Low-rank approximation of a matrix by means of random sampling has been consistently efficient in its empirical studies by many scientists who applied it with various sparse and structured multipliers, but adequate formal support for this empirical phenomenon has been missing so far. Our novel insight into the subject leads to such an elusive formal support and promises significant acceleration of the known algorithms for some fundamental problems of matrix computations and data mining and analysis. Our formal results and our numerical tests are in good accordance with each other. We also outline extensions of low-rank approximation algorithms and of our progress to the Least Squares Regression, the Fast Multipole Method, and the Conjugate Gradient algorithms.
Part (ii) of our Theorem is implied by [Theorem 10.8] HMT11 , but our specific supporting estimates are more compact, cover the case of any @math (whereas @cite_36 assumes that @math ), and we deduce them by using a shorter proof (see Remark ). Our approach, our results listed in Section , some of our techniques, e.g., expan -sion compres -sion in Section , and even the concept of factor-Gaussian matrices are new, and so are our families of multipliers and policies of their generation, combination, and application in Sections and as well. Moreover, our progress can be extended to a variety of important matrix computations. In Section (Conclusions) we outline such novel extensions to the highly popular and much studied computations for Least Squares Regression, the Fast Multipole Method and the Conjugate Gradient Algorithms. Hereafter we use the acronyms LSR" for Least Squares Regression", FMM" for Fast Multipole Method", and CG " for Conjugate Gradient". The extensions provide new insights and new opportunities and should motivate further effort and further progress.
@cite_36
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Section 1 of the paper contains a general discussion of robustness. In Section 2 the influence function of the Hampel-Rousseeuw least median of squares estimator is derived. Linearly invariant weak metrics are constructed in Section 3. It is shown in Section 4 that @math -estimators satisfy an exact Holder condition of order 1 2 at models with normal errors. In Section 5 the breakdown points of the Hampel-Krasker dispersion and regression functionals are shown to be 0. The exact breakdown point of the Krasker-Welsch dispersion functional is obtained as well as bounds for the corresponding regression functional. Section 6 contains the construction of a linearly equivariant, high breakdown and locally Lipschitz dispersion functional for any design distribution. In Section 7 it is shown that there is no inherent contradiction between efficiency and a high breakdown point. Section 8 contains a linearly equivariant, high breakdown regression functional which is Lipschitz continuous at models with normal errors.