}$. Barring a breakthrough in fast matrix multiplication, when $k$ is not too large, this requires $Ω(nnz(A)k)$ time where $nnz(A)$ is the number of non-zeros in $A$. This lower bound matches, in many parameter regimes, recent work on subquadratic time algorithms for low-rank approximation of general kernels [MM16,MW17], demonstrating that these algorithms are unlikely to be significantly improved, in particular to $O(nnz(A))$ input sparsity runtimes. At the same time there is hope: we show for the first time that $O(nnz(A))$ time approximation is possible for general radial basis function kernels (e.g., the Gaussian kernel) for the closely related problem of low-rank approximation of the kernelized dataset.
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