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<p>The so-called DCD-matrices consist of products of two diagonal matrices with a circulant matrix, merging diagonal and circulant matrices with rank-one matrices. By introducing a notion of double orthogonality for rectangular matrices, an iterative double orthogonalization process is devised for approximating a given <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(M\in \,\mathbb {C}^{n \times n}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mo>∈</mo> <mspace width="0.166667em" /> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mrow> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> with DCD-matrices. This simultaneous orthogonality of columns and rows leads to several approximation schemes of which the best rank-one approximation is just a special case. Expanding through summing yields, like principal component analysis (PCA), a novel optimal technique for reducing the dimensionality of datasets. Being notably general and flexible, this approach provides a natural way to merge fast Fourier methods with low rank matrix approximations. Least squares solution methods play a significant role in algorithms.</p>

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Best Low Rank Approximations are a Special Case of Best DCD-matrix Approximations

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年4月23日

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  • Marko Huhtanen,
  • Kai Noponen,
  • Charmin Pingamage Don

摘要

The so-called DCD-matrices consist of products of two diagonal matrices with a circulant matrix, merging diagonal and circulant matrices with rank-one matrices. By introducing a notion of double orthogonality for rectangular matrices, an iterative double orthogonalization process is devised for approximating a given \(M\in \,\mathbb {C}^{n \times n}\) M ∈ C n × n with DCD-matrices. This simultaneous orthogonality of columns and rows leads to several approximation schemes of which the best rank-one approximation is just a special case. Expanding through summing yields, like principal component analysis (PCA), a novel optimal technique for reducing the dimensionality of datasets. Being notably general and flexible, this approach provides a natural way to merge fast Fourier methods with low rank matrix approximations. Least squares solution methods play a significant role in algorithms.