Dominant subspace and low-rank approximations from block Krylov subspaces without a gap
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In this work we obtain results related to the approximation of h-dimensional dominant subspaces and low rank approximations of matrices 𝐀∈𝕂^m× n (where 𝕂=ℝ or ℂ) in case there is no singular gap, i.e. if σ_h=σ_h+1 (where σ_1≥…≥σ_p≥ 0 denote the singular values of 𝐀, and p=min{m,n}). In order to do this, we describe in a convenient way the class of h-dimensional right (respectively left) dominant subspaces. Then, we show that starting with a matrix 𝐗∈𝕂^n× r with r≥ h satisfying a compatibility assumption with some h-dimensional right dominant subspace, block Krylov methods produce arbitrarily good approximations for both problems mentioned above. Our approach is based on recent work by Drineas, Ipsen, Kontopoulou and Magdon-Ismail on approximation of structural left dominant subspaces; but instead of exploiting a singular gap at h (which is zero in this case) we exploit the nearest existing singular gaps.
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