Given @math distributed data streams @math , we consider the problem of estimating the number of unique identifiers in streams defined by set expressions over @math . We identify a broad class of algorithms for solving this problem, and show that the estimators output by any algorithm in this class are perfectly unbiased and satisfy strong variance bounds. Our analysis unifies and generalizes a variety of earlier results in the literature. To demonstrate its generality, we describe several novel sampling algorithms in our class, and show that they achieve a novel tradeoff between accuracy, space usage, update speed, and applicability.
This paper's Alpha Algorithm'' is inspired by the elegant Approximate Counting method of Morris @cite_13 , that has previously been applied to the estimation of the frequency moments @math , for @math . By contrast, our task is to estimate @math . The Alpha Algorithm is able to do this because its Approximate Counting process is tightly interleaved with another process that removes duplicates from the input stream while maintaining a small memory footprint by using feedback from the approximate counter.
@cite_13
2032665768
In this paper, a novel algorithm to blindly separate an instantaneous linear underdetermined mixture of nonstationary sources is proposed. It means that the number of sources exceeds the number of channels of the available data. The separation is based on the working assumption that the sources are piecewise stationary with a different variance in each block. It proceeds in two steps: 1) estimating the mixing matrix, and 2) computing the optimum beamformer in each block to maximize the signal-to-interference ratio of each separated signal with respect to the remaining signals. Estimating the mixing matrix is accomplished through a specialized tensor decomposition of the set of sample covariance matrices of the received mixture in each block. It utilizes optimum weighting, which allows statistically efficient (CRB attaining) estimation provided that the data obey the assumed Gaussian piecewise stationary model. In simulations, performance of the algorithm is successfully tested on blind separation of 16 speech signals from nine linear instantaneous mixtures of these signals.