In this paper, we study the problem of learning a mixture of Gaussians with streaming data: given a stream of N points in d dimensions generated by an unknown mixture of k spherical Gaussians, the goal is to estimate the model parameters using a single pass over the data stream. We analyze a streaming version of the popular Lloyd\u2019s heuristic and show that the algorithm estimates all the unknown centers of the component Gaussians accurately if they are suf\ufb01ciently separated. Assuming each pair of centers are C\u03c3 distant with C = \u2126((k log k)1\/4\u03c3) and where\u03c32 is the maximum variance of any Gaussian component, we show that asymptotically the algorithm estimates the centers optimally (up to certain constants); our center separation requirement matches the best known result for spherical Gaussians [16]. For \ufb01nite samples, we show that a bias term based on the initial estimate decreases at O(1\/poly(N)) rate while variance decreases at nearly optimal rate of \u03c32d\/N.<\/p>\n
Our analysis requires seeding the algorithm with a good initial estimate of the true cluster centers for which we provide an online PCA based clustering algorithm. Indeed, the asymptotic per-step time complexity of our algorithm is the optimal d\u00b7k while space complexity of our algorithm is O(dk log k). In addition to the bias and variance terms which tend to 0, the hard-thresholding based updates of streaming Lloyd\u2019s algorithm is agnostic to the data distribution and hence incurs an approximation error that cannot be avoided. However, by using a streaming version of the classical (soft-thresholding-based) EM method that exploits the Gaussian distribution explicitly, we show that for a mixture of two Gaussians the true means can be estimated consistently, with estimation error decreasing at nearly optimal rate, and tending to 0 for N \u2192\u221e.<\/p>\n","protected":false},"excerpt":{"rendered":"
In this paper, we study the problem of learning a mixture of Gaussians with streaming data: given a stream of N points in d dimensions generated by an unknown mixture of k spherical Gaussians, the goal is to estimate the model parameters using a single pass over the data stream. We analyze a streaming version 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