These techniques calibrate an initial non-metric distance matrix estimated on incomplete data to a distance metric.

The Tanimoto coefficient is commonly used to measure the similarity between molecules represented as discrete fingerprints, either as a distance metric or a positive definite kernel.

Federated learning has a significant advantage in protecting information privacy. Many scholars proposed various secure learning methods within the framework of federated learning but the study on secure federated unsupervised learning especially clustering is limited. We in this work propose a secure kernelized factorization method for federated spectral clustering on distributed dataset. The method is non-trivial because the kernel or similarity matrix for spectral clustering is computed by data pairs, which violates the principle of privacy protection. Our method implicitly constructs an approximation for the kernel matrix on distributed data such that we can perform spectral clustering under the constraint of privacy protection. We provide a convergence guarantee of the optimization algorithm, reconstruction error bounds of the Gaussian kernel matrix, and the sufficient condition of correct clustering of our method. We also present some results of differential privacy. Numerical results on synthetic and real datasets demonstrate that the proposed method is efficient and accurate in comparison to the baselines.

This paper investigates the impact of alignment between the target function of interest and the kernel matrix on a variety of kernel-based methods based on a general loss belonging to a rich loss function family, which covers many commonly used methods in regression and classification problems. We consider the truncated kernel-based method (TKM) which is estimated within a reduced function space constructed by using the spectral truncation of the kernel matrix and compare its theoretical behavior to that of the standard kernel-based method (KM) under various settings. By using the kernel complexity function that quantifies the complexity of the induced function space, we derive the upper bounds for both TKM and KM, and further reveal their dependencies on the degree of target-kernel alignment. Specifically, for the alignment with polynomial decay, the established results indicate that under the just-aligned and weakly-aligned regimes, TKM and KM share the same learning rate. Yet, under the strongly-aligned regime, KM suffers the saturation effect, while TKM can be continuously improved as the alignment becomes stronger. This further implies that TKM has a strong ability to capture the strong alignment and provide a theoretically guaranteed solution to eliminate the phenomena of saturation effect. The minimax lower bound is also established for the squared loss to confirm the optimality of TKM.

Structured kernel interpolation (SKI) accelerates Gaussian processes (GP) inference by interpolating the kernel covariance function using a dense grid of inducing points, whose corresponding kernel matrix is highly structured and thus amenable to fast linear algebra. Unfortunately, SKI scales poorly in the dimension of the input points, since the dense grid size grows exponentially with the dimension. To mitigate this issue, we propose the use of sparse grids within the SKI framework. These grids enable accurate interpolation, but with a number of points growing more slowly with dimension. We contribute a novel nearly linear time matrix-vector multiplication algorithm for the sparse grid kernel matrix. We also describe how sparse grids can be combined with an efficient interpolation scheme based on simplicial complexes. With these modifications, we demonstrate that SKI can be scaled to higher dimensions while maintaining accuracy, for both synthetic and real datasets.

Gaussian processes (GPs) produce good probabilistic models of functions, but most GP kernels require $O((n+m)n^2)$ time, where $n$ is the number of data points and $m$ the number of predictive locations. We present a new kernel that allows for Gaussian process regression in $O((n+m)\log(n+m))$ time.

We prove new explicit upper bounds on the leverage scores of Fourier sparse functions under both the Gaussian and Laplace measures.