The Modified Nystrom Method: Theories, Algorithms, and Extension

Many kernel methods suffer from high time and space complexities, so they are prohibitive in big-data applications. To tackle the computational challenge, the Nystr\"om method has been extensively used to reduce time and space complexities by sacrificing some accuracy. The Nystr\"om method speedups computation by constructing an approximation of the kernel matrix in question using only a few columns of the matrix. Recently, a variant of the Nystr\"om method called the modified Nystr\"om method has demonstrated significant improvement over the standard Nystr\"om method in approximation accuracy, both theoretically and empirically. In this paper we provide theoretical analysis, efficient algorithms, and a simple but highly accurate extension for the modified Nystr\"om method. First, we prove that the modified Nystr\"om method is exact under certain conditions, and we establish a lower error bound for the modified Nystr\"om method. Second, we develop two efficient algorithms to make the modified Nystr\"om method efficient and practical. We devise a simple column selection algorithm with a provable error bound. With the selected columns at hand, we propose an algorithm that computes the modified Nystr\"om approximation in lower time complexity than the approach in the previous work. Third, the extension which we call {\it the SS-Nystr\"om method} has much stronger error bound than the modified Nystr\"om method, especially when the spectrum of the kernel matrix decays slowly. Our proposed SS-Nystr\"om can be computed nearly as efficiently as the modified Nystr\"om method. Finally, experiments on real-world datasets demonstrate that the proposed column selection algorithm is both efficient and accurate and that the SS-Nystr\"om method always leads to much higher kernel approximation accuracy than the standard/modified Nystr\"om method.
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