Construction of generalized samplets in Banach spaces
This work addresses a theoretical extension for researchers in functional analysis and data science, but it appears incremental as it builds on existing samplet concepts.
The paper extends samplets, localized discrete signed measures with vanishing moments for feature detection and data compression, to functionals in Banach spaces more general than point evaluations, deriving an abstract localization result for coefficients and providing examples.
Recently, samplets have been introduced as localized discrete signed measures which are tailored to an underlying data set. Samplets exhibit vanishing moments, i.e., their measure integrals vanish for all polynomials up to a certain degree, which allows for feature detection and data compression. In the present article, we extend the different construction steps of samplets to functionals in Banach spaces more general than point evaluations. To obtain stable representations, we assume that these functionals form frames with square-summable coefficients or even Riesz bases with square-summable coefficients. In either case, the corresponding analysis operator is injective and we obtain samplet bases with the desired properties by means of constructing an isometry of the analysis operator's image. Making the assumption that the dual of the Banach space under consideration is imbedded into the space of compactly supported distributions, the multilevel hierarchy for the generalized samplet construction is obtained by spectral clustering of a similarity graph for the functionals' supports. Based on this multilevel hierarchy, generalized samplets exhibit vanishing moments with respect to a given set of primitives within the Banach space. We derive an abstract localization result for the generalized samplet coefficients with respect to the samplets' support sizes and the approximability of the Banach space elements by the chosen primitives. Finally, we present three examples showcasing the generalized samplet framework.