Efficient Algorithms and Error Analysis for the Modified Nystrom Method

Many kernel methods suffer from high time and space complexities and are thus prohibitive in big-data applications. To tackle the computational challenge, the Nystrom method has been extensively used to reduce time and space complexities by sacrificing some accuracy. The Nystrom method speedups computation by constructing an approximation of the kernel matrix using only a few columns of the matrix. Recently, a variant of the Nystrom method called the modified Nystrom method has demonstrated significant improvement over the standard Nystrom method in approximation accuracy, both theoretically and empirically. In this paper, we propose two algorithms that make the modified Nystrom method practical. First, we devise a simple column selection algorithm with a provable error bound. Our algorithm is more efficient and easier to implement than and nearly as accurate as the state-of-the-art algorithm. Second, with the selected columns at hand, we propose an algorithm that computes the approximation in lower time complexity than the approach in the previous work. Furthermore, we prove that the modified Nystrom method is exact under certain conditions, and we establish a lower error bound for the modified Nystrom method.