Likeverdiprojeksjoner

Likeverdiprojeksjoner is a term used in Norwegian-language mathematical and data-science contexts to describe a class of projection operators that aim to preserve a specified value when projecting from a larger space to a smaller one. In practice, a projection P from a vector space V to a subspace W is called a likeverdiprojeksjon with respect to a valuation φ if and only if φ(Px) = φ(x) for every x in V. This makes the projection “value-preserving” with respect to the chosen valuation.

Formal considerations depend on the nature of φ. If φ is linear, the condition imposes concrete constraints on P. If φ is the norm or an inner product-derived quantity, common realizations include orthogonal or isometric projections, which minimize or preserve specific measures of length or angle. In nonlinear settings, one may define generalized, approximate, or context-specific valuations and seek projections that preserve them approximately or under certain conditions.

Examples and variants: Orthogonal projections in Euclidean spaces preserve the inner product and, consequently, the norm, which can be viewed as a form of value preservation. In statistics and finance, projections that preserve expected value or risk measures under dimensionality reduction are discussed, though exact preservation is not always achievable with common techniques like PCA.

Applications and scope: Likeverdiprojeksjoner are relevant to dimensionality reduction, model reduction in control theory, data compression, and risk assessment, where retaining a core valuation is important while reducing complexity. The term is not widely standardized in English-language literature and may appear mainly in Norwegian technical writings, where it is used to describe projection concepts that prioritize preserving a designated value.