What does overcompleteness mean?

Definitions for overcompleteness
over·com·plete·ness

This dictionary definitions page includes all the possible meanings, example usage and translations of the word overcompleteness.

Wiktionary

  1. overcompletenessnoun

    The condition of being overcomplete

Wikipedia

  1. Overcompleteness

    Overcompleteness is a concept from linear algebra that is widely used in mathematics, computer science, engineering, and statistics (usually in the form of overcomplete frames). It was introduced by R. J. Duffin and A. C. Schaeffer in 1952.Formally, a subset of the vectors { ϕ i } i ∈ J {\displaystyle \{\phi _{i}\}_{i\in J}} of a Banach space X {\displaystyle X} , sometimes called a "system", is complete if every element in X {\displaystyle X} can be approximated arbitrarily well in norm by finite linear combinations of elements in { ϕ i } i ∈ J {\displaystyle \{\phi _{i}\}_{i\in J}} . A complete system is additionally overcomplete if there exists a ϕ j {\displaystyle \phi _{j}} which can be removed from the system while maintaining completeness (i.e., ∃ j : { ϕ i } i ∈ J ∖ { j } is complete {\displaystyle \exists j:\{\phi _{i}\}_{i\in J\backslash \{j\}}{\text{ is complete}}} ). In this sense, the system contains more vectors than necessary to be complete, hence "overcomplete". In research areas such as signal processing and function approximation, overcompleteness can help researchers to achieve a more stable, more robust, or more compact decomposition than using a basis.

Wikidata

  1. Overcompleteness

    A subset of a Banach space, sometimes called a "system", is complete if every element in can be approximated arbitrarily well in norm by finite linear combinations of elements in . Such a complete system is overcomplete if removal of a from the system results in a complete system. In different research, such as signal processing and function approximation, overcompleteness can help researchers to achieve a more stable, more robust, or more compact decomposition than using a basis. Overcomplete frames are widely used in mathematics, computer science, engineering, and statistics.

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Numerology

  1. Chaldean Numerology

    The numerical value of overcompleteness in Chaldean Numerology is: 3

  2. Pythagorean Numerology

    The numerical value of overcompleteness in Pythagorean Numerology is: 8


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"overcompleteness." Definitions.net. STANDS4 LLC, 2024. Web. 28 Mar. 2024. <https://www.definitions.net/definition/overcompleteness>.

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