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<p>In this note, we briefly present a generalized tensor CUR (GTCUR) approximation for tensor pairs <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((\underline{\textbf{X}},\underline{\textbf{Y}})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <munder> <mi mathvariant="bold">X</mi> <mo>̲</mo> </munder> <mo>,</mo> <munder> <mi mathvariant="bold">Y</mi> <mo>̲</mo> </munder> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and tensor triplets <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((\underline{\textbf{X}},\underline{\textbf{Y}},\underline{\textbf{Z}})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <munder> <mi mathvariant="bold">X</mi> <mo>̲</mo> </munder> <mo>,</mo> <munder> <mi mathvariant="bold">Y</mi> <mo>̲</mo> </munder> <mo>,</mo> <munder> <mi mathvariant="bold">Z</mi> <mo>̲</mo> </munder> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> based on the tubal product (t-product). We use the tensor Discrete Empirical Interpolation Method (TDEIM) to do these extensions. We demonstrate how the TDEIM can be applied to extend the traditional tensor CUR (TCUR) approximation, which operates on a single tensor, to simultaneously compute the TCUR approximations for two or three tensors. This method allows for the sampling of relevant lateral or horizontal slices from one data tensor in relation to one or two other data tensors. In certain special cases, the generalized TCUR (GTCUR) method simplifies to the classical TCUR approximations for both tensor pairs and tensor triplets, akin to the process shown for matrices.</p>

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A Note on Generalized Tensor CUR Approximation for Tensor Pairs and Tensor Triplets Based on the Tubal Product

  • 期刊论文

  • 发表日期：

    2026年1月8日

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  • Salman Ahmadi-Asl,
  • Naeim Rezaeian,
  • Keivan Ramazani

摘要

In this note, we briefly present a generalized tensor CUR (GTCUR) approximation for tensor pairs \((\underline{\textbf{X}},\underline{\textbf{Y}})\) ( X ̲ , Y ̲ ) and tensor triplets \((\underline{\textbf{X}},\underline{\textbf{Y}},\underline{\textbf{Z}})\) ( X ̲ , Y ̲ , Z ̲ ) based on the tubal product (t-product). We use the tensor Discrete Empirical Interpolation Method (TDEIM) to do these extensions. We demonstrate how the TDEIM can be applied to extend the traditional tensor CUR (TCUR) approximation, which operates on a single tensor, to simultaneously compute the TCUR approximations for two or three tensors. This method allows for the sampling of relevant lateral or horizontal slices from one data tensor in relation to one or two other data tensors. In certain special cases, the generalized TCUR (GTCUR) method simplifies to the classical TCUR approximations for both tensor pairs and tensor triplets, akin to the process shown for matrices.