Kernel Principal Component Analysis (KPCA) is a key machine learning algorithm for extracting nonlinear features from data. In the presence of a large volume of high dimensional data collected in a distributed fashion, it becomes very costly to communicate all of this data to a single data center and then perform kernel PCA. Can we perform kernel PCA on the entire dataset in a distributed and communication efficient fashion while maintaining provable and strong guarantees in solution quality? In this paper, we give an affirmative answer to the question by developing a communication efficient algorithm to perform kernel PCA in the distributed setting. The algorithm is a clever combination of subspace embedding and adaptive sampling techniques, and we show that the algorithm can take as input an arbitrary configuration of distributed datasets, and compute a set of global kernel principal components with relative error guarantees independent of the dimension of the feature space or the total number of data points. In particular, computing k principal components with relative error ε over s workers has communication cost Õ(sρκ/ε + sκ^2/ε^3) words, where ρ is the average number of nonzero entries in each data point. Furthermore, we experimented the algorithm with large-scale real world datasets. The experimental results showed that the algorithm produces a high quality kernel PCA solution while using significantly less communication than alternative approaches.

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