The Indispensability Index of Vector Sets
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| Natura: | Recurso digital |
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2025
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| author | SÉRGIO DE ANDRADE, PAULO |
| author_facet | SÉRGIO DE ANDRADE, PAULO |
| contents | This paper introduces the Indispensability Index (II), a novel metric designed to quantify the unique contribution of each vector within a finite set to the subspace it spans. Traditional linear algebra classifies vectors as either linearly independent or dependent, a binary distinction that does not capture the degree of a vector's importance, especially in cases of near-collinearity or within redundant spanning sets. The Indispensability Index provides a continuous measure, defined as the relative loss in the volume of the spanned parallelepiped upon the removal of a given vector. We formalize this concept using the Gram determinant, which allows the index to be computed for any subset of vectors in an inner product space. The methodology is detailed for both linearly independent and dependent sets, offering a unified framework. Theoretical properties of the index are explored, including its behavior in orthogonal and collinear cases. We present numerical results for various vector configurations in Euclidean spaces to illustrate the index's utility in identifying critical, redundant, and stabilizing vectors. The discussion highlights the potential applications of the II in fields such as feature selection, sensor placement, and the analysis of redundant systems like frames, where quantifying the contribution of individual components is crucial. The paper concludes that the Indispensability Index offers a more nuanced understanding of the internal structure of vector sets than traditional binary concepts. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17437146 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Indispensability Index of Vector Sets SÉRGIO DE ANDRADE, PAULO Indispensability Index Linear Algebra Vector Space Gram Determinant Linear Independence Spanning Sets Feature Selection This paper introduces the Indispensability Index (II), a novel metric designed to quantify the unique contribution of each vector within a finite set to the subspace it spans. Traditional linear algebra classifies vectors as either linearly independent or dependent, a binary distinction that does not capture the degree of a vector's importance, especially in cases of near-collinearity or within redundant spanning sets. The Indispensability Index provides a continuous measure, defined as the relative loss in the volume of the spanned parallelepiped upon the removal of a given vector. We formalize this concept using the Gram determinant, which allows the index to be computed for any subset of vectors in an inner product space. The methodology is detailed for both linearly independent and dependent sets, offering a unified framework. Theoretical properties of the index are explored, including its behavior in orthogonal and collinear cases. We present numerical results for various vector configurations in Euclidean spaces to illustrate the index's utility in identifying critical, redundant, and stabilizing vectors. The discussion highlights the potential applications of the II in fields such as feature selection, sensor placement, and the analysis of redundant systems like frames, where quantifying the contribution of individual components is crucial. The paper concludes that the Indispensability Index offers a more nuanced understanding of the internal structure of vector sets than traditional binary concepts. |
| title | The Indispensability Index of Vector Sets |
| topic | Indispensability Index Linear Algebra Vector Space Gram Determinant Linear Independence Spanning Sets Feature Selection |
| url | https://doi.org/10.5281/zenodo.17437146 |