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Low-rank approximation of a matrix by means of random sampling has beenconsistently efficient in its empirical studies by many scientists who appliedit with various sparse and structured multipliers, but adequate formal supportfor this empirical phenomenon has been missing so far. Our novel insight intothe subject leads to such an elusive formal support and promises significantacceleration of the known algorithms for some fundamental problems of matrixcomputations and data mining and analysis. Our formal results and our numericaltests are in good accordance with each other. We also outline extensions oflow-rank approximation algorithms and of our progress to the Least SquaresRegression, the Fast Multipole Method, and the Conjugate Gradient algorithms.

Low-rank Approximation of a Matrix: Novel Insights, New Progress, and Extensions