Abstract
The structure preserving rank reduction problem arises in many important applications. The singular value decomposition (SVD), while giving the closest low rank approximation to a given matrix in matrix L 2 norm and Frobenius norm, may not be appropriate for these applications since it does not preserve the given structure. We present a new method for structure preserving low rank approximation of a matrix, which is based on Structured Total Least Norm (STLN). The STLN is an efficient method for obtaining an approximate solution to an overdetermined linear system AX ≈ B, preserving the given linear structure in the perturbation [E F] such that (A + E)X = B + F. The approximate solution can be obtained to minimize the perturbation [E F] in the L p norm, where p = 1, 2, or ∞. An algorithm is described for Hankel structure preserving low rank approximation using STLN with L p norm. Computational results are presented, which show performances of the STLN based method for L 1 and L 2 norms for reduced rank approximation for Hankel matrices.
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Park, H., Zhang, L. & Rosen, J.B. Low Rank Approximation of a Hankel Matrix by Structured Total Least Norm. BIT Numerical Mathematics 39, 757–779 (1999). https://doi.org/10.1023/A:1022347425533
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DOI: https://doi.org/10.1023/A:1022347425533